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a/0tAyT4oBgHgl3EQf1PlJ/vector_store/index.pkl b/0tAyT4oBgHgl3EQf1PlJ/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..e6cf3fa4267bc69ce6d5acb52a9e075ffa636ab0 --- /dev/null +++ b/0tAyT4oBgHgl3EQf1PlJ/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:46334b2caee3aea666e27b99924269395e65189f98d10f10c98f6a9550895441 +size 124415 diff --git a/0tFRT4oBgHgl3EQfkzf9/content/tmp_files/2301.13597v1.pdf.txt b/0tFRT4oBgHgl3EQfkzf9/content/tmp_files/2301.13597v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3b208aaf2da8e09324303928879c63dfde1dee8f --- /dev/null +++ b/0tFRT4oBgHgl3EQfkzf9/content/tmp_files/2301.13597v1.pdf.txt @@ -0,0 +1,787 @@ +arXiv:2301.13597v1 [hep-ph] 31 Jan 2023 +The scalar exotic resonances X(3915), +X(3960), X(4140) +A.M.Badalian and Yu.A.Simonov +NRC “Kurchatov Institute” +Moscow, Russia +February 1, 2023 +Abstract +The scalar resonances X(3915), X(3960), X(4140) are considered +as exotic four-quark states: cq¯c¯q, cs¯c¯s, cs¯c¯s, while the X(3863) is proved +to be the c¯c, 2 3P0 state. The masses and the widths of these reso- +nances are calculated in the framework of the Extended Recoupling +Model, where a four-quark system is formed inside the bag and has +relatively small size (<∼ 1.0 fm). +Then the resonance X(3915) ap- +pears due to the transitions: J/ψω into D∗+D∗− (or D∗0 ¯D∗0) and +back, while the X(3960) is created due to the transitions D+ +s D− +s into +J/ψφ and back, and the X(4140) is formed in the transitions J/ψφ +into D∗+ +s D∗− +s +and back. The characteristic feature of the recoupling +mechanism is that this type of resonances can be predominantly in the +S-wave decay channels and has JP = 0+. In two-channel case the reso- +nance occurs to be just near the lower threshold, while due to coupling +to third channel (like the c¯c channel) it is shifted up and lies by (20– +30) MeV above the lower threshold. The following masses and widths +are calculated: M(X(3915)) = 3920 MeV, Γ(X(3915)) = 20 MeV; +M(X(3960)) = 3970 MeV, Γ(X(3960) = 45(5) MeV, M(X(4140)) = +4120(20) MeV, Γ(X(4140)) = 100 MeV, which are in good agreement +with experiment. +1 + +1 +Introduction +In the region (3.9–4.2) GeV there are now three scalar resonances and the +X(3915) was the first, observed by the Belle in the e+e− → J/ψωK process +[1]. +Later this resonance was confirmed by the BaBar [2] and in several +other experiments [3]), in particular, in two-photon collisions [4, 5]. +For +some years this resonance was assumed to be the conventional c¯c meson +– χco(2P), although this interpretation has called out some doubts [6, 7] +(see discussion in the reviews [8, 9]) and does not agree with predictions +in different relativistic potential models (RPM) [10]-[13]. The experimental +masses of the X(3915) and χc2(2P) were found to be almost equal, while +in the RPMs a smaller mass, M(2 3P0) ∼= 3870 ± 30 MeV, and much larger +mass difference, δ20(2P) = M(χc2(2P) − M(χc0(2P) ∼= (70 − 100) MeV, +were predicted. +Notice that large mass difference δ20 is kept even if the +coupling of the χc0(2P) to open channels is taken into account [14, 15]. Such +theoretical expectations were supported by the Belle observation of the wide +scalar X(3860) resonance [16], both in e+e− → J/ψD+D− and e+e− → +J/ψD0 ¯D0 decays, which has the mass M = 3862+26 +−32 ++40 +−82 MeV and large width +Γ ∼= 200 MeV. The existence of the scalar X(3860) resonance is confirmed +by the analysis of two-photon production, γγ → D ¯D in [17]. +Very recently the LHCb [18] has observed two more scalar resonances +X(3960), X(4140) in the D+ +s D− +s mass spectrum in the B+ → D+ +s D− +s K+ de- +cays with the parameters: M(X(3960)) = (3956±5±10) MeV, Γ(X(3960)) = +(43 ± 13 ± 8) MeV, M(X(4140)) = (4133 ± 6 ± 6) MeV, Γ(X(4140)) = +(67 ± 17 ± 7) MeV, both with JP C = 0++. These new scalar resonances evi- +dently look as exotic states and the X(3960) was interpreted as the molecular +D+ +s D− +s state within the QCD sum rules approach [19, 20] and in a coupled- +channel model [21]; in [22] it appears due to the triangle singularity, while +in [23] the parameters of the X(3960), as a diquark-antidiquark state, were +obtained in a good agreement with experiment, using the QCD sum rules +approach. Notice that the masses of the X(3960) and X(4140) resonances +lie by ∼ 20 MeV above the thresholds: D+ +s D− +s and J/ψφ, respectively. +In our paper we assume that the X(3915) and both the X(3960), X(4140) +belong to exotic four-quark states cq¯c¯q and cs¯c¯s and to define their parame- +ters we will use the Extended Recoupling Model (ERM), recently suggested +in [24], which develops the Recouplimg Model, presented earlier [25]. The +ERM allows to calculate the mass and width of a scalar four-quark states, +however, within suggested mechanism such resonances cannot exist in the +2 + +systems with two identical mesons, like D+ +s D+ +s , D∗+ +s , D∗+ +s . This theoretical +prediction is supported by the Belle experiment [26]. In the ERM the system +of two mesons, e.g. (J/ψ + φ), can transfer into another pair of the mesons +(D+ +s , D− +s ) by rearranging confining strings and back in the infinite chain of +transformations, like J/ψφ → (D+ +s ¯D− +s ) → J/ψφ → .... Note that such se- +quences can also be treated, for example, in the standard OBE approximation +with the meson exchanges, which, however, does not produce the singulari- +ties near the thresholds. In the coupled-channel models (CCM) [27, 28] the +interaction between hadrons, like D+ +s D− +s , J/ψφ, is usually neglected, while +in the ERM such interaction is taken into account, introducing the four- +quark bag. It is important that all hadrons involved have rather small sizes, +∼= (0.40 − 0.55) fm and only ω(1S) has a bit larger r.m.s. ∼ 0.7 fm. We +would like to underline the characteristic features of the ERM [24]: first, due +to the string rearrangement of a four-quark system the singularity lies close +to the lower threshold; second, this mechanism produces the resonance in +the S-wave hadron-hadron system and therefore, the quantum numbers of +these resonances JP C = 0++, 1++, 2++; third, a resonance does not appear, +if hadrons are identical. +In the literature there are still a controversy, concerning the X(3915), and +different interpretations were proposed. This resonance was considered in +tetraquark model within the Born–Oppenheimer approach in [29, 30, 31, 32], +due to the triangle singularity [22] and the threshold effects [33], as the +molecular Ds ¯Ds bound state [34] or the lightest cs¯c¯s state [35] and as the +diquark-antidiquark state, using the QCD sum rule method [23, 36]. In con- +trast to a molecular structure of four-quark states in the ERM these systems +are assumed to be compact systems, similar to the diquark-antidiquark states +studied in [37]. In such compact systems their wave functions at the origin +are not small and therefore they can be produced in the γγ transitions. +In our paper we will shortly discuss the higher scalars, X(4500), X(4700), +observed by the LHCb [38], which admit different interpretations. +The structure of the paper is as follows. In next section we shortly remind +the basic formulas in two-channel case and give the values of the parame- +ters, needed to define the masses and widths of the recoupled four-quark +resonances. In section 3 more general matrix representation of the ERM is +presented. In section 4 we calculate the transition amplitudes and give the +masses and widths of the scalar resonances, and compare them with exper- +imental data. In section 4 the masses of high X(4500), X(4700) resonances, +as the c¯c states, are discussed. Our conclusions are presented in section 5. +3 + +2 +The two-channel approach in the Extended +Recoupling Model +We study the experimental process where, among other products, two hadrons +are produced and one pair of hadrons (the pair 1) can transfer into another +pair of hadrons (the pair 2). In [24] the probability amplitude of this tran- +sition was denoted as V12(p1, p2), with p1, p2 – relative momenta of the +hadrons, referring to the pair 1 and 2. +If an infinite set of the transfor- +mations was supposed and the total production amplitude A2 of the pair +2 was written as a product of the slowly varying function F(E) and the +singular factor f12(E) = +1 +1−N , then the amplitude A2 = F(E)f12(E). This +definition of the transition amplitude V12 = V21 differs of that in other ap- +proaches, where one or more the OBE diagrams with meson exchanges are +taken. In the ERM [24] the process occurs through the intermediate stage of +the Quark Compound Bag (QCB) [39, 40], where all quarks and antiquarks +of two hadrons are participating in the string recoupling and, possibly, the +spin recoupling. Denoting the QCB wave functions as Φ(qi) (i = 1, 2, 3, 4) +and the two-hadron wave functions as Ψi(h1, h2), the amplitude V12 can be +written as, +V12 = (Ψ1(ha1hb1)Φ(qi))(Φ(qi)Ψ2(ha2hb2) = V1(p1)V2(p2), +(1) +i.e. the amplitude V12 = +1 +1−N acquires the factorized form: V12(p1, p2) = +v1(p1)v2(p2) with the factor N, written as +N = z(E)I1(E)I2(E). +(2) +Here z = z(E) can be called the transition probability, while I1(E), I2(E) +are the following integrals (see [24]): +Ii(E) = viGivi = +� +d3pi +(2π)3 +v2 +i (pi) +E′(pi) + E +′′(pi) − E , +(3) +where the hadron energies E′(pi), E +′′(pi) in the i-th pair near thresholds, +E′(p) = +p2 +2m′ + m′, include corresponding thresholds Eth +i +and the reduced +masses µi, namely, +Eth +i += m′(i) + m +′′(i), +µi = +m′(i)m +′′(i) +m′(i) + m +′′(i). +(4) +4 + +The result of the integration in Ii(E) can be approximated by the form: +Ii = consti +1 +νi − i +� +2µi(E − Eth +i ) +. +(5) +with µi, defined in (4), while νi is expressed via the parameters of the hadron +wave functions, which were calculated explicitly in [24]. Here we would like to +underline that the transition probability z(E) appears to be the only fitting +parameter in the ERM. +The whole series of the transitions from the pair 1 to 2 and back is summed +up to the amplitude f12, +f12(E) = +1 +1 − zI1I2 +, +Ii = +1 +νi − i +� +2µi(E − Eth +i ) +, +(6) +where νi are found from the four-quark wave functions, as in [37, 40]. The +form of Eq. (6) takes place for the energies E > E1, E2, while for E < +E1, E2, i.e. +below thresholds, the amplitude f1 = +� +1 +ν1+√ +2µ1(|E−E1|) +� +. +It +is important that in the ERM the process proceeds with the zero relative +angular momentum between two mesons, L = 0, otherwise the transition +probability z12(E) is much smaller and a resonance may not appear. +Note also that if the recoupling mechanism is instantaneous, or the tran- +sition from one pair of the mesons to another proceeds instantaneously, then +the transition amplitude V (12) does not factorize into V (1)V (2); such an +assumption was used in the original Recoupling Model [25]. However, in this +approximation, e.g. for the Tcc resonance agreement with experiment was +not reached [25]. On the contrary, in the ERM [24] the recoupling mecha- +nism proceeds in two stages: at first stage the hadrons h1, h2 collapse into +common “compound bag” [39, 40], where the four quarks are kept together +by the confining interaction between all possible quark pairs. This compound +bag has its own wave function Φi(q1, q2, q3, q4) and the probability amplitude +of the h1, h2 → Φ transition, which defines the factor V1(p1) in Eq. (2). In +a similar way the transition from the Bag state to the final hadrons h3, h4 +defines the factor V2(p2) and we obtain the relation: +v1(pi) = +� +d3q1...d3q4ψh1ψh2Φi(q1, ..q4), +(7) +and similar equation for v2(p2), replacing h1, h2 by h3, h4. From vi(pi) the +function Ii (3) is defined and using (6), one obtains νi. +5 + +Now we give experimental data and corresponding the ERM parame- +ters, referring to the four-quark systems, cq¯c¯q for X(3915) and cs¯c¯s for the +X(3960), X(4140). We give also the threshold energies E1, E2. +The parameters of the four-quark resonances +1) X(3915), JP = 0+, Γ(exp .) = 20(5) MeV [1, 3], J/ψω → D∗ ¯D∗, E1 = +3.880, E2 = 4020, µ1 = +M(J/ψ)M(ω) +M(J/ψ)+M(ω) = 0.624, +µ2 = +M(D∗)M( ¯D∗) +M(D∗)+M( ¯D∗) = +1.050 (all in GeV). From [24] ν1(J/ψω) = 0.21 GeV, ν2(D∗ ¯D∗) = +0.44 GeV. +2) X(3960), JP = 0+, Γ(exp .) = 43(21) MeV [18], [J/ψφ] → [D− +s D+ +s ], E1 = +3.936, E2 = 4116, µ1 = +MJ/ψMφ +MJ/ψ+Mφ = 0.767, +µ2 = +M(D+ +s )M(D− +s ) +M(D+ +s +M(D−) = +0.984; ν1(J/ψφ) = 0.265, +ν2 = 0.424 (all in GeV). +3) X(4140), JP = 0+, Γ(exp .) = 67(24) MeV[18], [J/ψφ] → [D∗− +s D∗+ +s ], E1 = +4.116, E2 = 4.224, µ1 = 0.767, +µ2 = 1.056, +ν1 = 0.265, +ν2 = 0.410 +(all in GeV). +Here q can be u, d quarks. To define the structure of the cross sections +we start with the value of the recoupling probability z = 0.2 GeV2 and the +parameters from the item 1) to obtain the distribution |f12(E)|2; the values +of |f12(E)|2 will be given in Section 4. In the amplitude f12(E) the resulting +singularity can be found in the form of (6) and for equal threshold masses +it produces a pole nearby thresholds; however, real distance between the +thresholds is large, ∼ 100 MeV and the actual singularity structure can be +more complicated. +3 +The matrix approach in the ERM +In previous Section we have presented the ERM equations in the case of two +channels, which are convenient to define the mass of a resonance. However, +they do not allow to study some details of the process, or to consider a larger +number of channels, which can have a influence at the properties of a four- +quark system. Therefore here we present a more general representation of +the amplitude using the unitarity relation, when the standard form of the +transition amplitudes fij(E) (for L = 0) is +fij − f ∗ +ji = +� +n +2iknfinf ∗ +jn, +(8) +6 + +or the unitarity relation can be realized through the M-matrix representation, +ˆfM = +1 +ˆ +M − iˆk +, +(9) +where ˆf, ˆ +M, ˆk are the matrices in the channel numbers [28]. In some cases +instead of the ˆ +M it is more convenient to use the ˆK matrix, ˆ +M = − ˆK−1, +where the matrix elements (m.e.) Mik(E) are the real analytic functions of +E with the dynamical cuts. For two-channel system ˆfM can be written as +ˆfM = +1 +ˆ +M − iˆk += +ˆN +D(E), +(10) +with +ˆN = +� +M22 − ik2 +−M21 +−M12 +M11 − ik1 +� +. +(11) +Here +D(E) = (M11 − ik1)(M22 − ik2) − M12M21. +(12) +One can easily establish the relation between the equations (10)- (12) and +the amplitude f12(ERM) (6) in two-channel case, which is a partial case of +these equations: +f12(ERM) = N11N22 +D(E) , +D(E) = (ν1 − ik1)(ν2 − ik2) − z, +(13) +and +z = M12M21, +νi ≡ Mii(E). +(14) +One can see that for z > 0 the values νi = Mii are real analytic functions +of E. In the ERM [24] νi were positive constants (defined via the parameters +of the compound bag model), while in general case Eqs. (12)-(14) include +other transition m.e.s fik. Later in our analysis we will be interested only in +the denominator D(E) (12) and the factors in (13), (14), which fully define +the position of a resonance. +The value of z, in principle, can be calculated within the ERM, however, +it can depend on many unknown parameters, and at the present stage we +prefer to keep z as a single fitting parameter. It can be shown that z depends +on the width of a resonance, but weakly depends on the resonance position. +Now we consider three channels case to study more realistic case and +choose the situation, when a resonance lies above the threshold 3. Here we do +7 + +not need to specify the channel 3, which for example, may be a conventional +c¯c state with JP C = 0++. We introduce the 3 × 3 amplitude ˆfM(E) with +three thresholds Ei (i = 1, 2, 3) and the momenta ki = +� +2µi(E − Ei), µi = +m1im2i +m1i+m2i, and Ei = m1i + m2i. Here m1i, m2i are the masses of two hadrons +in the channel i. In this case the form of Eq. (9) is kept, +ˆf3(E) = +ˆN3 +D3(E), D3(E) = ((M11−ik1)(M22−ik2)−M12M21))(M23−ik3)+∆M, +(15) +where ∆M is +∆M = M31M12M23+M32M21M13−M13M31(M22−ik2)−M32M23(M11−ik1). +(16) +For the energy E below the thresholds, 1 and 2, −ik1 = |k1|, −ik2 = |k2|, and +the factor ∆M is a real function of E. For the threshold 3 below thresholds +of 1 and 2 one can define the poles of the amplitude ˆf3, or the zeroes of +D3(E), and rewrite the Eq. (15) as, +D3 = (M11 − ik1)(M22 − ik2) − ˜z(E), +(17) +where the transition probability ˜z(E) +˜z(E) = M12M21 − ∆M(M33 + ik3) +M2 +33 + k2 +3 +(18) +One can see that ˜z(E) acquires imaginary part, which can be of both signs. +Therefore the influence of the third (or more) open channels, lying below +the thresholds E1, E2 in the 2 × 2 matrix f12(E), may be important in some +cases. The channel 3 can be taken into account, introducing complex values +of z(E), which can depend on the energy as in Eq. (18). +4 +The masses and widths of the scalar reso- +nances +We start with the X(3915) resonance and consider the following recoupling +process: J/ψω → D∗ ¯D∗. At first we look at two-channel situation and choose +the recoupling parameter z2 = 0.18 GeV2. For the X(3915) structure – cq¯c¯q +the parameters µi, νi, Ei are given in the item 1) of section 2. Then inserting +8 + +all parameters to the Eq. (13), one obtains the distribution |f12(E)|2 (f2 ≡ +f12). Its values for different E are given in Table 1, which show that the +maximum takes place at E = 3880 MeV, just near the lower threshold, and +Γ2 = Γ(2 − channels) ∼= 15 MeV. In experiment for this resonance, observed +by the Belle group in the process e+e− → e+e−J/ψω [1], the larger mass +M(exp .) = (3918.4 ± 1.9) MeV and Γ(exp .) = (20 ± 5) +MeV [3] were +obtained. +In the case of 3-channels, when e.g. the coupling to the c¯c channel is +taken into account, the factor z3(E) acquires an imaginary part. In this case +we calculate the amplitude f3(E), taking z3 = (0.18−i0.20) GeV2; the values +of |f3(E)|2 are given in Tab. 1. +Table 1: The values of the |f12(E)|2 for X(3915) +E(GeV) +3.85 +3.86 +3.88 +3.89 +3.90 +3.91 +3.915 +3.93 +|f2(E)|2 +3.04 +3.68 +63.08 +25.02 +8.33 +2.13 +1.65 +1.72 +|f3(E)|2 +1.82 +1.79 +1.03 +1.50 +3.30 +348.4 +360 +243 +From Table 1 one can see that in the 3-channel case the peak is shifted +up by ∼ 35 MeV and corresponds the mass ER ∼= 3.915 GeV and the width +Γ3 ∼= 20 MeV, which are in good agreement with the experimental mass and +Γ(exp.) = 20(5) MeV [3]. +The scalar resonance X(3960) with JP C = 0++ was recently observed by +the LHCb in the B+ → J/ψφK+ [18] and within the ERM it can be explained +due to the infinite chain of the transitions: J/ψφ → D+ +s D− +s and back. In +two-channel approximation the X(3960) parameters (νi, µi, Ei, (i = 1, 2) are +given in the item 2) (Section 2), which are used to define the amplitude (13). +First, we choose z2 = 0.30 GeV2 and calculate the transition amplitudes +|f12(E)|2; their values are given in the Table 2. +In the two-channel approximation the numbers from Table 2 show the +peak at E = 3940 MeV, near D+ +s D− +s threshold, and Γ(2 − ch.) ∼= 15 MeV. +In the 3-channel case the mass of the X(3960) resonance is shifted up to +the position M(3 − ch.) = 3970 MeV and the width increases to the value +Γ(th.) ∼= 45(5) MeV; these values are in agreement with the experimental +numbers: M(X(3960)) = 3956(15) MeV, Γ(X(3960)) = (43 ± 21) MeV [18]. +In [18] the LHCb has reported about another, the X(4140) resonance, +with JP C = 0++, in the B+ → D+ +s D− +s K+ decay. Its mass M(X(4140) = +9 + +Table 2: The transition probability |f12|2 as a function of the energy E for +the X(3960) resonance +E(GeV) +3.85 +3.88 +3.89 +3.92 +3.95 +3.97 +4.00 +4.05 +|f12|2(z = 0.30) +3.93 +28.6 +7.89 +3.20 +2.28 +2.00 +1.38 +1.50 +|f3|2(z = 0.30 − i0.30) +2.0 +1.43 +4.02 +23.7 +198 +500 +142.3 +42.2 +4133(12) MeV is close to the J/ψφ threshold. We consider this resonance as +the cs¯c¯s system and first calculate the squared amplitudes |f12(E)|2 in two- +channel case, taking the parameters µi, νi, Ei from the item 3) of Section 2. In +this 2-channel case: J/ψφ and D∗+ +s D∗− +s +the transition probability z2 = 0.35 +is taken and the calculated values of |f12|2 are given in Table 3. +In three-channel case the channel D+ +s D− +s is added as the third one, then +the values |f3|2 are calculated for z3 = 0.20 − i0.20 and given in Table 3. +Table 3: The values of the |f12(E)|2 and |f3(E)|2 for the X(4140) +E(GeV) +4.00 +4.07 +4.12 +4.17 +4.22 +|f12(E)|2(z = 0.35) +3.40 +8.67 +3.86 +1.27 +0.45 +|f3|2(z = 0.2 − i0.2) +4.54 +12.87 +32.12 +13.7 +0.66 +From Table 3 one can see the peak at ER = (4.09 ± 0.01) GeV, Γ(th.) = +60 MeV in two-channel approximation and the peak at ER = (4.12±0.02) GeV +with the width Γ(th.) ∼= 100 MeV in tree-channel case, which are in good +agreement with the experimental mass M(X(4140)) = (4133 ± 12) MeV and +Γ(X(4140)) = (67 ± 24) MeV [18]. +Our numbers in Tables 1–3 show that in two-channel case the resonance +always lies just near the lower threshold, however, if the coupling to the third +channel is taken into account, then it is shifted up and its position occurs to +be close to the experimental number. The masses and widths of the exotic +resonances, X(3915), X(3960), X(4140), defined in the ERM, are given in +the Table 4 together with experimental data. +From Table 4 one can see that in the ERM the predicted masses and +the widths of the scalar four-quark resonances are in good agreement with +10 + +Table 4: The ERM predictions for the masses and widths (in MeV) of exotic +resonances with JP C = 0++ +Resonance +M(th.) +M(exp.) +Γ(th.) +Γ(exp.) +X(3915) +3920 +3918 (2) +20 +20(5) [3] +X(3960) +3970 +3956(15) +45(5) +43(21) [18] +X(4140) +4120(20) +4133(12) +100 +67(24) [18] +experiment, if besides two channels, which creates the resonance, the coupling +of the resonance to third channel is taken into account. +Comparing our results with those in literature, one can notice that our +conclusions on the four-quark structure of the X(3915), X(3960, X(4140)) +also agree with the analysis in the paper [33], based on the coupled channel +model of the c¯c and meson-meson systems. Notice that the general structure +of the channel-coupling matrix elements in both approaches is similar. +5 +The scalar X(4500), X(4700) resonances +High scalar resonances X(4500), X(4700), or χc0(4500), χc0(4700), [38], were +studied in many papers and for them two interpretations were suggested. +First, the X(4500) and X(4700) are considered as the c¯c states – 4 3P0 and +5 3P0 and their masses were calculated in relativistic quark models, where +coupling to open channels was taken into account [14, 15, 41]. In [41] the +influence of open channels is studied using the so-called screened potential +[11], while in [13] the spectrum was calculated using the relativistic string +Hamiltonian [42] with the flattened confining potential [43]; this flattening +effect arises due to creation of virtual q¯q pairs. Notice that the flattened +confining potential appears to be universal for all types of the mesons and it +produces the hadronic shifts down ∼ (100 − 130) MeV for the 4P, 5P char- +monium states and gives the masses of the 4 3P0, 5 3P0 states in a reasonable +agreement with experiment [13]. On the contrary, in [44], within the +3P0 +model, much smaller shifts due to the coupled-channel effects, <∼ 30 MeV , +were obtained for the 4 3P0, 5 3P0 states, while in [41] these states acquire too +large mass shifts for the chosen screened potential. +Model-independent analysis of the c¯c spectrum can also be done by means +11 + +of the Regge trajectories, if they are defined not for the meson mass M(nL) +but for the excitation energy: E(nL) = M(nL) − 2 ¯mQ [45], where ¯mQ is the +current heavy quark mass [13]: +(M(n 3P0)−2 ¯mc)2 = 1.06+1.08nr, (inGeV2); n = nr +1, +¯mc = 1.20 GeV2. +(19) +This Regge trajectory gives M(4 3P0) = 4.474 GeV and M(5 3P0) = 4.719 GeV, +in good agreement with the LHCb data [38] (see Table 5). +Table 5: The Regge trajectory predictions for the masses of the charmonium +n 3P0 states (in MeV) +state +M(nP) +exp. mass +1 3P0 +3429 +3414.8(3)) +2 3P0 +3863 +3862+26 +−32 [16] +3 3P0 +4194 +abs. +4 3P0 +4473 +4474 ± 6 [38] +5 3P0 +4719 +4694 ± 4+16 +−3 [38] +6 3P0 +4941 +abs +In Table 5 the masses M(2 3P0) = 3863 MeV, M(4 3P0) = 4473 MeV and +M(5 3P0) = 4719 MeV, show very good agreement with those of χc0(3862) +[16], X(4500) and X(4700) [38]. +At present other high excitations with +JP = 1+, 2+ (n = 4, 5) are not yet found and their observation would be +very important to understand the fine-structure effects of high charmonium, +in particular, the fine-structure splitting have to decrease for a screened GE +potential. +Notice that the resonance X(4700) lies very close to the ψ(2S)φ threshold +and this fact indicates a possible connection between the c¯c and the cs¯c¯s +states. The four-quark interpretation of the X(4500), X(4700) was discussed +in different models [19],[46]-[49], where in the mass region (4.4–4.8) GeV the +radial or orbital excitations of a diquark-antidiquark systems can exist. +12 + +6 +Conclusions +In our paper the scalar resonances X(3915), X(3960), X(4140) are assumed +to be the four-quark states, produced due to recoupling mechanism, when +one pair of mesons can transform into another pair of mesons infinitely many +times. These resonances do not exist in the c¯c spectrum. As the four-quark +states they have several specific features: +1. The resonance appears only in the S-wave decay channel. +2. Within the ERM it lies rather close to the lower threshold. +3. The scalar four-quark resonance can be created in two channel case due +to transitions between channels, but it can also be coupled to another +channel 3, e.g. the c¯c channel. +4. 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(Belle collab.), arXiv:2301.09421 [hep-ex]. +16 + diff --git a/0tFRT4oBgHgl3EQfkzf9/content/tmp_files/load_file.txt b/0tFRT4oBgHgl3EQfkzf9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..34f4cde279fd8e64af6f259649c2f38b01d039cb --- /dev/null +++ b/0tFRT4oBgHgl3EQfkzf9/content/tmp_files/load_file.txt @@ -0,0 +1,750 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf,len=749 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='13597v1 [hep-ph] 31 Jan 2023 The scalar exotic resonances X(3915), X(3960), X(4140) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='Badalian and Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='Simonov NRC “Kurchatov Institute” Moscow, Russia February 1, 2023 Abstract The scalar resonances X(3915), X(3960), X(4140) are considered as exotic four-quark states: cq¯c¯q, cs¯c¯s, cs¯c¯s, while the X(3863) is proved to be the c¯c, 2 3P0 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The masses and the widths of these reso- nances are calculated in the framework of the Extended Recoupling Model, where a four-quark system is formed inside the bag and has relatively small size (<∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='0 fm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Then the resonance X(3915) ap- pears due to the transitions: J/ψω into D∗+D∗− (or D∗0 ¯D∗0) and back, while the X(3960) is created due to the transitions D+ s D− s into J/ψφ and back, and the X(4140) is formed in the transitions J/ψφ into D∗+ s D∗− s and back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The characteristic feature of the recoupling mechanism is that this type of resonances can be predominantly in the S-wave decay channels and has JP = 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In two-channel case the reso- nance occurs to be just near the lower threshold, while due to coupling to third channel (like the c¯c channel) it is shifted up and lies by (20– 30) MeV above the lower threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The following masses and widths are calculated: M(X(3915)) = 3920 MeV, Γ(X(3915)) = 20 MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' M(X(3960)) = 3970 MeV, Γ(X(3960) = 45(5) MeV, M(X(4140)) = 4120(20) MeV, Γ(X(4140)) = 100 MeV, which are in good agreement with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 1 1 Introduction In the region (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='9–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='2) GeV there are now three scalar resonances and the X(3915) was the first, observed by the Belle in the e+e− → J/ψωK process [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Later this resonance was confirmed by the BaBar [2] and in several other experiments [3]), in particular, in two-photon collisions [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' For some years this resonance was assumed to be the conventional c¯c meson – χco(2P), although this interpretation has called out some doubts [6, 7] (see discussion in the reviews [8, 9]) and does not agree with predictions in different relativistic potential models (RPM) [10]-[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The experimental masses of the X(3915) and χc2(2P) were found to be almost equal, while in the RPMs a smaller mass, M(2 3P0) ∼= 3870 ± 30 MeV, and much larger mass difference, δ20(2P) = M(χc2(2P) − M(χc0(2P) ∼= (70 − 100) MeV, were predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Notice that large mass difference δ20 is kept even if the coupling of the χc0(2P) to open channels is taken into account [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Such theoretical expectations were supported by the Belle observation of the wide scalar X(3860) resonance [16], both in e+e− → J/ψD+D− and e+e− → J/ψD0 ¯D0 decays, which has the mass M = 3862+26 −32 +40 −82 MeV and large width Γ ∼= 200 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The existence of the scalar X(3860) resonance is confirmed by the analysis of two-photon production, γγ → D ¯D in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Very recently the LHCb [18] has observed two more scalar resonances X(3960), X(4140) in the D+ s D− s mass spectrum in the B+ → D+ s D− s K+ de- cays with the parameters: M(X(3960)) = (3956±5±10) MeV, Γ(X(3960)) = (43 ± 13 ± 8) MeV, M(X(4140)) = (4133 ± 6 ± 6) MeV, Γ(X(4140)) = (67 ± 17 ± 7) MeV, both with JP C = 0++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' These new scalar resonances evi- dently look as exotic states and the X(3960) was interpreted as the molecular D+ s D− s state within the QCD sum rules approach [19, 20] and in a coupled- channel model [21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' in [22] it appears due to the triangle singularity, while in [23] the parameters of the X(3960), as a diquark-antidiquark state, were obtained in a good agreement with experiment, using the QCD sum rules approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Notice that the masses of the X(3960) and X(4140) resonances lie by ∼ 20 MeV above the thresholds: D+ s D− s and J/ψφ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In our paper we assume that the X(3915) and both the X(3960), X(4140) belong to exotic four-quark states cq¯c¯q and cs¯c¯s and to define their parame- ters we will use the Extended Recoupling Model (ERM), recently suggested in [24], which develops the Recouplimg Model, presented earlier [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The ERM allows to calculate the mass and width of a scalar four-quark states, however, within suggested mechanism such resonances cannot exist in the 2 systems with two identical mesons, like D+ s D+ s , D∗+ s , D∗+ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' This theoretical prediction is supported by the Belle experiment [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the ERM the system of two mesons, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (J/ψ + φ), can transfer into another pair of the mesons (D+ s , D− s ) by rearranging confining strings and back in the infinite chain of transformations, like J/ψφ → (D+ s ¯D− s ) → J/ψφ → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='. Note that such se- quences can also be treated, for example, in the standard OBE approximation with the meson exchanges, which, however, does not produce the singulari- ties near the thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the coupled-channel models (CCM) [27, 28] the interaction between hadrons, like D+ s D− s , J/ψφ, is usually neglected, while in the ERM such interaction is taken into account, introducing the four- quark bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' It is important that all hadrons involved have rather small sizes, ∼= (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='40 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='55) fm and only ω(1S) has a bit larger r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='7 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' We would like to underline the characteristic features of the ERM [24]: first, due to the string rearrangement of a four-quark system the singularity lies close to the lower threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' second, this mechanism produces the resonance in the S-wave hadron-hadron system and therefore, the quantum numbers of these resonances JP C = 0++, 1++, 2++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' third, a resonance does not appear, if hadrons are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the literature there are still a controversy, concerning the X(3915), and different interpretations were proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' This resonance was considered in tetraquark model within the Born–Oppenheimer approach in [29, 30, 31, 32], due to the triangle singularity [22] and the threshold effects [33], as the molecular Ds ¯Ds bound state [34] or the lightest cs¯c¯s state [35] and as the diquark-antidiquark state, using the QCD sum rule method [23, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In con- trast to a molecular structure of four-quark states in the ERM these systems are assumed to be compact systems, similar to the diquark-antidiquark states studied in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In such compact systems their wave functions at the origin are not small and therefore they can be produced in the γγ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In our paper we will shortly discuss the higher scalars, X(4500), X(4700), observed by the LHCb [38], which admit different interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In next section we shortly remind the basic formulas in two-channel case and give the values of the parame- ters, needed to define the masses and widths of the recoupled four-quark resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In section 3 more general matrix representation of the ERM is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In section 4 we calculate the transition amplitudes and give the masses and widths of the scalar resonances, and compare them with exper- imental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In section 4 the masses of high X(4500), X(4700) resonances, as the c¯c states, are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Our conclusions are presented in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 3 2 The two-channel approach in the Extended Recoupling Model We study the experimental process where, among other products, two hadrons are produced and one pair of hadrons (the pair 1) can transfer into another pair of hadrons (the pair 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In [24] the probability amplitude of this tran- sition was denoted as V12(p1, p2), with p1, p2 – relative momenta of the hadrons, referring to the pair 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' If an infinite set of the transfor- mations was supposed and the total production amplitude A2 of the pair 2 was written as a product of the slowly varying function F(E) and the singular factor f12(E) = 1 1−N , then the amplitude A2 = F(E)f12(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' This definition of the transition amplitude V12 = V21 differs of that in other ap- proaches, where one or more the OBE diagrams with meson exchanges are taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the ERM [24] the process occurs through the intermediate stage of the Quark Compound Bag (QCB) [39, 40], where all quarks and antiquarks of two hadrons are participating in the string recoupling and, possibly, the spin recoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Denoting the QCB wave functions as Φ(qi) (i = 1, 2, 3, 4) and the two-hadron wave functions as Ψi(h1, h2), the amplitude V12 can be written as, V12 = (Ψ1(ha1hb1)Φ(qi))(Φ(qi)Ψ2(ha2hb2) = V1(p1)V2(p2), (1) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' the amplitude V12 = 1 1−N acquires the factorized form: V12(p1, p2) = v1(p1)v2(p2) with the factor N, written as N = z(E)I1(E)I2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (2) Here z = z(E) can be called the transition probability, while I1(E), I2(E) are the following integrals (see [24]): Ii(E) = viGivi = � d3pi (2π)3 v2 i (pi) E′(pi) + E ′′(pi) − E , (3) where the hadron energies E′(pi), E ′′(pi) in the i-th pair near thresholds, E′(p) = p2 2m′ + m′, include corresponding thresholds Eth i and the reduced masses µi, namely, Eth i = m′(i) + m ′′(i), µi = m′(i)m ′′(i) m′(i) + m ′′(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (4) 4 The result of the integration in Ii(E) can be approximated by the form: Ii = consti 1 νi − i � 2µi(E − Eth i ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (5) with µi, defined in (4), while νi is expressed via the parameters of the hadron wave functions, which were calculated explicitly in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Here we would like to underline that the transition probability z(E) appears to be the only fitting parameter in the ERM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The whole series of the transitions from the pair 1 to 2 and back is summed up to the amplitude f12, f12(E) = 1 1 − zI1I2 , Ii = 1 νi − i � 2µi(E − Eth i ) , (6) where νi are found from the four-quark wave functions, as in [37, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (6) takes place for the energies E > E1, E2, while for E < E1, E2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' below thresholds, the amplitude f1 = � 1 ν1+√ 2µ1(|E−E1|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' It is important that in the ERM the process proceeds with the zero relative angular momentum between two mesons, L = 0, otherwise the transition probability z12(E) is much smaller and a resonance may not appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Note also that if the recoupling mechanism is instantaneous, or the tran- sition from one pair of the mesons to another proceeds instantaneously, then the transition amplitude V (12) does not factorize into V (1)V (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' such an assumption was used in the original Recoupling Model [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' However, in this approximation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' for the Tcc resonance agreement with experiment was not reached [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' On the contrary, in the ERM [24] the recoupling mecha- nism proceeds in two stages: at first stage the hadrons h1, h2 collapse into common “compound bag” [39, 40], where the four quarks are kept together by the confining interaction between all possible quark pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' This compound bag has its own wave function Φi(q1, q2, q3, q4) and the probability amplitude of the h1, h2 → Φ transition, which defines the factor V1(p1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In a similar way the transition from the Bag state to the final hadrons h3, h4 defines the factor V2(p2) and we obtain the relation: v1(pi) = � d3q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='d3q4ψh1ψh2Φi(q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='.q4), (7) and similar equation for v2(p2), replacing h1, h2 by h3, h4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' From vi(pi) the function Ii (3) is defined and using (6), one obtains νi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 5 Now we give experimental data and corresponding the ERM parame- ters, referring to the four-quark systems, cq¯c¯q for X(3915) and cs¯c¯s for the X(3960), X(4140).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' We give also the threshold energies E1, E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The parameters of the four-quark resonances 1) X(3915), JP = 0+, Γ(exp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = 20(5) MeV [1, 3], J/ψω → D∗ ¯D∗, E1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='880, E2 = 4020, µ1 = M(J/ψ)M(ω) M(J/ψ)+M(ω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='624, µ2 = M(D∗)M( ¯D∗) M(D∗)+M( ¯D∗) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='050 (all in GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' From [24] ν1(J/ψω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='21 GeV, ν2(D∗ ¯D∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='44 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 2) X(3960), JP = 0+, Γ(exp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = 43(21) MeV [18], [J/ψφ] → [D− s D+ s ], E1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='936, E2 = 4116, µ1 = MJ/ψMφ MJ/ψ+Mφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='767, µ2 = M(D+ s )M(D− s ) M(D+ s +M(D−) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' ν1(J/ψφ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='265, ν2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='424 (all in GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 3) X(4140), JP = 0+, Γ(exp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = 67(24) MeV[18], [J/ψφ] → [D∗− s D∗+ s ], E1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='116, E2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='224, µ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='767, µ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='056, ν1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='265, ν2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='410 (all in GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Here q can be u, d quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' To define the structure of the cross sections we start with the value of the recoupling probability z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='2 GeV2 and the parameters from the item 1) to obtain the distribution |f12(E)|2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' the values of |f12(E)|2 will be given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the amplitude f12(E) the resulting singularity can be found in the form of (6) and for equal threshold masses it produces a pole nearby thresholds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' however, real distance between the thresholds is large, ∼ 100 MeV and the actual singularity structure can be more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 3 The matrix approach in the ERM In previous Section we have presented the ERM equations in the case of two channels, which are convenient to define the mass of a resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' However, they do not allow to study some details of the process, or to consider a larger number of channels, which can have a influence at the properties of a four- quark system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Therefore here we present a more general representation of the amplitude using the unitarity relation, when the standard form of the transition amplitudes fij(E) (for L = 0) is fij − f ∗ ji = � n 2iknfinf ∗ jn, (8) 6 or the unitarity relation can be realized through the M-matrix representation, ˆfM = 1 ˆ M − iˆk , (9) where ˆf, ˆ M, ˆk are the matrices in the channel numbers [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In some cases instead of the ˆ M it is more convenient to use the ˆK matrix, ˆ M = − ˆK−1, where the matrix elements (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') Mik(E) are the real analytic functions of E with the dynamical cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' For two-channel system ˆfM can be written as ˆfM = 1 ˆ M − iˆk = ˆN D(E), (10) with ˆN = � M22 − ik2 −M21 −M12 M11 − ik1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (11) Here D(E) = (M11 − ik1)(M22 − ik2) − M12M21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (12) One can easily establish the relation between the equations (10)- (12) and the amplitude f12(ERM) (6) in two-channel case, which is a partial case of these equations: f12(ERM) = N11N22 D(E) , D(E) = (ν1 − ik1)(ν2 − ik2) − z, (13) and z = M12M21, νi ≡ Mii(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (14) One can see that for z > 0 the values νi = Mii are real analytic functions of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the ERM [24] νi were positive constants (defined via the parameters of the compound bag model), while in general case Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (12)-(14) include other transition m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='s fik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Later in our analysis we will be interested only in the denominator D(E) (12) and the factors in (13), (14), which fully define the position of a resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The value of z, in principle, can be calculated within the ERM, however, it can depend on many unknown parameters, and at the present stage we prefer to keep z as a single fitting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' It can be shown that z depends on the width of a resonance, but weakly depends on the resonance position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Now we consider three channels case to study more realistic case and choose the situation, when a resonance lies above the threshold 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Here we do 7 not need to specify the channel 3, which for example, may be a conventional c¯c state with JP C = 0++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' We introduce the 3 × 3 amplitude ˆfM(E) with three thresholds Ei (i = 1, 2, 3) and the momenta ki = � 2µi(E − Ei), µi = m1im2i m1i+m2i, and Ei = m1i + m2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Here m1i, m2i are the masses of two hadrons in the channel i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In this case the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (9) is kept, ˆf3(E) = ˆN3 D3(E), D3(E) = ((M11−ik1)(M22−ik2)−M12M21))(M23−ik3)+∆M, (15) where ∆M is ∆M = M31M12M23+M32M21M13−M13M31(M22−ik2)−M32M23(M11−ik1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (16) For the energy E below the thresholds, 1 and 2, −ik1 = |k1|, −ik2 = |k2|, and the factor ∆M is a real function of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' For the threshold 3 below thresholds of 1 and 2 one can define the poles of the amplitude ˆf3, or the zeroes of D3(E), and rewrite the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (15) as, D3 = (M11 − ik1)(M22 − ik2) − ˜z(E), (17) where the transition probability ˜z(E) ˜z(E) = M12M21 − ∆M(M33 + ik3) M2 33 + k2 3 (18) One can see that ˜z(E) acquires imaginary part, which can be of both signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Therefore the influence of the third (or more) open channels, lying below the thresholds E1, E2 in the 2 × 2 matrix f12(E), may be important in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The channel 3 can be taken into account, introducing complex values of z(E), which can depend on the energy as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 4 The masses and widths of the scalar reso- nances We start with the X(3915) resonance and consider the following recoupling process: J/ψω → D∗ ¯D∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' At first we look at two-channel situation and choose the recoupling parameter z2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='18 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' For the X(3915) structure – cq¯c¯q the parameters µi, νi, Ei are given in the item 1) of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Then inserting 8 all parameters to the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (13), one obtains the distribution |f12(E)|2 (f2 ≡ f12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Its values for different E are given in Table 1, which show that the maximum takes place at E = 3880 MeV, just near the lower threshold, and Γ2 = Γ(2 − channels) ∼= 15 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In experiment for this resonance, observed by the Belle group in the process e+e− → e+e−J/ψω [1], the larger mass M(exp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = (3918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='9) MeV and Γ(exp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = (20 ± 5) MeV [3] were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the case of 3-channels, when e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' the coupling to the c¯c channel is taken into account, the factor z3(E) acquires an imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In this case we calculate the amplitude f3(E), taking z3 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='18−i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='20) GeV2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' the values of |f3(E)|2 are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Table 1: The values of the |f12(E)|2 for X(3915) E(GeV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='86 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='915 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='93 |f2(E)|2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='68 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='08 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='72 |f3(E)|2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='30 348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='4 360 243 From Table 1 one can see that in the 3-channel case the peak is shifted up by ∼ 35 MeV and corresponds the mass ER ∼= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='915 GeV and the width Γ3 ∼= 20 MeV, which are in good agreement with the experimental mass and Γ(exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = 20(5) MeV [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The scalar resonance X(3960) with JP C = 0++ was recently observed by the LHCb in the B+ → J/ψφK+ [18] and within the ERM it can be explained due to the infinite chain of the transitions: J/ψφ → D+ s D− s and back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In two-channel approximation the X(3960) parameters (νi, µi, Ei, (i = 1, 2) are given in the item 2) (Section 2), which are used to define the amplitude (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' First, we choose z2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='30 GeV2 and calculate the transition amplitudes |f12(E)|2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' their values are given in the Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the two-channel approximation the numbers from Table 2 show the peak at E = 3940 MeV, near D+ s D− s threshold, and Γ(2 − ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') ∼= 15 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the 3-channel case the mass of the X(3960) resonance is shifted up to the position M(3 − ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = 3970 MeV and the width increases to the value Γ(th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') ∼= 45(5) MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' these values are in agreement with the experimental numbers: M(X(3960)) = 3956(15) MeV, Γ(X(3960)) = (43 ± 21) MeV [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In [18] the LHCb has reported about another, the X(4140) resonance, with JP C = 0++, in the B+ → D+ s D− s K+ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Its mass M(X(4140) = 9 Table 2: The transition probability |f12|2 as a function of the energy E for the X(3960) resonance E(GeV) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='92 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='97 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='05 |f12|2(z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='30) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='93 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='50 |f3|2(z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='30 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='30) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='43 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='02 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='7 198 500 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='2 4133(12) MeV is close to the J/ψφ threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' We consider this resonance as the cs¯c¯s system and first calculate the squared amplitudes |f12(E)|2 in two- channel case, taking the parameters µi, νi, Ei from the item 3) of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In this 2-channel case: J/ψφ and D∗+ s D∗− s the transition probability z2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='35 is taken and the calculated values of |f12|2 are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In three-channel case the channel D+ s D− s is added as the third one, then the values |f3|2 are calculated for z3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='20 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='20 and given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Table 3: The values of the |f12(E)|2 and |f3(E)|2 for the X(4140) E(GeV) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='07 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='22 |f12(E)|2(z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='35) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='40 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='45 |f3|2(z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='2 − i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='54 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='87 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='12 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='66 From Table 3 one can see the peak at ER = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='01) GeV, Γ(th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') = 60 MeV in two-channel approximation and the peak at ER = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='02) GeV with the width Γ(th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') ∼= 100 MeV in tree-channel case, which are in good agreement with the experimental mass M(X(4140)) = (4133 ± 12) MeV and Γ(X(4140)) = (67 ± 24) MeV [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Our numbers in Tables 1–3 show that in two-channel case the resonance always lies just near the lower threshold, however, if the coupling to the third channel is taken into account, then it is shifted up and its position occurs to be close to the experimental number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The masses and widths of the exotic resonances, X(3915), X(3960), X(4140), defined in the ERM, are given in the Table 4 together with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' From Table 4 one can see that in the ERM the predicted masses and the widths of the scalar four-quark resonances are in good agreement with 10 Table 4: The ERM predictions for the masses and widths (in MeV) of exotic resonances with JP C = 0++ Resonance M(th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') M(exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') Γ(th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') Γ(exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=') X(3915) 3920 3918 (2) 20 20(5) [3] X(3960) 3970 3956(15) 45(5) 43(21) [18] X(4140) 4120(20) 4133(12) 100 67(24) [18] experiment, if besides two channels, which creates the resonance, the coupling of the resonance to third channel is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Comparing our results with those in literature, one can notice that our conclusions on the four-quark structure of the X(3915), X(3960, X(4140)) also agree with the analysis in the paper [33], based on the coupled channel model of the c¯c and meson-meson systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Notice that the general structure of the channel-coupling matrix elements in both approaches is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 5 The scalar X(4500), X(4700) resonances High scalar resonances X(4500), X(4700), or χc0(4500), χc0(4700), [38], were studied in many papers and for them two interpretations were suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' First, the X(4500) and X(4700) are considered as the c¯c states – 4 3P0 and 5 3P0 and their masses were calculated in relativistic quark models, where coupling to open channels was taken into account [14, 15, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In [41] the influence of open channels is studied using the so-called screened potential [11], while in [13] the spectrum was calculated using the relativistic string Hamiltonian [42] with the flattened confining potential [43];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' this flattening effect arises due to creation of virtual q¯q pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Notice that the flattened confining potential appears to be universal for all types of the mesons and it produces the hadronic shifts down ∼ (100 − 130) MeV for the 4P, 5P char- monium states and gives the masses of the 4 3P0, 5 3P0 states in a reasonable agreement with experiment [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' On the contrary, in [44], within the 3P0 model, much smaller shifts due to the coupled-channel effects, <∼ 30 MeV , were obtained for the 4 3P0, 5 3P0 states, while in [41] these states acquire too large mass shifts for the chosen screened potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Model-independent analysis of the c¯c spectrum can also be done by means 11 of the Regge trajectories, if they are defined not for the meson mass M(nL) but for the excitation energy: E(nL) = M(nL) − 2 ¯mQ [45], where ¯mQ is the current heavy quark mass [13]: (M(n 3P0)−2 ¯mc)2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='06+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='08nr, (inGeV2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' n = nr +1, ¯mc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='20 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' (19) This Regge trajectory gives M(4 3P0) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='474 GeV and M(5 3P0) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='719 GeV, in good agreement with the LHCb data [38] (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Table 5: The Regge trajectory predictions for the masses of the charmonium n 3P0 states (in MeV) state M(nP) exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' mass 1 3P0 3429 3414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='8(3)) 2 3P0 3863 3862+26 −32 [16] 3 3P0 4194 abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 4 3P0 4473 4474 ± 6 [38] 5 3P0 4719 4694 ± 4+16 −3 [38] 6 3P0 4941 abs In Table 5 the masses M(2 3P0) = 3863 MeV, M(4 3P0) = 4473 MeV and M(5 3P0) = 4719 MeV, show very good agreement with those of χc0(3862) [16], X(4500) and X(4700) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' At present other high excitations with JP = 1+, 2+ (n = 4, 5) are not yet found and their observation would be very important to understand the fine-structure effects of high charmonium, in particular, the fine-structure splitting have to decrease for a screened GE potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Notice that the resonance X(4700) lies very close to the ψ(2S)φ threshold and this fact indicates a possible connection between the c¯c and the cs¯c¯s states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The four-quark interpretation of the X(4500), X(4700) was discussed in different models [19],[46]-[49], where in the mass region (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='4–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='8) GeV the radial or orbital excitations of a diquark-antidiquark systems can exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 12 6 Conclusions In our paper the scalar resonances X(3915), X(3960), X(4140) are assumed to be the four-quark states, produced due to recoupling mechanism, when one pair of mesons can transform into another pair of mesons infinitely many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' These resonances do not exist in the c¯c spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' As the four-quark states they have several specific features: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The resonance appears only in the S-wave decay channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Within the ERM it lies rather close to the lower threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The scalar four-quark resonance can be created in two channel case due to transitions between channels, but it can also be coupled to another channel 3, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' the c¯c channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' These resonances have no large sizes, being the compact systems, and this fact may be important for their observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' In the case of the X(3915) this statement is confirmed by the Belle analysis of the Q2 distribution of the X(3915) → J/ψω decays in [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The masses and widths of the X(3915), X(3960), X(4140), presented in Ta- ble 4, are obtained in a good agreement with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' The authors are grateful to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFRT4oBgHgl3EQfkzf9/content/2301.13597v1.pdf'} +page_content=' Igumnova for collaboration.' metadata={'source': 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full-field ultrasonic characterization +Yang Xu1, Fatemeh Pourahmadian1,2∗, Jian Song1, Conglin Wang3 +1 Department of Civil, Environmental & Architectural Engineering, University of Colorado Boulder, USA +2 Department of Applied Mathematics, University of Colorado Boulder, USA +3 Department of Physics, University of Colorado Boulder, USA +Abstract +This study takes advantage of recent advances in machine learning to establish a physics-based data analytic +platform for distributed reconstruction of mechanical properties in layered components from full waveform +data. In this vein, two logics, namely the direct inversion and physics-informed neural networks (PINNs), are +explored. The direct inversion entails three steps: (i) spectral denoising and differentiation of the full-field +data, (ii) building appropriate neural maps to approximate the profile of unknown physical and regularization +parameters on their respective domains, and (iii) simultaneous training of the neural networks by minimizing +the Tikhonov-regularized PDE loss using data from (i). PINNs furnish efficient surrogate models of complex +systems with predictive capabilities via multitask learning where the field variables are modeled by neural +maps endowed with (scaler or distributed) auxiliary parameters such as physical unknowns and loss function +weights. PINNs are then trained by minimizing a measure of data misfit subject to the underlying physical +laws as constraints. +In this study, to facilitate learning from ultrasonic data, the PINNs loss adopts (a) +wavenumber-dependent Sobolev norms to compute the data misfit, and (b) non-adaptive weights in a specific +scaling framework to naturally balance the loss objectives by leveraging the form of PDEs germane to elastic- +wave propagation. Both paradigms are examined via synthetic and laboratory test data. In the latter case, the +reconstructions are performed at multiple frequencies and the results are verified by a set of complementary +experiments highlighting the importance of verification and validation in data-driven modeling. +Keywords: +deep learning, ultrasonic testing, data-driven mechanics, full-wavefield inversion +1. Introduction +Recent advances in laser-based ultrasonic testing has led to the emergence of dense spatiotemporal datasets +which along with suitable data analytic solutions may lead to better understanding of the mechanics of complex +materials and components. This includes learning of distributed mechanical properties from test data which is +of interest in a wide spectrum of applications from medical diagnosis to additive manufacturing [1, 2, 3, 4, 5, +6, 7]. This work makes use of recent progress in deep learning [8, 9] germane to direct and inverse problems in +partial differential equations [10, 11, 12, 13] to develop a systematic full-field inversion framework to recover the +profile of pertinent physical quantities in layered components from laser ultrasonic measurements. The focus is +on two paradigms, namely: the direct inversion and physics-informed neural networks (PINNs) [14, 15, 16, 17]. +The direct inversion approach is in fact the authors’ rendition of elastography method [18, 19, 20] through the +prism of deep learning. To this end, tools of signal processing are deployed to (a) denoise the experimental +data, and (b) carefully compute the required field derivatives as per the governing equations. In parallel, +the unknown distribution of PDE parameters in space-frequency are identified by neural networks which are +then trained by minimizing the single-objective elastography loss. The learning process is stabilized via the +Tikhonov regularization [21, 22] where the regularization parameter is defined in a distributed sense as a +separate neural network which is simultaneously trained with the sought-for physical quantities. This unique +∗Corresponding author: tel. 303-492-2027, email fatemeh.pourahmadian@colorado.edu +Preprint submitted to Elsevier +January 9, 2023 +arXiv:2301.02378v1 [math.NA] 6 Jan 2023 + +exercise of learning the regularization field without a-priori estimates, thanks to neural networks, proved to +be convenient, effective, and remarkably insightful in inversion of multi-fidelity experimental data. +PINNs have recently come under the spotlight for offering efficient, yet predictive, models of complex +PDE systems [10] that has so far been backed by rigorous theoretical justification within the context of linear +elliptic and parabolic PDEs [23]. Given the multitask nature of training for these networks and the existing +challenges with modeling stiff and highly oscillatory PDEs [12, 24], much of the most recent efforts has been +focused on (a) adaptive gauging of the loss function [12, 25, 26, 27, 28, 29, 13], and (b) addressing the gradient +pathologies [24, 13] e.g., via learning rate annealing [30] and customizing the network architecture [11, 31, 32]. +In this study, our initially austere implementations of PINNs using both synthetic and experimental waveforms +led almost invariably to failure which further investigation attributed to the following impediments: (a) high- +norm gradient fields due to large wavenumbers, (b) high-order governing PDEs in the case of laboratory +experiments, and (c) imbalanced objectives in the loss function. +These problems were further magnified +by our attempts for distributed reconstruction of discontinuous PDE parameters – in the case of laboratory +experiments, from contaminated and non-smooth measurements. The following measures proved to be effective +in addressing some of these challenges: (i) training PINNs in a specific scaling framework where the dominant +wavenumber is the reference length scale, (ii) using the wavenumber-dependent Sobolev norms in quantifying +the data misfit, (iii) taking advantage of the inertia term in the governing PDEs to naturally balance the +objectives in the loss function, and (iv) denoising of the experimental data prior to training. +This paper is organized as follows. +Section 2 formulates the direct scattering problem related to the +synthetic and laboratory experiments, and provides an overview of the data inversion logic. Section 3 presents +the computational implementation of direct inversion and PINNs to reconstruct the distribution of L´ame +parameters in homogeneous and heterogeneous models from in-plane displacement fields. Section 4 provides +a detailed account of laboratory experiments, scaling, signal processing, and inversion of antiplane particle +velocity fields to recover the distribution of a physical parameter affiliated with flexural waves in thin plates. +The reconstruction results are then verified by a set of complementary experiments. +2. Concept +This section provides (i) a generic formalism for the direct scattering problem pertinent to the ensuing +(synthetic and experimental) full-field characterizations, and (ii) data inversion logic. +2.1. Forward scattering problem +Consider ultrasonic tests where the specimen Π ⊂ Rd, d = 2, 3, is subject to (boundary or internal) +excitation over the incident surface Sinc ⊂ Π and the induced (particle displacement or velocity) field u: Π × +[0 T] → RNΛ (NΛ ⩽ d) is captured over the observation surface Sobs ⊂ Π in a timeframe of length T. Here, Π +is an open set whose closure is denoted by Π, and the sensing configuration is such that Sinc ∩ Sobs = ∅. In +this setting, the spectrum of observed waveforms ˆu: Sobs × Ω → CNΛ is governed by +Λ[ˆu; ϑ](ξ, ω) = 0, +ˆu := F[u](ξ, ω), +ξ ∈ Sobs, ω ∈ Ω, +(1) +where Λ of size NΛ×1 designates a differential operator in frequency-space; F represents the temporal Fourier +transform; ϑ of dimension Nϑ×1 is the vector of relevant geometric and elastic parameters e.g., Lam´e constants +and mass density; ξ ∈ Rd is the position vector; and ω > 0 is the frequency of wave motion within the specified +bandwidth Ω. +2.2. Dimensional platform +All quantities in (1) are rendered dimensionless by identifying ρ◦, σ◦, and ℓ◦ as the respective reference +scales [33] for mass density, elastic modulus, and length whose explicit values will be later specified. +2.3. Data inversion +Given the full waveform data ˆu on Sobs × Ω, the goal is to identify the distribution of material properties +over Sobs. +For this purpose, two reconstruction paradigms based on neural networks are adopted in this +study, namely: (i) direct inversion, and (ii) physics-based neural networks. +Inspired by the elastography +2 + +method [18, 19], quantities of interest in (i) are identified by neural maps over Sobs × Ω that minimize a +regularized measure of Λ in (1). The neural networks in (ii), however, are by design predictive maps of the +waveform data (i.e., ˆu) obtained by minimizing the data mismatch subject to (1) as a soft or hard constraint. +In this setting, the unknown properties of Λ may be recovered as distributed parameters of the (data) network +during training via multitask optimization. +In what follows, a detailed description of the deployed cost +functions in (i) and (ii) is provided after a brief review of the affiliated networks. +2.3.1. Waveform and parameter networks +Laser-based ultrasonic experiments furnish a dense dataset on Sobs × Ω. Based on this, multilayer per- +ceptrons (MLPs) owing to their dense range [34] may be appropriate for approximating complex wavefields +and distributed PDE parameters. Moreover, this architecture has proven successful in numerous applications +within the PINN framework [15]. +In this study, MLPs serve as both data and property maps where the +input consists of discretized space and frequency coordinates (ξi, ωj), i = 1, 2, . . . , Nξ, j = 1, 2, . . . , Nω, as +well as distinct experimental parameters, e.g., the source location, distilled as one vector τk on domain T +with k = 1, 2, . . . , Nτ, while the output represents waveform data Dijk = [Rˆu, Iˆu](ξi, ωj; τk) ∈ RNΛ × RNΛ, +and/or the sought-for mechanical properties Pijn = [Rϑn, Iϑn](ξi, ωj) ∈ R × R, n = 1, 2, . . . , Nϑ. Note that +following [35], the real R and imaginary I parts of (1) and every complex-valued variable are separated such +that both direct and inverse problems are reformulated in terms of real-valued quantities. In this setting, each +fully-connected MLP layer with Nl neurons is associated with the forward map Υl : RNl−1 → RNl, +Υl(xl−1) = tanh(W lxl−1 + bl), +xl−1 ∈ RNl−1, +(2) +where W l ∈ RNl×Nl−1 and bl ∈ RNl respectively denote the lth layer’s weight and bias. Consecutive compo- +sition of Υl for l = 1, 2, . . . , Nm builds the network map wherein Nm designates the number of layers. +2.3.2. Direct inversion +Logically driven by the elastography method, the direct inversion approach depicted in Fig. 1 takes advan- +tage of the leading-order physical principles underpinning the test data to recover the distribution of relevant +physical quantities in space-frequency i.e., over the measurement domain. +The ML-based direct inversion +entails three steps: (a) spectral denoising and differentiation of (n-differentiable) waveforms ˆu over Sobs × Ω +according to the (n-th order) governing PDEs in (1), (b) building appropriate MLP maps to estimate the +profile of unknown physical parameters of the forward problem and regularization parameters of the inverse +solution, and (c) learning the MLPs through regularized fitting of data to the germane PDEs. +Note that synthetic datasets – generated via e.g., computer modeling or the method of manufactured +solutions, may directly lend themselves to the fitting process in (c) as they are typically smooth by virtue +Figure 1: Direct inversion: (a) FFT-based spatial differentiation of the full-field data as per operator Λ, (b) MLP-based approx- +imation of the unknown PDE and regularization parameters (ϑ, α) on their respective domains, and (c) training the MLPs via +minimizing the elastography loss Lε according to (3). +3 + +MLP +ultrasonic test data +u(S,w; T) +N(s,w) +spectral differentiation +3 +Vu(s,w; T) +Mα(s, w) +VVu($, w; T) +: +(a) +(b) +M +(9*,α*) := (Ng, ) +) = arg min L(u, *;α*) +(c) +9*,α*of numerical integration or analytical form of the postulated solution. Laboratory test data, however, are +generally contaminated by noise and uncertainties, and thus, spectral differentiation is critical to achieve the +smoothness requirements in (c). The four-tier signal processing of experimental data follows closely that of [36, +Section 3.1] which for completeness is summarized here: (1) a band-pass filter consistent with the frequency +spectrum of excitation is applied to the measured time signals at every receiver point, (2) the obtained +temporally smooth signals are then differentiated or integrated to obtain the pertinent field variables, (3) +spatial smoothing is implemented at every snapshot in time via application of median and moving average +filters followed by computing the Fourier representation of the processed waveforms in space, (4) the resulting +smooth fields may be differentiated (analytically in the Fourier space) as many times as needed based on the +underlying physical laws in preparation for the full-field reconstruction in step (c). It should be mentioned +that the experimental data may feature intrinsic discontinuities e.g., due to material heterogeneities or contact +interfaces. In this case, the spatial smoothing in (3) must be implemented in a piecewise manner after the +geometric reconstruction of discontinuity surfaces in Sobs which is quite straightforward thanks to the full-field +measurements, see e.g., [36, section 3.2]. +Next, the unknown PDE parameters ϑ are approximated by a fully connected MLP network ϑ⋆ := Nϑ(ξ, ω) +as per Section 2.3.1. The network is trained by minimizing the loss function +Lε(ˆu, ϑ⋆; α) = ∥Λ(ˆu; ϑ⋆)∥2 +L2(Sobs×Ω×T )NΛ + ∥α1ϑ ⊙ ϑ⋆∥2 +L2(Sobs×Ω)Nϑ , +(3) +where 1ϑ indicates an all-ones vector of dimension Nϑ × 1, and ⊙ designates the (element-wise) Hadamard +product. Here, the PDE residual based on (1) is penalized by the norm of unknown parameters. Observe +that the latter is a function of the weights and biases of the neural network which may help stabilize the MLP +estimates during optimization. Such Tikhonov-type functionals are quite common in waveform tomography +applications [37, 38, 39] owing to their well-established regularizing properties [21, 22]. Within this framework, +R ∋ α > 0 is the regularization parameter which may be determined by three means, namely: (i) the Morozov +discrepancy principle [40, 41], (ii) its formulation as a (constant or distributed) parameter of the ϑ⋆ network +which could then be learned during training, and (iii) its independent reconstruction as a separate MLP +network α⋆ := Nα(ξ, ω) illustrated in Fig. 1 (b) that is simultaneously trained along with ϑ⋆ by minimizing (3). +In this study, direct inversion is applied to synthetic and laboratory test data with both α = 0 and α > 0, +based on (ii) and (iii). It was consistently observed that the regularization parameter α plays a key role in +controlling the MLP estimates. This is particularly the case in situations where the field ˆu is strongly polarized +or near-zero in certain neighborhoods which brings about instability i.e., very large estimates for ϑ⋆ in these +areas. In light of this, all direct inversion results in this paper correspond to the case of α > 0 identified by +the MLP network α⋆. +2.3.3. Physics-informed neural networks +By deploying the knowledge of underlying physics, PINNs [14, 15] furnish efficient neural models of complex +PDE systems with predictive capabilities. +In this vein, a multitask learning process is devised according +to Fig. 2 where (a) the field variable ˆu – i.e., measured data on Sobs × Ω × T , is modeled by the MLP +map ˆu⋆ : = Nˆu(ξ, ω; τ) endowed with the auxiliary parameter γ(ξ, ω; τ) related to the loss function (4), +(b) the physical unknowns ϑ could be defined either as parameters of ˆu⋆ as in Fig. 2 (i), or as a separate +MLP ϑ⋆ : = Nϑ(ξ, ω) as shown in Fig. 2 (ii), and (c) learning the MLPs and affiliated parameters through +minimizing a measure of data misfit subject to the governing PDEs as soft/hard constraints wherein the spatial +derivatives of ˆu⋆ are computed via automatic differentiation [42]. It should be mentioned that in this study +all MLP networks are defined on (a subset of) Sobs × Ω × T where Sobs ∩ ∂Π = ∅. Hence, the initial and +boundary conditions – which could be specified as additional constraints in the loss function [15], are ignored. +In this setting, the PINNs loss takes the form +Lϖ(ˆu⋆, ϑ⋆|γ) = ∥ˆu − ˆu⋆∥2 +N(Sobs×Ω×T )NΛ + ∥γ1Λ ⊙ Λ(ˆu⋆; ϑ⋆)∥2 +L2(Sobs×Ω×T )NΛ, N = L2, �Hι, ι ⩽ n, (4) +where 1Λ is a NΛ× 1 vector of ones; n is the order of Λ, and �Hι denotes the adaptive Hι norm defined by +4 + +Figure 2: Two logics for the physics-informed neural networks (PINNs) with distributed parameters: (i) the test data ˆu(ξ, ω; τ) +are modeled by a MLP map, while the unknown physical parameters ϑ – on Sobs × Ω, and the loss function weight γ – on +Sobs × Ω × T , are defined as network parameters, and (ii) ˆu(ξ, ω; τ) and ϑ(ξ, ω) are identified by separate MLPs, while γ is a +parameter of Nˆu. The MLP(s) in (i) and (ii) are then trained by minimizing Lϖ of (4) in the space of data and PDE parameters. +∥ · ∥ � +Hι := +� +� +1⩽|e|⩽ ι +γe ∥∇e(·)∥2 +L2 + ∥·∥2 +L2, +∇e = +∂|e| +∂ξe1 +1 ∂ξe2 +2 ··· ∂ξed +d +, +|e| := +d +� +i=1 +ei. +(5) +Here, e:= {e1, e2, . . . ed} is a vector of integers ei ⩾ 0. Provided that ∀e, γe = 1, then �Hι is by definition +equal to Hι [43]. Note however that at high wavenumbers, Hι is dominated by the highest derivatives ∇eˆu⋆, +|e| = ι, which may complicate (or even lead to the failure of) the training process due to uncontrolled error +amplification by automatic differentiation particularly in earlier epochs. This issue may be addressed through +proper weighting of derivatives in (5). In light of the frequency-dependent Sobolev norms in [44, 37], one +potential strategy is to adopt the wavenumber-dependent weights as the following +γe = +� +1 +κe1 +1 κe2 +2 ··· κed +d +�2 +, +1 ⩽ |e| ⩽ ι, +wherein κi is a measure of wavenumber along ξi for i = 1, . . . , d. +In this setting, the weighted norms of +derivatives in (5) remain approximately within the same order as the L2 norm of data misfit. Another way to +automatically achieve the latter is to set the reference scale ℓ◦ such that κi ∼1. Note that the �Hι norms directly +inform the PINNs about the “expected” field derivatives – while preventing their uncontrolled magnification. +This may help stabilize the learning process as such derivatives are intrinsically involved in the PINNs loss via +Λ(ˆu⋆; ϑ⋆). It should be mentioned that when N = �Hι in (4), the “true” estimates for derivatives ∇eˆu may +be obtained via spectral differentiation as per Section 2.3.2. +The Lagrange multiplier [45, 46] γ(ξ, ω; τ) in (4) is critical for balancing the loss components during +training. Its optimal value, however, highly depends on (a) the nature of Λ [12], and (b) the distribution +of unknown parameters ϑ. +It should be mentioned that setting γ = 1 led to failure in almost all of the +synthetic and experimental implementations of PINNs in this study. Gauging of loss function weights has +been the subject of extensive recent studies [12, 25, 47, 26, 27, 28]. One systematic approach is the adaptive +SA-PINNs [12] where the multiplier γ(ξ, ω; τ) is a distributed parameter of ˆu⋆ whose value is updated in +each epoch according to a minimax weighting paradigm. Within this framework, the data (and parameter) +networks are trained by minimizing Lϖ with respect to ˆu⋆ and ϑ⋆, while maximizing the loss with respect to +γ as shown in Fig. 2. +Depending on the primary objective for PINNs, one may choose nonadaptive or adaptive weighting. More +specifically, if the purpose is high-fidelity forward modeling via neural networks where ϑ is known a-priori and +PINNs are intended to serve as predictive surrogate models of Λ, then ideas rooted in constrained optimization +e.g., minimax weighting is theoretically sound. However, if the inverse solution i.e., identification of ϑ(ξ, ω) +from “real-world” or laboratory test data is the main goal particularly in a situation where any assumption on +the smoothness of ϑ and/or applicability of Λ may be (at least locally) violated e.g., due to unknown material +5 + +MLP +network parameters +(i) +(ii) +9*($, w) +9*:= +(S,w; T) +N(s, w) +? +↑ +automatic +E +3 +differentiation +α*:= +V*(S,w; T) +α*:= +T +3 +Na(S, w; T) +VVu*($, w; T) +Na(S, w; T) +T +: +MLP +↑ +(S,w; T) +u* += arg min max Lw(u*, *I) +*,9*heterogeneities or interfacial discontinuities, then trying to enforce Λ everywhere on Sobs × Ω × T (via point- +wise adaptive weighting) may lead to instability and failure of data inversion. In such cases, nonadaptive +weighting may be more appropriate. In light of this, in what follows, γ is a non-adaptive weight specified by +taking advantage of the PDE structure to naturally balance the loss objectives. +3. Synthetic implementation +Full-field characterization via the direct inversion and physics-informed neural networks are examined +through a set of numerical experiments. The waveform data in this section are generated via a FreeFem++ [48] +code developed as part of [49]. +3.1. Problem statement +Plane-strain wave motion in two linear, elastic, piecewise homogeneous, and isotropic samples is modeled +according to Fig. 3 (a). On denoting the frequency of excitation by ω, let ℓr = 2π +ω +� +µr/ρr, ρr = 1, and µr = 1 +be the reference scales for length, mass density, and stress, respectively. In this framework, both specimens +are of size 16×16 and uniform density ρ = 1. The first sample Π1 ⊂ R2 is characterized by the constant Lam´e +parameters µ◦ = 1 and λ◦ = 0.47, while the second sample Π2 ⊂ R2 is comprised of four perfectly bonded +homogenous components Π2j of µj = j and λj = 2j/3, j = {1, 2, 3, 4} such that Π2 = �4 +j=1 Π2j. Accordingly, +the shear and compressional wave speeds read c◦ +s = 1, c◦ +p = 1.57 in Π1, and cj +s = √j, cj +p = 1.63√j in Π2j. +Every numerical experiment entails an in-plane harmonic excitation at ω = 3.91 via a point source on Sinc +(the perimeter of a 14 × 14 square centered at the origin). The resulting displacement field uα = (uα +1 , uα +2 ), +α = 1, 2, is then computed in Πα over Sobs (a concentric square of dimension 8 ×8) such that +µα∆uα(ξ) + (λα + µα)∇∇ · uα(ξ) + ρω2uα(ξ) = δ(ξ − x)d, +ξ ∈ Πα, x ∈ Sinc, +� +λα∇ · uα(ξ)I2 + 2µα∇symuα(ξ) +� +· n(ξ) = 0, +ξ ∈ ∂Πα, +(6) +where x and d respectively indicate the source location and polarization vector; n is the unit outward normal +to the specimen’s exterior, and +� +µα = µ◦, λα = λ◦, +α = 1 +µα = µj, λα = λj, +α = 2 ∧ ξ ∈ Π2j∈{1,2,3,4} +. +Figure 3: synthetic experiments simulating plane-strain wave motion in homogeneous (top-left) and heterogeneous (bottom-left) +specimens: (a) testing configuration where the model is harmonically excited at frequency ω by a point source on Sinc, and the +induced displacement field u is computed over Sobs along ξ1 and ξ2 as shown in (b) and (c), respectively. +6 + +TT1 +μo,\。 +u1 +μ3,^3 +μ4，^4 +TT2 +W2 +(a) +(b)When α = 2, the first of (6) should be understood as a shorthand for the set of four governing equations +over Π2j, j = {1, 2, 3, 4}, supplemented by the continuity conditions for displacement and traction across +∂Π2j\∂Π2 as applicable. +In this setting, the generic form (1) may be identified as the following +Λ = Λα := µα∆ + (λα + µα)∇∇ · + ρω2I2, +α = 1, 2, +ˆu = uα(ξ, ω; τ), +ϑ = [µα, λα](ξ, ω), +ξ ∈ Sobs, ω ∈ Ω, τ ∈ T , +(7) +wherein I2 is the second-order identity tensor; τ = (x, d) ∈ Sinc × B1 = T with B1 denoting the unit circle +of polarization directions. Note that ρ is treated here as a known parameter. +In the numerical experiments, Sinc (resp. Sobs) is discretized by a uniform grid of 32 (resp. 50×50) points, +while Ω and B1 are respectively sampled at ω = 3.91 and d = (1, 0). +All inversions in this study are implemented within the PyTorch framework [50]. +3.2. Direct inversion +The three-tier logic of Section 2.3.2 is employed to reconstruct the distribution of µα and λα, α = 1, 2, +over Sobs, entailing: (a) spectral differentiation of the displacement field uα in order to compute ∆uα and +∇∇ · uα as per (6), (b) construction of three positive-definite MLP networks µ⋆, λ⋆, and α⋆; each of which +is comprised of one hidden layer of 64 neurons, and (c) training the MLPs by minimizing Lε as in (3) +and (7) by way of the ADAM algorithm [51]. To avoid near-boundary errors affiliated with the one-sided FFT +differentiation in ∆uα and ∇∇·uα, a concentric 40×40 subset of collocation points sampling Sobs is deployed +for training purposes. It should also be mentioned that in the heterogeneous case, i.e., α = 2, the discontinuity +of derivatives across ∂Π2j∈{1,2,3,4} calls for piecewise spectral differentiation. According to Section 2.3.1, the +input to P⋆ = NP(ξ, ω), P = µ, λ, and α⋆ = Nα(ξ, ω) is of size NξNτ × Nω = 1600Ns × 1 where Ns ⩽ 32 +is the number of simulations i.e., source locations used to generate distinct waveforms for training. In this +setting, since the physical quantities of interest are independent of τ, the real-valued output of MLPs is of +dimension 1600 × 1 furnishing a local estimate of the L´ame and regularization parameters at the specified +sampling points on Sobs. Each epoch makes use of the full dataset and the learning rate is 0.005. +In this work, the reconstruction error is measured in terms of the normal misfit +Ξ(q⋆) = ∥q⋆ − q ∥L2 +∥q ∥L∞ +, +(8) +where q⋆ is an MLP estimate for a quantity with the “true” value q. +Let Sinc be sampled at one point i.e., Ns = 1 so that a single forward simulation in Πα, α = 1, 2, generates +the training dataset. The resulting reconstructions are shown in Figs. 4 and 5. It is evident from both figures +that the single-source reconstruction fails at the loci of near-zero displacement which may explain the relatively +high values of the recovered regularization parameter α⋆. Table 1 details the true values as well as mean and +standard deviation of the reconstructed L´ame distributions ϑ⋆ = (µ⋆, λ⋆) in Π1 (resp. Π2j for j = 1, 2, 3, 4) +according to Fig. 4 (resp. Fig. 5). +This problem may be addressed by enriching the training dataset e.g., via increasing Ns. Figs. 6 and 7 +illustrate the reconstruction results when Sinc is sampled at Ns = 5 source points. The mean and standard +deviation of the reconstructed distributions are provided in Table 2. It is worth noting that in this case the +identified regularization parameter α⋆ assumes much smaller values – compared to that of Figs. 4 and 5. This +is closer to the scale of computational errors in the forward simulations. +To examine the impact of noise on the reconstruction, the multisource dataset used to generate Figs. 6 +and 7 are perturbed with 5% white noise. The subsequent direct inversions from noisy data are displayed in +Figs. 8 and 9, and the associated statistics are presented in Table 3. Note that spectral differentiation as the +first step in direct inversion plays a critical role in denoising the waveforms, and subsequently regularizing the +reconstruction process. This may substantiate the low magnitude of MLP-recovered α⋆ in the case of noisy +data in Figs. 8 and 9. The presence of noise, nonetheless, affects the magnitude and thus composition of terms +in the Fourier representation of the processed displacement fields in space which is used for differentiation. +This may in turn lead to the emergence of fluctuations in the reconstructed fields. +7 + +Figure 4: Direct inversion of the L´ame parameters in Π1 using noiseless data from a single source: (a) MLP-predicted distributions +µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µ◦ = 1 and λ◦ = 0.47, (c) MLP-recovered distribution of +the regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne in the log = log10 scale. +Figure 5: Direct inversion of the L´ame parameters in Π2 using noiseless data from a single source: (a) MLP-predicted distributions +µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µj = j and λj = 2j/3, j = {1, 2, 3, 4}, (c) MLP-recovered +regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne. +Table 1: Mean ⟨·⟩D and standard deviation σ(·|D) of the reconstructed L´ame distributions in D = Π1, Π2j=1,2,3,4. Here, +the direct inversion is applied to noiseless data from a single source as shown in Figs. 4 and 5. +D +Π1 +Π21 +Π22 +Π23 +Π24 +µ +µ◦ = 1 +µ1 = 1 +µ2 = 2 +µ3 = 3 +µ4 = 4 +⟨µ⋆⟩D +0.998 +0.991 +1.983 +2.825 +3.835 +σ(µ⋆|D) +0.024 +0.083 +0.182 +0.441 +0.325 +λ +λ◦ = 0.47 +λ1 = 0.67 +λ2 = 1.33 +λ3 = 2 +λ4 = 2.66 +⟨λ⋆⟩D +0.376 +0.615 +0.850 +1.746 +1.412 +σ(λ⋆|D) +0.128 +0.161 +0.399 +0.486 +0.864 +8 + +(a) +(b) +1.2 +0.2 +1.1 +0.15 +0.1 +×10-2 +(c) +(d) +1.4 +log(Le) +0.9 +0.05 +1 +1 +0 +0.8 +0 +0.7 +0.2 +0.6 +-1 +（? +0.6 +0.15 +-2 +0.2 +0.5 +0.1 +0 +0.5 +×104 +1 +Ne +0.4 +0.05 +0.3 +0(a) +(b) +三(μ*) +0.8 +3 +(c) +×10-2 +(d) +0.4 +2 +log(Le) +1 +2 +0 +0.8 +0 +0.5 +1 ×104 +0.4 +NeFigure 6: Direct inversion of the L´ame parameters in Π1 using noiseless data from five distinct simulations: (a) MLP-predicted +distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µ◦ = 1 and λ◦ = 0.47, (c) MLP-recovered +regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne. +Figure 7: Direct inversion of the L´ame parameters in Π2 using five noiseless datasets: (a) MLP-predicted distributions µ⋆ and +λ⋆, (b) reconstruction error (8) with respect to the true values µj = j and λj = 2j/3, j = {1, 2, 3, 4}, (c) MLP-recovered +regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne. +Table 2: Mean and standard deviation of the reconstructed L´ame distributions from five distinct noiseless datasets +according to Figs. 6 and 7. +D +Π1 +Π21 +Π22 +Π23 +Π24 +µ +1 +1 +2 +3 +4 +⟨µ⋆⟩D +1.000 +0.999 +2.003 +2.999 +3.999 +σ(µ⋆|D) +0.001 +0.012 +0.011 +0.012 +0.016 +λ +0.47 +0.67 +1.33 +2 +2.66 +⟨λ⋆⟩D +0.464 +0.660 +1.302 +1.997 +2.635 +σ(λ⋆|D) +0.012 +0.039 +0.071 +0.048 +0.068 +9 + +(a) +(b) +2 +u* +1.02 +(×)m +1 +(c) +×10-3 +(d) +1 +log(Le) +5 +1 +0.98 +×10-2 +0 +3 +-1 +7.5 +^* +0.5 +三(\*) +-2 +5 +0.45 +0 +0.5 +1 ×104 +Ne +2.5 +0.4 +/×10-2(a) +(b) +4 +1.75 +3 +(c) +×10-3 +(d) +0.75 +1.4 +2 +log(Le) +×10-2 +0 +0.8 +0.2 -2 +2 +0.5 +×104 +0 +0.4 +Ne +×10-1Figure 8: Direct inversion of the L´ame parameters in Π1 using five datasets perturbed with 5% white noise: (a) MLP-predicted +distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µ◦ = 1 and λ◦ = 0.47, (c) MLP-recovered +regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne. +Figure 9: Direct inversion of the L´ame parameters in Π2 using five datasets perturbed with 5% white noise: (a) MLP-predicted +distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µj = j and λj = 2j/3, j = {1, 2, 3, 4}, (c) +MLP-recovered regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne. +Table 3: Mean and standard deviation of the reconstructed L´ame distributions from noisy data according to Figs. 8 +and 9. +D +Π1 +Π21 +Π22 +Π23 +Π24 +µ +1 +1 +2 +3 +4 +⟨µ⋆⟩D +1.001 +1.002 +2.005 +2.996 +3.996 +σ(µ⋆|D) +0.005 +0.016 +0.035 +0.054 +0.088 +λ +0.47 +0.67 +1.33 +2 +2.66 +⟨λ⋆⟩D +0.462 +0.650 +1.263 +2.006 +2.654 +σ(λ⋆|D) +0.042 +0.051 +0.225 +0.182 +0.300 +10 + +(a) +(b) +1.05 +(r)= +L￥ +2 +×10-3 +(d) +(c) +1 +4 +log(Le) +3 +0.95 +×10-2 +0 +2 +0.5 +三()*) +0.35 +1 +-2 +0.25 +0.45 +0 +0.5 +1 ×104 +0.15 +Ne +0.05 +0.4(a) +(b) +4 +E(μ*) +0.8 +3 +(c) +×10-3 +(d) +0.4 +1.4 +2 +log(Le) +1 +×10-1 +0 +0.5 +0.6 +-1 +0.2 +2 +0.3 +0 +0.5 +Ne +0.13.3. Physics-informed neural networks +The learning process of Section 2.3.3 is performed as follows: (a) the MLP network uα⋆ = Nuα(ξ, ω, x|γ, ϑ⋆) +endowed with the positive-definite parameters γ and ϑ⋆ = (µ⋆, λ⋆) is constructed such that the input x labels +the source location and the auxiliary weight γ is a nonadaptive scaler, (b) µ⋆ and λ⋆ may be specified as scaler +or distributed parameters of the network according to Fig. 2 (i), and (c) uα⋆ is trained by minimizing Lϖ +in (4) via the ADAM optimizer using the synthetic waveforms of Section 3.1. Reconstructions are performed +on the same set of collocation points sampling Sobs×Ω×T as in Section 3.2. Accordingly, the input to uα⋆ is +of size Nξ×Nω×Nτ = 1600×1×Ns, while its output is of dimension (1600×1×Ns)2 modeling the displacement +field along ξ1 and ξ2 in the sampling region. Similar to Section 3.2, each epoch makes use of the full dataset for +training and the learning rate is 0.005. The PyTorch implementation of PINNs in this section is accomplished +by building upon the available codes on the Github repository [52]. +The MLP network u1⋆ = u1⋆(ξ, ω, x|γ, ϑ⋆) with three hidden layers of respectively 20, 40, and 20 neurons +is employed to map the displacement field u1 (in Π1) associated with a single point source of frequency +ω = 3.91 at x = x1 ∈ Sinc. +The L´ame constants are defined as the unknown scaler parameters of the +network i.e., ϑ⋆ = {µ⋆, λ⋆}, and the Lagrange multiplier γ is specified per the following argument. Within +the dimensional framework of this section and with reference to (7), observe that on setting γ = +1 +ρω2 (i.e., +γ = 0.065), both (the PDE residue and data misfit) components of the loss function Lϖ in 4 emerge as some +form of balance in terms of the displacement field. This may naturally facilitate maintaining of the same scale +for the loss terms during training, and thus, simplifying the learning process by dispensing with the need to +tune an additional parameter γ. Keep in mind that the input to u1⋆ is of size 1600×1×1, while its output is +of dimension (1600×1×1)2. In this setting, the training objective is two-fold: (a) construction of a surrogate +map for u1, and (b) identification of µ⋆ and λ⋆. +Fig. 10 showcases (i) the accuracy of PINN estimates based on noiseless data in terms of the vertical +component of displacement field u1 +2 in Π1, and (ii) the performance of automatic differentiation [42] in capturing +the field derivatives in terms of components that appear in the governing PDE 7 i.e., u1 +2,ij = ∂2u1 +2/(∂ξi∂ξj), +i, j = 1, 2. +The comparative analysis in (ii) is against the spectral derivates of FEM fields according to +Section 2.3.2. It is worth noting that similar to Fourier-based differentiation, the most pronounced errors +in automatic differentiation occur in the near-boundary region i.e., the support of one-sided derivatives. It +is observed that the magnitude of such discrepancies may be reduced remarkably by increasing the number +of epochs. Nonetheless, the loci of notable errors remain at the vicinity of specimen’s external boundary or +internal discontinuities such as cracks or material interfaces. Fig. 10 is complemented with the reconstruction +results of Fig. 11 indicating (µ⋆, λ⋆) = (1.000, 0.486) for the homogenous specimen Π1 with the true L´ame +constants (µ◦, λ◦) = (1, 0.47). The impact of noise on training is examined by perturbing the noiseless data +related to Fig. 10 with 5% white noise, which led to (µ⋆, λ⋆) = (0.999, 0.510) as shown in Fig. 12. +Next, the PINN u2⋆ = u2⋆(ξ, ω, x|ϑ⋆) with three hidden layers of respectively 120, 120, and 80 neurons +is created to reconstruct (i) displacement field u2 in the heterogeneous specimen Π2, and (ii) distribution of +the L´ame parameters over the observation surface. In this vein, synthetic waveform data associated with five +point sources {xi} ∈ Sinc, i = 1, 2, . . . , 5 at ω = 3.91 is used for training. Here, ϑ⋆ is the network’s unknown +distributed parameter, of dimension (40×40)2, and the nonadaptive scaler weight γ = 0.065 in light of the +sample’s uniform density ρ = 1. In this setting, the input to u2⋆ is of size 1600×1×5, while its output is +of dimension (1600×1×5)2. Fig. 13 provides a comparative analysis between the FEM and PINN maps of +horizontal displacement u1 +2 in Π2 and its spatial derivatives computed by spectral and automatic differentiation +respectively. +Table 4: Mean and standard deviation of the PINN-reconstructed L´ame distributions from five distinct noiseless datasets +according to Fig. 14. +D +Π21 +Π22 +Π23 +Π24 +⟨µ⋆⟩D +0.975 +1.973 +2.941 +. 3.918 +σ(µ⋆|D) +0.054 +0.123 +0.135 +0.226 +⟨λ⋆⟩D +0.686 +1.250 +2.045 +2.065 +σ(λ⋆|D) +0.247 +0.400 +0.520 +0.857 +11 + +Figure 10: PINN vs. FEM maps of vertical displacement and its derivatives in Π1: (a) MLP estimates, from noiseless data, for +{u1 +2 +⋆, u1⋆ +2,11, u1⋆ +2,22, u1⋆ +2,12} wherein the derivatives u1⋆ +2,ij, i, j = 1, 2, are obtained by automatic differentiation, (b) FEM displacement +solution and its spectral derivatives for {u1 +2, u1 +2,11, u1 +2,22, u1 +2,12}, and (c) normal misfit 8 between (a) and (b). +Figure 11: PINN reconstruction of L´ame constants in the homogeneous plate Π1 from noiseless data: (a) µ⋆ vs. number of epochs +Ne, (b) λ⋆ vs. Ne, and (c) total loss Lϖ and its components (the PDE residue and data misfit) vs. Ne in log scale. +Figure 12: PINN reconstruction of L´ame constants in Π1 from noisy data: (a) µ⋆ vs. number of epochs Ne, (b) λ⋆ vs. Ne, and +(c) total loss Lϖ and its components (the PDE residue and data misfit) vs. Ne in log scale. +12 + +2 +0.2 +? +U2,22 +1 +1 +0.5 +0.1 +(a) +0 +0 +0 +0 +-0.1 +-0.5 +-1 +-0.2 +2 +0.2 +I +u2,11 +u2,22 +2,12 +1 +1 +0.5 +(b) +0 +0 +0 +0 +-0.5 +-1 +-1 +-0.2 +三(u2 +7 +E(u2,11) +三(u2,22) +0.3 +三(u2,12) +0.2 +?L +1.2 +5 +0.2 +(c) +0.8 +0.1 +3 +0.1 +0.4 +1 +×10-2 +×10-1(a) +(b) +(c) +0.8 +\* +- PDE loss +1.2 +0 +.- data loss + total loss +0.8 +0.4 +-2 +0.4 +-4 +Ne +Ne +0 +0 +×105 +×105 +×105 +0 +0.4 +0.8 +1.2 +1.6 +2. +0 +0.4 +0.8 +1.2 +1.6 +2 +0 +0.4 +0.8 +1.2 +1.6 +2(a) +(b) +(c) +\* +PDE loss +L* +1.2 +0.6 + data loss +0 + total loss +0.8 +0.4 +-2 +0.4 +0.2 +Ne +Ne +UN +0 +0 +×105 +×105 +×105 +0 +0.4 +0.8 +1.2 +1.6 +2. +0 +0.4 +0.8 +1.2 +1.6 +2 +0 +0.4 +0.8 +1.2 +1.6 +2Figure 13: PINN vs. FEM maps of horizontal displacement and its derivatives in Π2: (a) PINN estimates, from noiseless data, for +{u2 +1 +⋆, u2⋆ +1,11, u2⋆ +1,22, u2⋆ +1,12} wherein the derivatives u2⋆ +1,ij, i, j = 1, 2, are obtained by automatic differentiation, (b) FEM displacement +solution and its spectral derivatives for {u2 +1, u2 +1,11, u2 +1,22, u2 +1,12}, and (c) normal misfit 8 between (a) and (b). +Figure 14: PINN reconstruction of L´ame parameters in Π2 using five noiseless datasets: (a) PINN-predicted distributions µ⋆ and +λ⋆, (b) reconstruction error (8) with respect to the true values µj = j and λj = 2j/3, j = {1, 2, 3, 4}, (c) total loss Lϖ and its +components (the PDE residue and data misfit) vs. Ne in log scale. +The PINN-reconstructed distribution of PDE parameters is illustrated in Fig. 14 whose statistics is +detailed in Table 4. +It is worth mentioning that the learning process is repeated for a suit of weights +γ = {0.01, 0.025, 0.1, 0.25, 0.5, 1.5, 2, 5, 10, 15}. In all cases, the results are either quite similar or worse than +that of Figs. 13 and 14. +13 + +2 * +0.4 +2* +2* +2* +ui +ui,11 +ui,22 +ui,12 +3 +3 +2 +0 +1 +1 +(a) +0 +-0.4 +-1 +-1 +2 +-3 +-3 +-0.8 +0.4 +ui,11 +2 +ui,12 +3 +3 +2 +1 +1 +(b) +0 +-0.4 +-1 +-1 +-2 +-3 +-3 +-0.8 +5 +三(ui +三(ui,11) +2 +三(ui,22) +2* +E(ui,12) +2* +2 +2 +3 +(c) +1 +L +1 +×10-3 +×10-2 +×10-2 +×10-2(a) +(b) +4 +三(μ*) +0.4 +3 +(c) +0.2 +2 +PDE loss +0 +data loss +total loss +0 +-2 +三(\*) +-4 +0.4 +2 +-6 +×106 +0 +0.4 +0.8 +1.2 +1.6 +2 +0.2 +Ne +14. Laboratory implementation +This section examines the performance of direct inversion and PINNs for full-field ultrasonic character- +ization in a laboratory setting. In what follows, experimental data are processed prior to inversion as per +Section 2.3.2 which summarizes the detailed procedure in [36]. To verify the inversion results, quantities of +interest are also reconstructed through dispersion analysis, separately, from a set of auxiliary experiments. +4.1. Test set-up +Experiments are performed on two (homogeneous and heterogeneous) specimens: Π +exp +1 +which is a 27 cm +×27 cm×1.5 mm sheet of T6 6061 aluminum, and Π +exp +2 +composed of (a) 5 cm×27 cm×1.5 mm sheet of Grade +2 titanium, (b) 2.5 cm×27 cm×1.5 mm sheet of 4130 steel, and (c) 5 cm×27 cm×1.5 mm sheet of 260-H02 +brass, connected via metal epoxy. For future reference, the density ρµ, Young’s modulus Eµ, and Poisson’s +ratio νµ for µ = {Al, Ti, St, Br} are listed in Table 5 as per the manufacturer. +Ultrasonic experiments on both samples are performed in a similar setting in terms of the sensing config- +uration and illuminating wavelet. In both cases, the specimen is excited by an antiplane shear wave from a +designated source location Sinc, shown in Fig. 15, by a 0.5 MHz p-wave piezoceramic transducer (V101RB by +Olympus Inc.). The incident signal is a five-cycle burst of the form +H(fct) H(5−fct) sin +� +0.2πfct +� +sin +� +2πfct +� +, +(9) +where H denotes the Heaviside step function, and the center frequency fcis set at 165 kHz (resp. {80, 300} kHz) +in Π +exp +1 +(resp. Π +exp +2 ). The induced wave motion is measured in terms of the particle velocity vβ, β = 1, 2, on the +scan grids Gβ sampling Sobs where Sobs ∩Sinc = Sobs ∩∂Π +exp +β = ∅. A laser Doppler vibrometer (LDV) which is +mounted on a 2D robotic translation frame (for scanning) is deployed for measurements. The VibroFlex Xtra +VFX-I-120 LDV system by Polytec Inc. is capable of capturing particle velocity within the frequency range +∼ DC − 24 MHz along the laser beam which in this study is normal to the specimen’s surface. +The scanning grid G1 ⊂ Π +exp +1 +is identified by a 2 cm×2 cm square sampled by 100×100 uniformly spaced +measurement points. This amounts to a spatial resolution of 0.2 mm in both spatial directions. In parallel, +G2 ⊂ Π +exp +2 +is a 2.5 cm×7.5 cm rectangle positioned according to Fig. 15 (b) and sampled by a uniform grid of +180×60 scan points associated with the spatial resolution of 0.42 mm. At every scan point, the data acquisition +is conducted for a time period of 400 µs at the sampling rate of 250 MHz. To minimize the impact of optical +and mechanical noise in the system, the measurements are averaged over an ensemble of 80 realizations at +each scan point. Bear in mind that both the direct inversion and PINNs deploy the spectra of normalized +velocity fields vobs for data inversion. Such distributions of out-of-plane particle velocity at 165 kHz (resp. 80 +kHz) in Π +exp +1 +(resp. Π +exp +2 ) is displayed in Fig. 15. +It should be mentioned that in the above experiments, the magnitude of measured signals in terms of +displacement is of O(nm) so that it may be appropriate to assume a linear regime of propagation. The nature +of antiplane wave motion is dispersive nonetheless. Therefore, to determine the relevant length scales in each +component, the associated dispersion curves are obtained as in Fig. 19 via a set of complementary experiments +described in Section 4.4.1. Accordingly, for excitations of center frequency {fc1, fc2, fc3} = {165, 80, 300} kHz, +the affiliated phase velocity cµ and wavelength λµ for µ = {Al, Ti, St, Br} is identified in Table 6. +Figure 15: Test set-ups for ultrasonic full-field characterization: (a) an Al plate Π +exp +1 +is subject to antiplane shear waves at 165 +kHz by a piezoelectric transducer; the out-of-plane particle velocity field is then captured by a laser Doppler vibrometer scanning +on a robot over the observation surface, and (b) a Ti-St-Br plate Π +exp +2 +undergoes a similar test at 80 kHz and 300 kHz. +14 + +exp +2 +exp +1..239 +Ti +St +Br +(a) +(b)4.2. Dimensional framework +On recalling Section 2.2, let ℓr : = λAl = 0.01 m, µr : = EAl = 68.9 GPA, and ρr : = ρAl = 2700 kg/m3 be +the reference scales for length, stress, and mass density, respectively. In this setting, the following maps take +the physical quantities to their dimensionless values +(ρµ, Eµ, νµ) → (ρµ, Eµ, νµ) := +� 1 +ρr +ρµ, 1 +µr +Eµ, νµ +� +, +µ = {Al, Ti, St, Br}, +(fcι, λµ, cµ) → (fcι, λµ, cµ) := +� +ℓr +� ρr +µr +fcι, 1 +ℓr +λµ, +� ρr +µr +cµ +� +, +ι = 1, 2, 3, +(h, f, vβ) → (h, f, vβ) := +� 1 +ℓr +h, ℓr +� ρr +µr +f, +� ρr +µr +vβ� +, +β = 1, 2, +(10) +where h = 1.5 mm and f respectively indicate the specimen’s thickness and cyclic frequency of wave motion. +Table 5 (resp. Table 6) details the normal values for the first (resp. second) of (10). The normal thickness and +center frequencies are as follows, +{fc1, fc2, fc3} = {0.33, 0.16, 0.59}, +h = 0.15. +(11) +Table 5: Properties of the aluminum, titanium, steel and brass sheets as per the manufacturer. Here, χµ := Eµ/ρµ. +physical +µ +Al +Ti +St +Br +Eµ [GPA] +68.9 +105 +199.95 +110 +quantity +ρµ [kg/m3] +2700 +4510 +7850 +8530 +νµ +0.33 +0.34 +0.29 +0.31 +normal +Eµ +1 +1.52 +2.90 +1.60 +value +ρµ +1 +1.67 +2.91 +3.16 +χµ +1 +0.91 +1 +0.51 +Table 6: Phase velocity cµ and wavelength λµ in µ = {Al, Ti, St, Br} at {fc1, fc2, fc3} = {165, 80, 300} kHz as per Fig. 19, +and their normalized counterparts according to (10). +physical quantity +µ +Al +Ti +St +Br +λµ(fc1) [cm] +1 +− +− +− +cµ(fc1) [m/s] +1610.4 +− +− +− +λµ(fc2) [cm] +− +1.4 +1.4 +1.17 +cµ(fc2) [m/s] +− +1140 +1126 +936 +λµ(fc3) [cm] +− +0.65 +0.64 +0.5 +cµ(fc3) [m/s] +− +1960.8 +1929 +1501.6 +normal value +µ +Al +Ti +St +Br +λµ(fc1) +1 +− +− +− +cµ(fc1) +0.32 +− +− +− +λµ(fc2) +− +1.4 +1.4 +1.17 +cµ(fc2) +− +0.23 +0.22 +0.19 +λµ(fc3) +− +0.65 +0.64 +0.5 +cµ(fc3) +− +0.39 +0.38 +0.3 +4.3. Governing equation +In light of (11) and Table 6, observe that in all tests the wavelength-to-thickness ratio λµ +h ∈ [3.33 9.33], +µ = {Al, Ti, St, Br}. Therefore, one may invoke the equation governing flexural waves in thin plates [53] to +approximate the physics of measured data. In this framework, (1) may be recast as +Λ = Λβ := +χβh3 +12(1 − ν2 +β)∇4 − h(2πf)2, +χβ := Eβ +ρβ +, β = 1, 2, +ˆu = vβ(ξ, f; τ), +ϑ = χβ(ξ, f), +ξ ∈ Sobs, τ ∈ Sinc, f ∈ [0.8 1.2]fcι, ι = 1, 2, 3, +(12) +where ρβ, Eβ, νβ respectively denote the normal density, Young’s modulus, and Poisson’s ratio in Π +exp +β , β = +1, 2, and τ indicates the source location. Note that νβ ∼ 0.32 according to Table 5 and Λ, related to 1 − ν2 +β, +15 + +shows little sensitivity to small variations in the Poisson’s ratio. Thus, in what follows, νβ is treated as a +known parameter. Provided vβ(ξ, f; τ), the objective is to reconstruct χβ(ξ, f). +4.4. Direct inversion +Following the reconstruction procedure of Section 3.2, the distribution of χβ in Gβ, β = 1, 2, is obtained +at specific frequencies. In this vein, the positive-definite MLP networks χ⋆ +β = Nχβ(ξ, ω) and α⋆ = Nα(ξ, ω) +comprised of three hidden layers of respectively 20, 40, and 20 neurons are constructed according to Fig. 1. +In all MLP trainings of this section, each epoch makes use of the full dataset and the learning rate is 0.005. +When β = 1, the inversion is conducted at f1 = 0.336. Sinc is sampled at one point i.e., the piezoelectric +transducer remains fixed during the test on Al plate, and thus, Nτ = 1, while a concentric 60×60 subset +of collocation points sampling Sobs is deployed for training. In this setting, the input to χ⋆ +1 and α⋆ is of +size NξNτ × Nω = 3600 × 1, and their real-valued outputs are of the same size. The results are shown in +Fig. 16. When β = 2, the direct inversion is conducted at f2 = 0.17 and f3 = 0.61. For the low-frequency +reconstruction, Sinc is sampled at one point, while a 40×120 subset of scan points in G2 is used for training +so that the input/output size for χ⋆ +2 and α⋆ is 4600×1. The recovered fields and associated normal error are +provided in Fig. 17. Table 7 enlists the true values as well as mean and standard deviation of the reconstructed +distributions χ⋆ +β in Π +exp +β , β = 1, 2, according to Figs. 16 and 17. For the high-frequency reconstruction, when +β = 2, Sinc is sampled at three points i.e., experiments are performed for three distinct positions of the +piezoelectric transducer, while the same subset of scan points is used for training. In this case, the input to +χ⋆ +2 and α⋆ is 13800×1, while their output is of dimension 4600×1. The high-frequency reconstruction results +are illustrated in Fig. 18, and the affiliated means and standard deviations are provided in Table 8. It should +be mentioned that the computed normal errors in Figs. 16, 17, and 18 are with respect to the verified values +of Section 4.4.1. Note that the recovered α⋆s from laboratory test data are much smoother than the ones +reconstructed from synthetic data in Section 3.2. This could be attributed to the scaler nature of (12) with a +single unknown parameter – as opposed to the vector equations governing the in-plane wave motion with two +unknown parameters. More specifically, here, α⋆ controls the weights and biases of a single network χ⋆ +β, while +in Section 3.2, α⋆ simultaneously controls the parameters of two separate networks µ⋆ and λ⋆. A comparative +analysis of Figs. 17 and 18 reveals that (a) enriching the waveform data by increasing the number of sources +remarkably decrease the reconstruction error, (b) the regularization parameter α in (3) is truly distributed +in nature as the magnitude of the recovered α⋆ in brass is ten times greater than that of titanium and steel +which is due to the difference in the level of noise in measurements related to distinct material surfaces, and +(c) the recovered field χ⋆ +2 – which according to (12) is a material property E2/ρ2, demonstrates a significant +dependence to the reconstruction frequency. The latter calls for proper verification of the results which is the +subject of Section 4.4.1. +4.4.1. Verification +To shine some light on the nature discrepancies between the low- and high- frequency reconstructions in +Figure 16: Direct inversion of the PDE parameter χ1 in Π +exp +1 +using test data from a single source at frequency f1 = 0.336: (a) MLP- +predicted distribution χ1(ξ, f1) in ξ ∈ G1, (b) reconstruction error (8) with respect to the true value χ1 = χAl = 1, (c) MLP- +recovered distribution of the regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne in log scale. +16 + +(a) +(b) +(c) +×10-3 +(d) +三(x1) +X1 +α* +6 +1.06 +0.06 +log(Le) +4 +1.04 +4 +0.04 +-5 +1.02 +0.02 +2 +-6 +Ne +ELLLEFE +0 +2 +×103 +4Figure 17: Direct inversion of the PDE parameter χ2 in Π +exp +2 +using test data from a single source at frequency f2 = 0.17: (a) MLP- +predicted distribution χ2(ξ, f2) in ξ ∈ G2, (b) reconstruction error (8) with respect to the true value χ2 ∈ {χTi, χSt, χBr} = +{0.91, 1, 0.51} as per Table 5, (c) MLP-recovered distribution of the regularization parameter α⋆, and (d) loss function Lε vs. the +number of epochs Ne in log scale. +Table 7: Mean and standard deviation of the reconstructed distributions in Figs. 16 and 17 via the direct inversion of +single-source test data. +β +1 +2Ti +2St +2Br +χβ +1 +0.91 +1 +0.51 +⟨χ⋆ +β⟩Πexp +β +1.041 +0.872 +0.978 +0.443 +σ(χ⋆ +β|Πexp +β ) +0.017 +0.044 +0.060 +0.052 +Figure 18: Direct inversion of the PDE parameter χ2 in Π +exp +2 +using test data from three source locations at frequency f3 = +0.61: (a) MLP-predicted distribution χ2(ξ, f3) in ξ ∈ G2, (b) reconstruction error (8) with respect to the related estimates +{0.57, 0.59, 0.24} as per Fig. 20, (c) MLP-recovered distribution of the regularization parameter α⋆, and (d) loss function Lε +vs. the number of epochs Ne in log scale. +Table 8: Mean and standard deviation of the reconstructed distributions in Fig. 18 via the direct inversion applied to +high-frequency test data from three distinct sources. +β +2Ti +2St +2Br +χ′ +β +0.57 +0.59 +0.24 +⟨χ⋆ +β⟩Πexp +β +0.585 0.606 0.227 +σ(χ⋆ +β|Πexp +β ) +0.015 0.029 0.016 +Figs. 17 and 18, a set of secondary tests are performed to obtain the dispersion curve for each component of +the test setup. For this purpose, antiplane shear waves of form (9) are induced at fcj = 50j kHz, j = 1, 2, . . . , 7, +17 + +(a) +x2 +0.9 +(d) +0.7 +(c) +log(Le) +2 +-3 +0.5 +*Φ +(b) +1 +-3.5 +三(x2) +-4 +0.2 +×10-3 +0 +8 +×103 +0.1 +Ne(a) +0.6 +x2 +(d) +0.4 +(c) +log(Le) +0.2 +*0 +1.2 +(b) +0.6 +-2 +三(x2) +0.08 +×10-3 +0 +4 +8 +×103 +0.04 +NeFigure 19: Experimental vs. theoretical dispersion curves f(λ−1 +µ ) for µ = {Al, Ti, St, Br}. Analytical curves (solid lines) are +computed from (13) using the pertinent properties in Table 5. +Figure 20: Discrepancy in the balance law (12) at f3 = 0.61: (a) elastic force field T1 +µ, µ = {Ti, St, Br}, according to (14) with +adjusted coefficients {χTi, χSt, χBr} = {0.57, 0.59, 0.24}, (b) the inertia field T2 +µ, and (c) normal discrepancy Dµ. +in 60 cm × 60 cm cuts of aluminum, titanium, steel, and brass sheets used in the primary tests of Fig. 15. +In each experiment, the piezoelectric transducer is placed in the middle of specimen (far from the external +boundary), and the out-of-plane wave motion is captured in the immediate vicinity of the transducer along +a straight line of length 8 cm sampled at 400 scan points. The Fourier-transformed signals in time-space +furnish the dispersion relations of Fig. 19. In parallel, the theoretical dispersion curves affiliated with (12) are +computed according to +f = 2π(λµ)−2 +� +χµh2 +12(1 − ν2µ), +χµ = Eµ +ρµ +, +µ = {Al, Ti, St, Br}, +(13) +using the values of Table 5 for χµ and νµ and h = 1.5mm. A comparison between the experimental and +theoretical dispersion curves f(λ−1 +µ ) in Fig. 19 verifies the theory and the values of Table 5 for χµ in the low- +frequency regime of wave motion. This is also in agreement with the direct inversion results of Figs. 16 and 17. +Moreover, Fig. 19 suggests that at approximately fµ = {170, 200, 120, 110} kHz for µ = {Al, Ti, St, Br} the +governing PDE (12) with physical coefficients fails to predict the experimental results which may provide an +insight regarding the high-frequency reconstruction results in Fig. 18. Further investigation of the balance +law (12), as illustrated in Fig. 20, shows that the test data at 312 kHz satisfy – with less than 10 − 20% +discrepancy depending on the material – a PDE of form (12) with modified coefficients. More specifically, +Fig. 20 demonstrates the achievable balance between the elastic force distribution T1 +µ and inertia field T2 +µ +in (12) by directly adjusting the PDE parameter χ′ +2 to minimize the discrepancy Dµ according to +T1 +µ := +χ′ +2h3 +12(1 − ν2 +2 )∇4v2, +T2 +µ := h(2πf)2v2, +Dµ := |T1 +µ − T2 +µ| +max |T2µ| . +(14) +18 + +×106 +1 +T +Br +Al +0.8 +0.6 +f [s-1] +0.4 +0.2 +0 +0 +0.2 +0.4 +0.6 +0.8 +1 0 +0.2 +0.4 +0.60.810 +0.2 +0.6 +0.8 +10 +0.2 +0.4 +0.60.8 +1 +×103 +入=1 [m-1](a) +(c) +2 +①μ +0 +(b) +Ti +St +Br +2 +×10-1 +Ti +St +BrWith reference to Table 8, the recovered coefficients χ′ +2 at f = f3 = 0.61 verify the direct inversion results of +Fig. 18. This implies that the direct inversion (or PINNs) may lead to non-physical reconstructions in order to +attain the best fit for the data to the “perceived”” underlying physics. Thus, it is imperative to establish the +range of validity of the prescribed physical principles in data-driven modeling. Here, the physics of the system +at f3 is in transition, yet close enough to the leading-order approximation (12) that the discrepancy is less +than 20%. It is unclear, however, if this equation with non-physical coefficients may be used as a predictive +tool. It would be interesting to further investigate the results through the prism of higher-order continuum +theories and a set of independent experiments for validation which could be the subject of a future study. +4.5. Physics-informed neural networks +Following Section 3.3, PINNs are built and trained using experimental test data of Section 4.4. The MLP +network v1⋆ = v1⋆(ξ, f, x|γ, χ⋆ +1) with six hidden layers of respectively 40, 40, 120, 80, 40, and 40 neurons is +constructed to map the out-of-plane velocity field v1 (in Π +exp +1 ) related to a single transducer location x1 and +frequency f1 = 0.336. The PDE parameter χ1 is defined as the unknown scaler parameter of the network, and +following the argument of Section 3.3, the Lagrange multiplier γ is specified as a nonadaptive scaler weight of +magnitude +1 +h(2πf1)2 = 1.5. The input/output dimension for v1⋆ is Nξ×Nω×Nτ = 3600×1×1, and each epoch +makes use of the full dataset for training and the learning rate is 0.005. Keep in mind that the objective here +is to (a) construct a surrogate map for v1, and (b) identify χ⋆ +1. +Fig. 21 demonstrates (a) the accuracy of PINN-estimated field v1⋆ compared to the test data v1, (b) +performance of automatic differentiation in capturing the fourth-order field derivatives e.g., v1⋆ +,1111 that appear +in the governing PDE (12), and (c) the evolution of parameter χ⋆ +1. The comparison in (b) is with respect to the +spectral derivates of test data according to Section 2.3.2. It is no surprise that the automatic differentiation +incurs greater errors in estimating the higher order derivatives involved in the antiplane wave motion compared +to the second-order derivatives of Section 3.3. +In addition, the PINN v2⋆ = v2⋆(ξ, f, x|γ, χ⋆ +2) with seven hidden layers of respectively 40, 40, 120, 120, 80, +40, and 40 neurons is created to reconstruct (i) particle velocity field v2 in the layered specimen Π +exp +2 , and (ii) +distribution of the PDE parameter χ2 in the sampling area. The latter is defined as an unknown parameter +of the network with dimension 40×120, and the scaler weight γ is set to +1 +h(2πf2)2 = 5.84 for the low-frequency +reconstruction. In this setting, the input/output dimension for v2⋆ reads 4800×1×1. Fig. 22 provides a +comparative analysis between the experimental and PINN-predicted maps of velocity and PDE parameter. +The associated statistics are provided in Table 9. It is evident from the waveform in Fig. 22 (a) that the most +pronounced errors in Fig. 22 (d) occur at the loci of vanishing particle velocity. Similar to Section 3.2, this +could be potentially addressed by enriching the test data. +5. Conclusions +The ML-based direct inversion and physics-informed neural networks are investigated for full-field ultra- +sonic characterization of layered components. Direct inversion makes use of signal processing tools to directly +compute the field derivatives from dense datasets furnished by laser-based ultrasonic experiments. This allows +for a simplified and controlled learning process that specifically recovers the sought-for physical fields through +minimizing a single-objective loss function. PINNs are by design more versatile and particularly advantageous +with limited test data where waveform completion is desired (or required) for mechanical characterization. +PINNs multi-objective learning from ultrasonic data may be more complex but can be accomplished via +carefully gauged loss functions. +In direct inversion, Tikhonov regularization is critical for stable reconstruction of distributed PDE param- +eters from limited or multi-fidelity experimental data. In this vein, deep learning offers a unique opportunity +to simultaneously recover the regularization parameter as an auxiliary field which proved to be particularly +insightful in inversion of experimental data. +In training PINNs, two strategies were remarkably helpful: (1) identifying the reference length scale by the +dominant wavelength in an effort to control the norm of spatial derivatives – which turned out to be crucial in +the case of flexural waves in thin plates with the higher order PDE, and (2) estimating the Lagrange multiplier +by taking advantage of the inertia term in the governing PDEs. +19 + +Figure +21: +PINN +vs. +experimental +maps +of +particle +velocity +and +its +derivatives +in +Π +exp +1 +: +(a) +PINN +estimates +for +{v1⋆, v1⋆ +,1111, v1⋆ +,2222, v1⋆ +,1122} wherein the derivatives are obtained by automatic differentiation, (b) normalized LDV-captured par- +ticle velocity field v1 and its corresponding spectral derivatives, (c) normal misfit 8 between (a) and (b), (d) PINN-reconstructed +PDE parameter χ⋆ +1 vs. the number of epochs Ne, and (e) total loss Lϖ and its components (the PDE residue and data misfit) +vs. Ne in log scale. +Laboratory implementations at multiple frequencies exposed that verification and validation are indis- +pensable for predictive data-driven modeling. More specifically, both direct inversion and PINNs recover the +unknown “physical” quantities that best fit the data to specific equations (with often unspecified range of va- +lidity). This may lead to mathematically decent but physically incompatible reconstructions especially when +the perceived physical laws are near their limits such that the discrepancy in capturing the actual physics +is significant. In which case, the inversion algorithms try to compensate for this discrepancy by adjusting +the PDE parameters which leads to non-physical reconstructions. Thus, it is paramount to conduct comple- +mentary experiments to (a) establish the applicability of prescribed PDEs, and (b) validate the predictive +capabilities of the reconstructed models. +Authors’ contributions +Y.X. investigation, methodology, data curation, software, visualization, writing – original draft; F.P. con- +ceptualization, methodology, funding acquisition, supervision, writing – original draft; J.S. experimental data +curation; C.W. experimental data curation. +20 + +1× +.1* +1 * +1* +0.4 +2 +1 +0.4 +(a) +0 +0 +0 +0 +-2 +-0.4 +-0.4 +v,1111 +V,2222 +v,1122 +0.4 +2 +1 +0.4 +(b) +0 +0 +0 +0 +-2 +-0.4 +-0.4 +-1 +8 +三(v,1111) +3 +?L +1 * +6 +6 +6 +4 +4 +(c) +4 +2 +2 +2 +×10-3 +×10-1 +×10-1 +×10-1 +1 +x1 +PDE loss +2 + data loss +0.8 +total loss +0.6 +(d) +(e) +-4 +0.4 +MA +0.2 +-6 +Ne +Ne +0 +×105 +0 +0.2 +0.4 +0.6 +0.8 +0 +0.2 +0.4 +0.6 +0.8 +1Figure 22: Low-frequency PINN reconstruction in Π +exp +2 +using test data from a single source at f2 = 0.17: (a) PINN-predicted distri- +bution of particle velocity v2⋆, (b) normalized LDV-captured particle velocity v2, (c) normal misfit between (a) and (b), (d) PINN- +predicted distribution of the PDE parameter χ⋆ +2, and (e) total loss Lϖ and its components (the PDE residue and data misfit) +vs. the number of epochs Ne in log scale. +Table 9: Mean and standard deviation of the PINN-reconstructed distributions in Fig. 22 from a single-source, low- +frequency test data. +β +2Ti +2St +2Br +χβ +0.91 +1 +0.51 +⟨χ⋆ +β⟩Πexp +β +0.790 +0.890 +0.414 +σ(χ⋆ +β|Πexp +β ) +0.214 +0.356 +0.134 +Acknowledgments +This study was funded by the National Science Foundation (Grant No. 1944812) and the University of +Colorado Boulder through FP’s startup. 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Graff, Wave motion in elastic solids, Courier Corporation, 2012. +24 + diff --git a/1tE0T4oBgHgl3EQfdgCu/content/tmp_files/load_file.txt b/1tE0T4oBgHgl3EQfdgCu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29a3d8c69967c400a6179582fe7d3f53651f053f --- /dev/null +++ b/1tE0T4oBgHgl3EQfdgCu/content/tmp_files/load_file.txt @@ -0,0 +1,1297 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf,len=1296 +page_content='Deep learning for full-field ultrasonic characterization Yang Xu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Fatemeh Pourahmadian1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Jian Song1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Conglin Wang3 1 Department of Civil,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Environmental & Architectural Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' USA 2 Department of Applied Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' USA 3 Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' USA Abstract This study takes advantage of recent advances in machine learning to establish a physics-based data analytic platform for distributed reconstruction of mechanical properties in layered components from full waveform data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this vein, two logics, namely the direct inversion and physics-informed neural networks (PINNs), are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The direct inversion entails three steps: (i) spectral denoising and differentiation of the full-field data, (ii) building appropriate neural maps to approximate the profile of unknown physical and regularization parameters on their respective domains, and (iii) simultaneous training of the neural networks by minimizing the Tikhonov-regularized PDE loss using data from (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' PINNs furnish efficient surrogate models of complex systems with predictive capabilities via multitask learning where the field variables are modeled by neural maps endowed with (scaler or distributed) auxiliary parameters such as physical unknowns and loss function weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' PINNs are then trained by minimizing a measure of data misfit subject to the underlying physical laws as constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this study, to facilitate learning from ultrasonic data, the PINNs loss adopts (a) wavenumber-dependent Sobolev norms to compute the data misfit, and (b) non-adaptive weights in a specific scaling framework to naturally balance the loss objectives by leveraging the form of PDEs germane to elastic- wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Both paradigms are examined via synthetic and laboratory test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In the latter case, the reconstructions are performed at multiple frequencies and the results are verified by a set of complementary experiments highlighting the importance of verification and validation in data-driven modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Keywords: deep learning, ultrasonic testing, data-driven mechanics, full-wavefield inversion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Introduction Recent advances in laser-based ultrasonic testing has led to the emergence of dense spatiotemporal datasets which along with suitable data analytic solutions may lead to better understanding of the mechanics of complex materials and components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This includes learning of distributed mechanical properties from test data which is of interest in a wide spectrum of applications from medical diagnosis to additive manufacturing [1, 2, 3, 4, 5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This work makes use of recent progress in deep learning [8, 9] germane to direct and inverse problems in partial differential equations [10, 11, 12, 13] to develop a systematic full-field inversion framework to recover the profile of pertinent physical quantities in layered components from laser ultrasonic measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The focus is on two paradigms, namely: the direct inversion and physics-informed neural networks (PINNs) [14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The direct inversion approach is in fact the authors’ rendition of elastography method [18, 19, 20] through the prism of deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' To this end, tools of signal processing are deployed to (a) denoise the experimental data, and (b) carefully compute the required field derivatives as per the governing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In parallel, the unknown distribution of PDE parameters in space-frequency are identified by neural networks which are then trained by minimizing the single-objective elastography loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The learning process is stabilized via the Tikhonov regularization [21, 22] where the regularization parameter is defined in a distributed sense as a separate neural network which is simultaneously trained with the sought-for physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This unique ∗Corresponding author: tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 303-492-2027, email fatemeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='pourahmadian@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='edu Preprint submitted to Elsevier January 9, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='02378v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='NA] 6 Jan 2023 exercise of learning the regularization field without a-priori estimates, thanks to neural networks, proved to be convenient, effective, and remarkably insightful in inversion of multi-fidelity experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' PINNs have recently come under the spotlight for offering efficient, yet predictive, models of complex PDE systems [10] that has so far been backed by rigorous theoretical justification within the context of linear elliptic and parabolic PDEs [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Given the multitask nature of training for these networks and the existing challenges with modeling stiff and highly oscillatory PDEs [12, 24], much of the most recent efforts has been focused on (a) adaptive gauging of the loss function [12, 25, 26, 27, 28, 29, 13], and (b) addressing the gradient pathologies [24, 13] e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', via learning rate annealing [30] and customizing the network architecture [11, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this study, our initially austere implementations of PINNs using both synthetic and experimental waveforms led almost invariably to failure which further investigation attributed to the following impediments: (a) high- norm gradient fields due to large wavenumbers, (b) high-order governing PDEs in the case of laboratory experiments, and (c) imbalanced objectives in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' These problems were further magnified by our attempts for distributed reconstruction of discontinuous PDE parameters – in the case of laboratory experiments, from contaminated and non-smooth measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The following measures proved to be effective in addressing some of these challenges: (i) training PINNs in a specific scaling framework where the dominant wavenumber is the reference length scale, (ii) using the wavenumber-dependent Sobolev norms in quantifying the data misfit, (iii) taking advantage of the inertia term in the governing PDEs to naturally balance the objectives in the loss function, and (iv) denoising of the experimental data prior to training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Section 2 formulates the direct scattering problem related to the synthetic and laboratory experiments, and provides an overview of the data inversion logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Section 3 presents the computational implementation of direct inversion and PINNs to reconstruct the distribution of L´ame parameters in homogeneous and heterogeneous models from in-plane displacement fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Section 4 provides a detailed account of laboratory experiments, scaling, signal processing, and inversion of antiplane particle velocity fields to recover the distribution of a physical parameter affiliated with flexural waves in thin plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The reconstruction results are then verified by a set of complementary experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Concept This section provides (i) a generic formalism for the direct scattering problem pertinent to the ensuing (synthetic and experimental) full-field characterizations, and (ii) data inversion logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Forward scattering problem Consider ultrasonic tests where the specimen Π ⊂ Rd, d = 2, 3, is subject to (boundary or internal) excitation over the incident surface Sinc ⊂ Π and the induced (particle displacement or velocity) field u: Π × [0 T] → RNΛ (NΛ ⩽ d) is captured over the observation surface Sobs ⊂ Π in a timeframe of length T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Here, Π is an open set whose closure is denoted by Π, and the sensing configuration is such that Sinc ∩ Sobs = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this setting, the spectrum of observed waveforms ˆu: Sobs × Ω → CNΛ is governed by Λ[ˆu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' ϑ](ξ, ω) = 0, ˆu := F[u](ξ, ω), ξ ∈ Sobs, ω ∈ Ω, (1) where Λ of size NΛ×1 designates a differential operator in frequency-space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' F represents the temporal Fourier transform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' ϑ of dimension Nϑ×1 is the vector of relevant geometric and elastic parameters e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', Lam´e constants and mass density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' ξ ∈ Rd is the position vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' and ω > 0 is the frequency of wave motion within the specified bandwidth Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Dimensional platform All quantities in (1) are rendered dimensionless by identifying ρ◦, σ◦, and ℓ◦ as the respective reference scales [33] for mass density, elastic modulus, and length whose explicit values will be later specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Data inversion Given the full waveform data ˆu on Sobs × Ω, the goal is to identify the distribution of material properties over Sobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' For this purpose, two reconstruction paradigms based on neural networks are adopted in this study, namely: (i) direct inversion, and (ii) physics-based neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Inspired by the elastography 2 method [18, 19], quantities of interest in (i) are identified by neural maps over Sobs × Ω that minimize a regularized measure of Λ in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The neural networks in (ii), however, are by design predictive maps of the waveform data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', ˆu) obtained by minimizing the data mismatch subject to (1) as a soft or hard constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this setting, the unknown properties of Λ may be recovered as distributed parameters of the (data) network during training via multitask optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In what follows, a detailed description of the deployed cost functions in (i) and (ii) is provided after a brief review of the affiliated networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Waveform and parameter networks Laser-based ultrasonic experiments furnish a dense dataset on Sobs × Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Based on this, multilayer per- ceptrons (MLPs) owing to their dense range [34] may be appropriate for approximating complex wavefields and distributed PDE parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Moreover, this architecture has proven successful in numerous applications within the PINN framework [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this study, MLPs serve as both data and property maps where the input consists of discretized space and frequency coordinates (ξi, ωj), i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' , Nξ, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' , Nω, as well as distinct experimental parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', the source location, distilled as one vector τk on domain T with k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' , Nτ, while the output represents waveform data Dijk = [Rˆu, Iˆu](ξi, ωj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' τk) ∈ RNΛ × RNΛ, and/or the sought-for mechanical properties Pijn = [Rϑn, Iϑn](ξi, ωj) ∈ R × R, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' , Nϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Note that following [35], the real R and imaginary I parts of (1) and every complex-valued variable are separated such that both direct and inverse problems are reformulated in terms of real-valued quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this setting, each fully-connected MLP layer with Nl neurons is associated with the forward map Υl : RNl−1 → RNl, Υl(xl−1) = tanh(W lxl−1 + bl), xl−1 ∈ RNl−1, (2) where W l ∈ RNl×Nl−1 and bl ∈ RNl respectively denote the lth layer’s weight and bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Consecutive compo- sition of Υl for l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' , Nm builds the network map wherein Nm designates the number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Direct inversion Logically driven by the elastography method, the direct inversion approach depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 1 takes advan- tage of the leading-order physical principles underpinning the test data to recover the distribution of relevant physical quantities in space-frequency i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', over the measurement domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The ML-based direct inversion entails three steps: (a) spectral denoising and differentiation of (n-differentiable) waveforms ˆu over Sobs × Ω according to the (n-th order) governing PDEs in (1), (b) building appropriate MLP maps to estimate the profile of unknown physical parameters of the forward problem and regularization parameters of the inverse solution, and (c) learning the MLPs through regularized fitting of data to the germane PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Note that synthetic datasets – generated via e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', computer modeling or the method of manufactured solutions, may directly lend themselves to the fitting process in (c) as they are typically smooth by virtue Figure 1: Direct inversion: (a) FFT-based spatial differentiation of the full-field data as per operator Λ, (b) MLP-based approx- imation of the unknown PDE and regularization parameters (ϑ, α) on their respective domains, and (c) training the MLPs via minimizing the elastography loss Lε according to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 3 MLP ultrasonic test data u(S,w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' T) N(s,w) spectral differentiation 3 Vu(s,w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' T) Mα(s, w) VVu($, w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' T) : (a) (b) M (9*,α*) := (Ng, ) ) = arg min L(u, *;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='α*) (c) 9*,α*of numerical integration or analytical form of the postulated solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Laboratory test data, however, are generally contaminated by noise and uncertainties, and thus, spectral differentiation is critical to achieve the smoothness requirements in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The four-tier signal processing of experimental data follows closely that of [36, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1] which for completeness is summarized here: (1) a band-pass filter consistent with the frequency spectrum of excitation is applied to the measured time signals at every receiver point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' (2) the obtained temporally smooth signals are then differentiated or integrated to obtain the pertinent field variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' (3) spatial smoothing is implemented at every snapshot in time via application of median and moving average filters followed by computing the Fourier representation of the processed waveforms in space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' (4) the resulting smooth fields may be differentiated (analytically in the Fourier space) as many times as needed based on the underlying physical laws in preparation for the full-field reconstruction in step (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It should be mentioned that the experimental data may feature intrinsic discontinuities e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', due to material heterogeneities or contact interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this case, the spatial smoothing in (3) must be implemented in a piecewise manner after the geometric reconstruction of discontinuity surfaces in Sobs which is quite straightforward thanks to the full-field measurements, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', [36, section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Next, the unknown PDE parameters ϑ are approximated by a fully connected MLP network ϑ⋆ := Nϑ(ξ, ω) as per Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The network is trained by minimizing the loss function Lε(ˆu, ϑ⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' α) = ∥Λ(ˆu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' ϑ⋆)∥2 L2(Sobs×Ω×T )NΛ + ∥α1ϑ ⊙ ϑ⋆∥2 L2(Sobs×Ω)Nϑ , (3) where 1ϑ indicates an all-ones vector of dimension Nϑ × 1, and ⊙ designates the (element-wise) Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Here, the PDE residual based on (1) is penalized by the norm of unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Observe that the latter is a function of the weights and biases of the neural network which may help stabilize the MLP estimates during optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Such Tikhonov-type functionals are quite common in waveform tomography applications [37, 38, 39] owing to their well-established regularizing properties [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Within this framework, R ∋ α > 0 is the regularization parameter which may be determined by three means, namely: (i) the Morozov discrepancy principle [40, 41], (ii) its formulation as a (constant or distributed) parameter of the ϑ⋆ network which could then be learned during training, and (iii) its independent reconstruction as a separate MLP network α⋆ := Nα(ξ, ω) illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 1 (b) that is simultaneously trained along with ϑ⋆ by minimizing (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this study, direct inversion is applied to synthetic and laboratory test data with both α = 0 and α > 0, based on (ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It was consistently observed that the regularization parameter α plays a key role in controlling the MLP estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This is particularly the case in situations where the field ˆu is strongly polarized or near-zero in certain neighborhoods which brings about instability i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', very large estimates for ϑ⋆ in these areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In light of this, all direct inversion results in this paper correspond to the case of α > 0 identified by the MLP network α⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Physics-informed neural networks By deploying the knowledge of underlying physics, PINNs [14, 15] furnish efficient neural models of complex PDE systems with predictive capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this vein, a multitask learning process is devised according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 2 where (a) the field variable ˆu – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', measured data on Sobs × Ω × T , is modeled by the MLP map ˆu⋆ : = Nˆu(ξ, ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' τ) endowed with the auxiliary parameter γ(ξ, ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' τ) related to the loss function (4), (b) the physical unknowns ϑ could be defined either as parameters of ˆu⋆ as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 2 (i), or as a separate MLP ϑ⋆ : = Nϑ(ξ, ω) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 2 (ii), and (c) learning the MLPs and affiliated parameters through minimizing a measure of data misfit subject to the governing PDEs as soft/hard constraints wherein the spatial derivatives of ˆu⋆ are computed via automatic differentiation [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It should be mentioned that in this study all MLP networks are defined on (a subset of) Sobs × Ω × T where Sobs ∩ ∂Π = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Hence, the initial and boundary conditions – which could be specified as additional constraints in the loss function [15], are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this setting, the PINNs loss takes the form Lϖ(ˆu⋆, ϑ⋆|γ) = ∥ˆu − ˆu⋆∥2 N(Sobs×Ω×T )NΛ + ∥γ1Λ ⊙ Λ(ˆu⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' ϑ⋆)∥2 L2(Sobs×Ω×T )NΛ, N = L2, �Hι, ι ⩽ n, (4) where 1Λ is a NΛ× 1 vector of ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' n is the order of Λ, and �Hι denotes the adaptive Hι norm defined by 4 Figure 2: Two logics for the physics-informed neural networks (PINNs) with distributed parameters: (i) the test data ˆu(ξ, ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' τ) are modeled by a MLP map, while the unknown physical parameters ϑ – on Sobs × Ω, and the loss function weight γ – on Sobs × Ω × T , are defined as network parameters, and (ii) ˆu(ξ, ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' τ) and ϑ(ξ, ω) are identified by separate MLPs, while γ is a parameter of Nˆu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The MLP(s) in (i) and (ii) are then trained by minimizing Lϖ of (4) in the space of data and PDE parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' ∥ · ∥ � Hι := � � 1⩽|e|⩽ ι γe ∥∇e(·)∥2 L2 + ∥·∥2 L2, ∇e = ∂|e| ∂ξe1 1 ∂ξe2 2 ··· ∂ξed d , |e| := d � i=1 ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' (5) Here, e:= {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' ed} is a vector of integers ei ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Provided that ∀e, γe = 1, then �Hι is by definition equal to Hι [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Note however that at high wavenumbers, Hι is dominated by the highest derivatives ∇eˆu⋆, |e| = ι, which may complicate (or even lead to the failure of) the training process due to uncontrolled error amplification by automatic differentiation particularly in earlier epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This issue may be addressed through proper weighting of derivatives in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In light of the frequency-dependent Sobolev norms in [44, 37], one potential strategy is to adopt the wavenumber-dependent weights as the following γe = � 1 κe1 1 κe2 2 ··· κed d �2 , 1 ⩽ |e| ⩽ ι, wherein κi is a measure of wavenumber along ξi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this setting, the weighted norms of derivatives in (5) remain approximately within the same order as the L2 norm of data misfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Another way to automatically achieve the latter is to set the reference scale ℓ◦ such that κi ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Note that the �Hι norms directly inform the PINNs about the “expected” field derivatives – while preventing their uncontrolled magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This may help stabilize the learning process as such derivatives are intrinsically involved in the PINNs loss via Λ(ˆu⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' ϑ⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It should be mentioned that when N = �Hι in (4), the “true” estimates for derivatives ∇eˆu may be obtained via spectral differentiation as per Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The Lagrange multiplier [45, 46] γ(ξ, ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' τ) in (4) is critical for balancing the loss components during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Its optimal value, however, highly depends on (a) the nature of Λ [12], and (b) the distribution of unknown parameters ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It should be mentioned that setting γ = 1 led to failure in almost all of the synthetic and experimental implementations of PINNs in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Gauging of loss function weights has been the subject of extensive recent studies [12, 25, 47, 26, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' One systematic approach is the adaptive SA-PINNs [12] where the multiplier γ(ξ, ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' τ) is a distributed parameter of ˆu⋆ whose value is updated in each epoch according to a minimax weighting paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Within this framework, the data (and parameter) networks are trained by minimizing Lϖ with respect to ˆu⋆ and ϑ⋆, while maximizing the loss with respect to γ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Depending on the primary objective for PINNs, one may choose nonadaptive or adaptive weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' More specifically, if the purpose is high-fidelity forward modeling via neural networks where ϑ is known a-priori and PINNs are intended to serve as predictive surrogate models of Λ, then ideas rooted in constrained optimization e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', minimax weighting is theoretically sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' However, if the inverse solution i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', identification of ϑ(ξ, ω) from “real-world” or laboratory test data is the main goal particularly in a situation where any assumption on the smoothness of ϑ and/or applicability of Λ may be (at least locally) violated e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', due to unknown material 5 MLP network parameters (i) (ii) 9*($, w) 9*:= (S,w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' T) N(s, w) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' ↑ automatic E 3 differentiation α*:= V*(S,w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' T) α*:= T 3 Na(S, w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' T) VVu*($, w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' T) Na(S, w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' T) T : MLP ↑ (S,w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' T) u* = arg min max Lw(u*, *I) ,9*heterogeneities or interfacial discontinuities, then trying to enforce Λ everywhere on Sobs × Ω × T (via point- wise adaptive weighting) may lead to instability and failure of data inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In such cases, nonadaptive weighting may be more appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In light of this, in what follows, γ is a non-adaptive weight specified by taking advantage of the PDE structure to naturally balance the loss objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Synthetic implementation Full-field characterization via the direct inversion and physics-informed neural networks are examined through a set of numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The waveform data in this section are generated via a FreeFem++ [48] code developed as part of [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Problem statement Plane-strain wave motion in two linear, elastic, piecewise homogeneous, and isotropic samples is modeled according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' On denoting the frequency of excitation by ω, let ℓr = 2π ω � µr/ρr, ρr = 1, and µr = 1 be the reference scales for length, mass density, and stress, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this framework, both specimens are of size 16×16 and uniform density ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The first sample Π1 ⊂ R2 is characterized by the constant Lam´e parameters µ◦ = 1 and λ◦ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='47, while the second sample Π2 ⊂ R2 is comprised of four perfectly bonded homogenous components Π2j of µj = j and λj = 2j/3, j = {1, 2, 3, 4} such that Π2 = �4 j=1 Π2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Accordingly, the shear and compressional wave speeds read c◦ s = 1, c◦ p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='57 in Π1, and cj s = √j, cj p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='63√j in Π2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Every numerical experiment entails an in-plane harmonic excitation at ω = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='91 via a point source on Sinc (the perimeter of a 14 × 14 square centered at the origin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The resulting displacement field uα = (uα 1 , uα 2 ), α = 1, 2, is then computed in Πα over Sobs (a concentric square of dimension 8 ×8) such that µα∆uα(ξ) + (λα + µα)∇∇ · uα(ξ) + ρω2uα(ξ) = δ(ξ − x)d, ξ ∈ Πα, x ∈ Sinc, � λα∇ · uα(ξ)I2 + 2µα∇symuα(ξ) � n(ξ) = 0, ξ ∈ ∂Πα, (6) where x and d respectively indicate the source location and polarization vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' n is the unit outward normal to the specimen’s exterior, and � µα = µ◦, λα = λ◦, α = 1 µα = µj, λα = λj, α = 2 ∧ ξ ∈ Π2j∈{1,2,3,4} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Figure 3: synthetic experiments simulating plane-strain wave motion in homogeneous (top-left) and heterogeneous (bottom-left) specimens: (a) testing configuration where the model is harmonically excited at frequency ω by a point source on Sinc, and the induced displacement field u is computed over Sobs along ξ1 and ξ2 as shown in (b) and (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 6 TT1 μo,\\。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' u1 μ3,^3 μ4，^4 TT2 W2 (a) (b)When α = 2, the first of (6) should be understood as a shorthand for the set of four governing equations over Π2j, j = {1, 2, 3, 4}, supplemented by the continuity conditions for displacement and traction across ∂Π2j\\∂Π2 as applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this setting, the generic form (1) may be identified as the following Λ = Λα := µα∆ + (λα + µα)∇∇ · + ρω2I2, α = 1, 2, ˆu = uα(ξ, ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' τ), ϑ = [µα, λα](ξ, ω), ξ ∈ Sobs, ω ∈ Ω, τ ∈ T , (7) wherein I2 is the second-order identity tensor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' τ = (x, d) ∈ Sinc × B1 = T with B1 denoting the unit circle of polarization directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Note that ρ is treated here as a known parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In the numerical experiments, Sinc (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Sobs) is discretized by a uniform grid of 32 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 50×50) points, while Ω and B1 are respectively sampled at ω = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='91 and d = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' All inversions in this study are implemented within the PyTorch framework [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Direct inversion The three-tier logic of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 is employed to reconstruct the distribution of µα and λα, α = 1, 2, over Sobs, entailing: (a) spectral differentiation of the displacement field uα in order to compute ∆uα and ∇∇ · uα as per (6), (b) construction of three positive-definite MLP networks µ⋆, λ⋆, and α⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' each of which is comprised of one hidden layer of 64 neurons, and (c) training the MLPs by minimizing Lε as in (3) and (7) by way of the ADAM algorithm [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' To avoid near-boundary errors affiliated with the one-sided FFT differentiation in ∆uα and ∇∇·uα, a concentric 40×40 subset of collocation points sampling Sobs is deployed for training purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It should also be mentioned that in the heterogeneous case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', α = 2, the discontinuity of derivatives across ∂Π2j∈{1,2,3,4} calls for piecewise spectral differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' According to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1, the input to P⋆ = NP(ξ, ω), P = µ, λ, and α⋆ = Nα(ξ, ω) is of size NξNτ × Nω = 1600Ns × 1 where Ns ⩽ 32 is the number of simulations i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', source locations used to generate distinct waveforms for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this setting, since the physical quantities of interest are independent of τ, the real-valued output of MLPs is of dimension 1600 × 1 furnishing a local estimate of the L´ame and regularization parameters at the specified sampling points on Sobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Each epoch makes use of the full dataset and the learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this work, the reconstruction error is measured in terms of the normal misfit Ξ(q⋆) = ∥q⋆ − q ∥L2 ∥q ∥L∞ , (8) where q⋆ is an MLP estimate for a quantity with the “true” value q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Let Sinc be sampled at one point i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', Ns = 1 so that a single forward simulation in Πα, α = 1, 2, generates the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The resulting reconstructions are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It is evident from both figures that the single-source reconstruction fails at the loci of near-zero displacement which may explain the relatively high values of the recovered regularization parameter α⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Table 1 details the true values as well as mean and standard deviation of the reconstructed L´ame distributions ϑ⋆ = (µ⋆, λ⋆) in Π1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Π2j for j = 1, 2, 3, 4) according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 4 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This problem may be addressed by enriching the training dataset e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', via increasing Ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 6 and 7 illustrate the reconstruction results when Sinc is sampled at Ns = 5 source points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The mean and standard deviation of the reconstructed distributions are provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It is worth noting that in this case the identified regularization parameter α⋆ assumes much smaller values – compared to that of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This is closer to the scale of computational errors in the forward simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' To examine the impact of noise on the reconstruction, the multisource dataset used to generate Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 6 and 7 are perturbed with 5% white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The subsequent direct inversions from noisy data are displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 8 and 9, and the associated statistics are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Note that spectral differentiation as the first step in direct inversion plays a critical role in denoising the waveforms, and subsequently regularizing the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This may substantiate the low magnitude of MLP-recovered α⋆ in the case of noisy data in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The presence of noise, nonetheless, affects the magnitude and thus composition of terms in the Fourier representation of the processed displacement fields in space which is used for differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This may in turn lead to the emergence of fluctuations in the reconstructed fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 7 Figure 4: Direct inversion of the L´ame parameters in Π1 using noiseless data from a single source: (a) MLP-predicted distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µ◦ = 1 and λ◦ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='47, (c) MLP-recovered distribution of the regularization parameter α⋆, and (d) loss function Lε vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' the number of epochs Ne in the log = log10 scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Figure 5: Direct inversion of the L´ame parameters in Π2 using noiseless data from a single source: (a) MLP-predicted distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µj = j and λj = 2j/3, j = {1, 2, 3, 4}, (c) MLP-recovered regularization parameter α⋆, and (d) loss function Lε vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' the number of epochs Ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Table 1: Mean ⟨·⟩D and standard deviation σ(·|D) of the reconstructed L´ame distributions in D = Π1, Π2j=1,2,3,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Here, the direct inversion is applied to noiseless data from a single source as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' D Π1 Π21 Π22 Π23 Π24 µ µ◦ = 1 µ1 = 1 µ2 = 2 µ3 = 3 µ4 = 4 ⟨µ⋆⟩D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='991 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='983 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='825 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='835 σ(µ⋆|D) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='441 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='325 λ λ◦ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='47 λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='67 λ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='33 λ3 = 2 λ4 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='66 ⟨λ⋆⟩D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='376 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='850 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='746 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='412 σ(λ⋆|D) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='399 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='864 8 (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1 ×10-2 (c) (d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 log(Le) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='05 1 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 1 （?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='15 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 ×104 1 Ne 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3 0(a) (b) 三(μ*) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 3 (c) ×10-2 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 2 log(Le) 1 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 1 ×104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 NeFigure 6: Direct inversion of the L´ame parameters in Π1 using noiseless data from five distinct simulations: (a) MLP-predicted distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µ◦ = 1 and λ◦ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='47, (c) MLP-recovered regularization parameter α⋆, and (d) loss function Lε vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' the number of epochs Ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Figure 7: Direct inversion of the L´ame parameters in Π2 using five noiseless datasets: (a) MLP-predicted distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µj = j and λj = 2j/3, j = {1, 2, 3, 4}, (c) MLP-recovered regularization parameter α⋆, and (d) loss function Lε vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' the number of epochs Ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Table 2: Mean and standard deviation of the reconstructed L´ame distributions from five distinct noiseless datasets according to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' D Π1 Π21 Π22 Π23 Π24 µ 1 1 2 3 4 ⟨µ⋆⟩D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='999 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='003 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='999 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='999 σ(µ⋆|D) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='016 λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='33 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='66 ⟨λ⋆⟩D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='464 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='660 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='302 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='997 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='635 σ(λ⋆|D) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='068 9 (a) (b) 2 u* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='02 (×)m 1 (c) ×10-3 (d) 1 log(Le) 5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='98 ×10-2 0 3 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 ^* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 三(\\*) 2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='45 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 1 ×104 Ne 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 /×10-2(a) (b) 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='75 3 (c) ×10-3 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 2 log(Le) ×10-2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 -2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 ×104 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 Ne ×10-1Figure 8: Direct inversion of the L´ame parameters in Π1 using five datasets perturbed with 5% white noise: (a) MLP-predicted distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µ◦ = 1 and λ◦ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='47, (c) MLP-recovered regularization parameter α⋆, and (d) loss function Lε vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' the number of epochs Ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Figure 9: Direct inversion of the L´ame parameters in Π2 using five datasets perturbed with 5% white noise: (a) MLP-predicted distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µj = j and λj = 2j/3, j = {1, 2, 3, 4}, (c) MLP-recovered regularization parameter α⋆, and (d) loss function Lε vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' the number of epochs Ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Table 3: Mean and standard deviation of the reconstructed L´ame distributions from noisy data according to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' D Π1 Π21 Π22 Π23 Π24 µ 1 1 2 3 4 ⟨µ⋆⟩D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='002 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='005 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='996 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='996 σ(µ⋆|D) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='088 λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='33 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='66 ⟨λ⋆⟩D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='650 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='263 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='006 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='654 σ(λ⋆|D) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='300 10 (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='05 (r)= L￥ 2 ×10-3 (d) (c) 1 4 log(Le) 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='95 ×10-2 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 三()*) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='35 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='45 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 1 ×104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='15 Ne 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4(a) (b) 4 E(μ*) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 3 (c) ×10-3 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 2 log(Le) 1 ×10-1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 Ne 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Physics-informed neural networks The learning process of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3 is performed as follows: (a) the MLP network uα⋆ = Nuα(ξ, ω, x|γ, ϑ⋆) endowed with the positive-definite parameters γ and ϑ⋆ = (µ⋆, λ⋆) is constructed such that the input x labels the source location and the auxiliary weight γ is a nonadaptive scaler, (b) µ⋆ and λ⋆ may be specified as scaler or distributed parameters of the network according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 2 (i), and (c) uα⋆ is trained by minimizing Lϖ in (4) via the ADAM optimizer using the synthetic waveforms of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Reconstructions are performed on the same set of collocation points sampling Sobs×Ω×T as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Accordingly, the input to uα⋆ is of size Nξ×Nω×Nτ = 1600×1×Ns, while its output is of dimension (1600×1×Ns)2 modeling the displacement field along ξ1 and ξ2 in the sampling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Similar to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2, each epoch makes use of the full dataset for training and the learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The PyTorch implementation of PINNs in this section is accomplished by building upon the available codes on the Github repository [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The MLP network u1⋆ = u1⋆(ξ, ω, x|γ, ϑ⋆) with three hidden layers of respectively 20, 40, and 20 neurons is employed to map the displacement field u1 (in Π1) associated with a single point source of frequency ω = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='91 at x = x1 ∈ Sinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The L´ame constants are defined as the unknown scaler parameters of the network i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', ϑ⋆ = {µ⋆, λ⋆}, and the Lagrange multiplier γ is specified per the following argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Within the dimensional framework of this section and with reference to (7), observe that on setting γ = 1 ρω2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='065), both (the PDE residue and data misfit) components of the loss function Lϖ in 4 emerge as some form of balance in terms of the displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This may naturally facilitate maintaining of the same scale for the loss terms during training, and thus, simplifying the learning process by dispensing with the need to tune an additional parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Keep in mind that the input to u1⋆ is of size 1600×1×1, while its output is of dimension (1600×1×1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this setting, the training objective is two-fold: (a) construction of a surrogate map for u1, and (b) identification of µ⋆ and λ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 10 showcases (i) the accuracy of PINN estimates based on noiseless data in terms of the vertical component of displacement field u1 2 in Π1, and (ii) the performance of automatic differentiation [42] in capturing the field derivatives in terms of components that appear in the governing PDE 7 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', u1 2,ij = ∂2u1 2/(∂ξi∂ξj), i, j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The comparative analysis in (ii) is against the spectral derivates of FEM fields according to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It is worth noting that similar to Fourier-based differentiation, the most pronounced errors in automatic differentiation occur in the near-boundary region i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', the support of one-sided derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It is observed that the magnitude of such discrepancies may be reduced remarkably by increasing the number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Nonetheless, the loci of notable errors remain at the vicinity of specimen’s external boundary or internal discontinuities such as cracks or material interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 10 is complemented with the reconstruction results of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 11 indicating (µ⋆, λ⋆) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='486) for the homogenous specimen Π1 with the true L´ame constants (µ◦, λ◦) = (1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The impact of noise on training is examined by perturbing the noiseless data related to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 10 with 5% white noise, which led to (µ⋆, λ⋆) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='999, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='510) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Next, the PINN u2⋆ = u2⋆(ξ, ω, x|ϑ⋆) with three hidden layers of respectively 120, 120, and 80 neurons is created to reconstruct (i) displacement field u2 in the heterogeneous specimen Π2, and (ii) distribution of the L´ame parameters over the observation surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this vein, synthetic waveform data associated with five point sources {xi} ∈ Sinc, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' , 5 at ω = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='91 is used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Here, ϑ⋆ is the network’s unknown distributed parameter, of dimension (40×40)2, and the nonadaptive scaler weight γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='065 in light of the sample’s uniform density ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this setting, the input to u2⋆ is of size 1600×1×5, while its output is of dimension (1600×1×5)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 13 provides a comparative analysis between the FEM and PINN maps of horizontal displacement u1 2 in Π2 and its spatial derivatives computed by spectral and automatic differentiation respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Table 4: Mean and standard deviation of the PINN-reconstructed L´ame distributions from five distinct noiseless datasets according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' D Π21 Π22 Π23 Π24 ⟨µ⋆⟩D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='975 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='973 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='941 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='918 σ(µ⋆|D) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='226 ⟨λ⋆⟩D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='686 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='250 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='045 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='065 σ(λ⋆|D) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='857 11 Figure 10: PINN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' FEM maps of vertical displacement and its derivatives in Π1: (a) MLP estimates, from noiseless data, for {u1 2 ⋆, u1⋆ 2,11, u1⋆ 2,22, u1⋆ 2,12} wherein the derivatives u1⋆ 2,ij, i, j = 1, 2, are obtained by automatic differentiation, (b) FEM displacement solution and its spectral derivatives for {u1 2, u1 2,11, u1 2,22, u1 2,12}, and (c) normal misfit 8 between (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Figure 11: PINN reconstruction of L´ame constants in the homogeneous plate Π1 from noiseless data: (a) µ⋆ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' number of epochs Ne, (b) λ⋆ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Ne, and (c) total loss Lϖ and its components (the PDE residue and data misfit) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Ne in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Figure 12: PINN reconstruction of L´ame constants in Π1 from noisy data: (a) µ⋆ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' number of epochs Ne, (b) λ⋆ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Ne, and (c) total loss Lϖ and its components (the PDE residue and data misfit) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Ne in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 12 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' U2,22 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1 (a) 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 I u2,11 u2,22 2,12 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 (b) 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 三(u2 7 E(u2,11) 三(u2,22) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3 三(u2,12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='L 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 1 ×10-2 ×10-1(a) (b) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 \\* PDE loss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='- data loss total loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 4 Ne Ne 0 0 ×105 ×105 ×105 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 2(a) (b) (c) \\* PDE loss L* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 data loss 0 total loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 Ne Ne UN 0 0 ×105 ×105 ×105 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 2Figure 13: PINN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' FEM maps of horizontal displacement and its derivatives in Π2: (a) PINN estimates, from noiseless data, for {u2 1 ⋆, u2⋆ 1,11, u2⋆ 1,22, u2⋆ 1,12} wherein the derivatives u2⋆ 1,ij, i, j = 1, 2, are obtained by automatic differentiation, (b) FEM displacement solution and its spectral derivatives for {u2 1, u2 1,11, u2 1,22, u2 1,12}, and (c) normal misfit 8 between (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Figure 14: PINN reconstruction of L´ame parameters in Π2 using five noiseless datasets: (a) PINN-predicted distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µj = j and λj = 2j/3, j = {1, 2, 3, 4}, (c) total loss Lϖ and its components (the PDE residue and data misfit) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Ne in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The PINN-reconstructed distribution of PDE parameters is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 14 whose statistics is detailed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It is worth mentioning that the learning process is repeated for a suit of weights γ = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5, 2, 5, 10, 15}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In all cases, the results are either quite similar or worse than that of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 13 2 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 2* 2* 2* ui ui,11 ui,22 ui,12 3 3 2 0 1 1 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 1 1 2 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 ui,11 2 ui,12 3 3 2 1 1 (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 1 1 2 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 5 三(ui 三(ui,11) 2 三(ui,22) 2* E(ui,12) 2* 2 2 3 (c) 1 L 1 ×10-3 ×10-2 ×10-2 ×10-2(a) (b) 4 三(μ*) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 3 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 2 PDE loss 0 data loss total loss 0 2 三(\\*) 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 2 6 ×106 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 Ne 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Laboratory implementation This section examines the performance of direct inversion and PINNs for full-field ultrasonic character- ization in a laboratory setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In what follows, experimental data are processed prior to inversion as per Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 which summarizes the detailed procedure in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' To verify the inversion results, quantities of interest are also reconstructed through dispersion analysis, separately, from a set of auxiliary experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Test set-up Experiments are performed on two (homogeneous and heterogeneous) specimens: Π exp 1 which is a 27 cm ×27 cm×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 mm sheet of T6 6061 aluminum, and Π exp 2 composed of (a) 5 cm×27 cm×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 mm sheet of Grade 2 titanium, (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 cm×27 cm×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 mm sheet of 4130 steel, and (c) 5 cm×27 cm×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 mm sheet of 260-H02 brass, connected via metal epoxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' For future reference, the density ρµ, Young’s modulus Eµ, and Poisson’s ratio νµ for µ = {Al, Ti, St, Br} are listed in Table 5 as per the manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Ultrasonic experiments on both samples are performed in a similar setting in terms of the sensing config- uration and illuminating wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In both cases, the specimen is excited by an antiplane shear wave from a designated source location Sinc, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 15, by a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 MHz p-wave piezoceramic transducer (V101RB by Olympus Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The incident signal is a five-cycle burst of the form H(fct) H(5−fct) sin � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2πfct � sin � 2πfct � , (9) where H denotes the Heaviside step function, and the center frequency fcis set at 165 kHz (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' {80, 300} kHz) in Π exp 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Π exp 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The induced wave motion is measured in terms of the particle velocity vβ, β = 1, 2, on the scan grids Gβ sampling Sobs where Sobs ∩Sinc = Sobs ∩∂Π exp β = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' A laser Doppler vibrometer (LDV) which is mounted on a 2D robotic translation frame (for scanning) is deployed for measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The VibroFlex Xtra VFX-I-120 LDV system by Polytec Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' is capable of capturing particle velocity within the frequency range ∼ DC − 24 MHz along the laser beam which in this study is normal to the specimen’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The scanning grid G1 ⊂ Π exp 1 is identified by a 2 cm×2 cm square sampled by 100×100 uniformly spaced measurement points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This amounts to a spatial resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 mm in both spatial directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In parallel, G2 ⊂ Π exp 2 is a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 cm×7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 cm rectangle positioned according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 15 (b) and sampled by a uniform grid of 180×60 scan points associated with the spatial resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='42 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' At every scan point, the data acquisition is conducted for a time period of 400 µs at the sampling rate of 250 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' To minimize the impact of optical and mechanical noise in the system, the measurements are averaged over an ensemble of 80 realizations at each scan point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Bear in mind that both the direct inversion and PINNs deploy the spectra of normalized velocity fields vobs for data inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Such distributions of out-of-plane particle velocity at 165 kHz (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 80 kHz) in Π exp 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Π exp 2 ) is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It should be mentioned that in the above experiments, the magnitude of measured signals in terms of displacement is of O(nm) so that it may be appropriate to assume a linear regime of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The nature of antiplane wave motion is dispersive nonetheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Therefore, to determine the relevant length scales in each component, the associated dispersion curves are obtained as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 19 via a set of complementary experiments described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Accordingly, for excitations of center frequency {fc1, fc2, fc3} = {165, 80, 300} kHz, the affiliated phase velocity cµ and wavelength λµ for µ = {Al, Ti, St, Br} is identified in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Figure 15: Test set-ups for ultrasonic full-field characterization: (a) an Al plate Π exp 1 is subject to antiplane shear waves at 165 kHz by a piezoelectric transducer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' the out-of-plane particle velocity field is then captured by a laser Doppler vibrometer scanning on a robot over the observation surface, and (b) a Ti-St-Br plate Π exp 2 undergoes a similar test at 80 kHz and 300 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 14 exp 2 exp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='.239 Ti St Br (a) (b)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Dimensional framework On recalling Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2, let ℓr : = λAl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='01 m, µr : = EAl = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='9 GPA, and ρr : = ρAl = 2700 kg/m3 be the reference scales for length, stress, and mass density, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this setting, the following maps take the physical quantities to their dimensionless values (ρµ, Eµ, νµ) → (ρµ, Eµ, νµ) := � 1 ρr ρµ, 1 µr Eµ, νµ � , µ = {Al, Ti, St, Br}, (fcι, λµ, cµ) → (fcι, λµ, cµ) := � ℓr � ρr µr fcι, 1 ℓr λµ, � ρr µr cµ � , ι = 1, 2, 3, (h, f, vβ) → (h, f, vβ) := � 1 ℓr h, ℓr � ρr µr f, � ρr µr vβ� , β = 1, 2, (10) where h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 mm and f respectively indicate the specimen’s thickness and cyclic frequency of wave motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Table 5 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Table 6) details the normal values for the first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' second) of (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The normal thickness and center frequencies are as follows, {fc1, fc2, fc3} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='16, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='59}, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' (11) Table 5: Properties of the aluminum, titanium, steel and brass sheets as per the manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Here, χµ := Eµ/ρµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' physical µ Al Ti St Br Eµ [GPA] 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='9 105 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='95 110 quantity ρµ [kg/m3] 2700 4510 7850 8530 νµ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='31 normal Eµ 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='60 value ρµ 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='67 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='16 χµ 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='91 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='51 Table 6: Phase velocity cµ and wavelength λµ in µ = {Al, Ti, St, Br} at {fc1, fc2, fc3} = {165, 80, 300} kHz as per Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 19, and their normalized counterparts according to (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' physical quantity µ Al Ti St Br λµ(fc1) [cm] 1 − − − cµ(fc1) [m/s] 1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 − − − λµ(fc2) [cm] − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='17 cµ(fc2) [m/s] − 1140 1126 936 λµ(fc3) [cm] − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 cµ(fc3) [m/s] − 1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 1929 1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 normal value µ Al Ti St Br λµ(fc1) 1 − − − cµ(fc1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='32 − − − λµ(fc2) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='17 cµ(fc2) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='19 λµ(fc3) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 cµ(fc3) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Governing equation In light of (11) and Table 6, observe that in all tests the wavelength-to-thickness ratio λµ h ∈ [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='33 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='33], µ = {Al, Ti, St, Br}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Therefore, one may invoke the equation governing flexural waves in thin plates [53] to approximate the physics of measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this framework, (1) may be recast as Λ = Λβ := χβh3 12(1 − ν2 β)∇4 − h(2πf)2, χβ := Eβ ρβ , β = 1, 2, ˆu = vβ(ξ, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' τ), ϑ = χβ(ξ, f), ξ ∈ Sobs, τ ∈ Sinc, f ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2]fcι, ι = 1, 2, 3, (12) where ρβ, Eβ, νβ respectively denote the normal density, Young’s modulus, and Poisson’s ratio in Π exp β , β = 1, 2, and τ indicates the source location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Note that νβ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='32 according to Table 5 and Λ, related to 1 − ν2 β, 15 shows little sensitivity to small variations in the Poisson’s ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Thus, in what follows, νβ is treated as a known parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Provided vβ(ξ, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' τ), the objective is to reconstruct χβ(ξ, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Direct inversion Following the reconstruction procedure of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2, the distribution of χβ in Gβ, β = 1, 2, is obtained at specific frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this vein, the positive-definite MLP networks χ⋆ β = Nχβ(ξ, ω) and α⋆ = Nα(ξ, ω) comprised of three hidden layers of respectively 20, 40, and 20 neurons are constructed according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In all MLP trainings of this section, each epoch makes use of the full dataset and the learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' When β = 1, the inversion is conducted at f1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Sinc is sampled at one point i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', the piezoelectric transducer remains fixed during the test on Al plate, and thus, Nτ = 1, while a concentric 60×60 subset of collocation points sampling Sobs is deployed for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this setting, the input to χ⋆ 1 and α⋆ is of size NξNτ × Nω = 3600 × 1, and their real-valued outputs are of the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' When β = 2, the direct inversion is conducted at f2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='17 and f3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' For the low-frequency reconstruction, Sinc is sampled at one point, while a 40×120 subset of scan points in G2 is used for training so that the input/output size for χ⋆ 2 and α⋆ is 4600×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The recovered fields and associated normal error are provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Table 7 enlists the true values as well as mean and standard deviation of the reconstructed distributions χ⋆ β in Π exp β , β = 1, 2, according to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 16 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' For the high-frequency reconstruction, when β = 2, Sinc is sampled at three points i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', experiments are performed for three distinct positions of the piezoelectric transducer, while the same subset of scan points is used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this case, the input to χ⋆ 2 and α⋆ is 13800×1, while their output is of dimension 4600×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The high-frequency reconstruction results are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 18, and the affiliated means and standard deviations are provided in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It should be mentioned that the computed normal errors in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 16, 17, and 18 are with respect to the verified values of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Note that the recovered α⋆s from laboratory test data are much smoother than the ones reconstructed from synthetic data in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This could be attributed to the scaler nature of (12) with a single unknown parameter – as opposed to the vector equations governing the in-plane wave motion with two unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' More specifically, here, α⋆ controls the weights and biases of a single network χ⋆ β, while in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2, α⋆ simultaneously controls the parameters of two separate networks µ⋆ and λ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' A comparative analysis of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 17 and 18 reveals that (a) enriching the waveform data by increasing the number of sources remarkably decrease the reconstruction error,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' (b) the regularization parameter α in (3) is truly distributed in nature as the magnitude of the recovered α⋆ in brass is ten times greater than that of titanium and steel which is due to the difference in the level of noise in measurements related to distinct material surfaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' and (c) the recovered field χ⋆ 2 – which according to (12) is a material property E2/ρ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' demonstrates a significant dependence to the reconstruction frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The latter calls for proper verification of the results which is the subject of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Verification To shine some light on the nature discrepancies between the low- and high- frequency reconstructions in Figure 16: Direct inversion of the PDE parameter χ1 in Π exp 1 using test data from a single source at frequency f1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='336: (a) MLP- predicted distribution χ1(ξ, f1) in ξ ∈ G1, (b) reconstruction error (8) with respect to the true value χ1 = χAl = 1, (c) MLP- recovered distribution of the regularization parameter α⋆, and (d) loss function Lε vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' the number of epochs Ne in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 16 (a) (b) (c) ×10-3 (d) 三(x1) X1 α* 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='06 log(Le) 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='04 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='04 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='02 2 6 Ne ELLLEFE 0 2 ×103 4Figure 17: Direct inversion of the PDE parameter χ2 in Π exp 2 using test data from a single source at frequency f2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='17: (a) MLP- predicted distribution χ2(ξ, f2) in ξ ∈ G2, (b) reconstruction error (8) with respect to the true value χ2 ∈ {χTi, χSt, χBr} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='91, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='51} as per Table 5, (c) MLP-recovered distribution of the regularization parameter α⋆, and (d) loss function Lε vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' the number of epochs Ne in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Table 7: Mean and standard deviation of the reconstructed distributions in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 16 and 17 via the direct inversion of single-source test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' β 1 2Ti 2St 2Br χβ 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='91 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='51 ⟨χ⋆ β⟩Πexp β 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='872 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='443 σ(χ⋆ β|Πexp β ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='052 Figure 18: Direct inversion of the PDE parameter χ2 in Π exp 2 using test data from three source locations at frequency f3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='61: (a) MLP-predicted distribution χ2(ξ, f3) in ξ ∈ G2, (b) reconstruction error (8) with respect to the related estimates {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='57, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='59, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='24} as per Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 20, (c) MLP-recovered distribution of the regularization parameter α⋆, and (d) loss function Lε vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' the number of epochs Ne in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Table 8: Mean and standard deviation of the reconstructed distributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 18 via the direct inversion applied to high-frequency test data from three distinct sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' β 2Ti 2St 2Br χ′ β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='24 ⟨χ⋆ β⟩Πexp β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='585 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='227 σ(χ⋆ β|Πexp β ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='016 Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 17 and 18, a set of secondary tests are performed to obtain the dispersion curve for each component of the test setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' For this purpose, antiplane shear waves of form (9) are induced at fcj = 50j kHz, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' , 7, 17 (a) x2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='9 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='7 (c) log(Le) 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 Φ (b) 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5 三(x2) 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 ×10-3 0 8 ×103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1 Ne(a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 x2 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 (c) log(Le) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 2 三(x2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='08 ×10-3 0 4 8 ×103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='04 NeFigure 19: Experimental vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' theoretical dispersion curves f(λ−1 µ ) for µ = {Al, Ti, St, Br}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Analytical curves (solid lines) are computed from (13) using the pertinent properties in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Figure 20: Discrepancy in the balance law (12) at f3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='61: (a) elastic force field T1 µ, µ = {Ti, St, Br}, according to (14) with adjusted coefficients {χTi, χSt, χBr} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='57, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='59, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='24}, (b) the inertia field T2 µ, and (c) normal discrepancy Dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' in 60 cm × 60 cm cuts of aluminum, titanium, steel, and brass sheets used in the primary tests of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In each experiment, the piezoelectric transducer is placed in the middle of specimen (far from the external boundary), and the out-of-plane wave motion is captured in the immediate vicinity of the transducer along a straight line of length 8 cm sampled at 400 scan points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The Fourier-transformed signals in time-space furnish the dispersion relations of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In parallel, the theoretical dispersion curves affiliated with (12) are computed according to f = 2π(λµ)−2 � χµh2 12(1 − ν2µ), χµ = Eµ ρµ , µ = {Al, Ti, St, Br}, (13) using the values of Table 5 for χµ and νµ and h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' A comparison between the experimental and theoretical dispersion curves f(λ−1 µ ) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 19 verifies the theory and the values of Table 5 for χµ in the low- frequency regime of wave motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This is also in agreement with the direct inversion results of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 16 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Moreover, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 19 suggests that at approximately fµ = {170, 200, 120, 110} kHz for µ = {Al, Ti, St, Br} the governing PDE (12) with physical coefficients fails to predict the experimental results which may provide an insight regarding the high-frequency reconstruction results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Further investigation of the balance law (12), as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 20, shows that the test data at 312 kHz satisfy – with less than 10 − 20% discrepancy depending on the material – a PDE of form (12) with modified coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' More specifically, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 20 demonstrates the achievable balance between the elastic force distribution T1 µ and inertia field T2 µ in (12) by directly adjusting the PDE parameter χ′ 2 to minimize the discrepancy Dµ according to T1 µ := χ′ 2h3 12(1 − ν2 2 )∇4v2, T2 µ := h(2πf)2v2, Dµ := |T1 µ − T2 µ| max |T2µ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' (14) 18 ×106 1 T Br Al 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 f [s-1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='810 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 1 ×103 入=1 [m-1](a) (c) 2 ①μ 0 (b) Ti St Br 2 ×10-1 Ti St BrWith reference to Table 8, the recovered coefficients χ′ 2 at f = f3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='61 verify the direct inversion results of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This implies that the direct inversion (or PINNs) may lead to non-physical reconstructions in order to attain the best fit for the data to the “perceived”” underlying physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Thus, it is imperative to establish the range of validity of the prescribed physical principles in data-driven modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Here, the physics of the system at f3 is in transition, yet close enough to the leading-order approximation (12) that the discrepancy is less than 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It is unclear, however, if this equation with non-physical coefficients may be used as a predictive tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It would be interesting to further investigate the results through the prism of higher-order continuum theories and a set of independent experiments for validation which could be the subject of a future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Physics-informed neural networks Following Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3, PINNs are built and trained using experimental test data of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The MLP network v1⋆ = v1⋆(ξ, f, x|γ, χ⋆ 1) with six hidden layers of respectively 40, 40, 120, 80, 40, and 40 neurons is constructed to map the out-of-plane velocity field v1 (in Π exp 1 ) related to a single transducer location x1 and frequency f1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The PDE parameter χ1 is defined as the unknown scaler parameter of the network, and following the argument of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3, the Lagrange multiplier γ is specified as a nonadaptive scaler weight of magnitude 1 h(2πf1)2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The input/output dimension for v1⋆ is Nξ×Nω×Nτ = 3600×1×1, and each epoch makes use of the full dataset for training and the learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Keep in mind that the objective here is to (a) construct a surrogate map for v1, and (b) identify χ⋆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 21 demonstrates (a) the accuracy of PINN-estimated field v1⋆ compared to the test data v1, (b) performance of automatic differentiation in capturing the fourth-order field derivatives e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=', v1⋆ ,1111 that appear in the governing PDE (12), and (c) the evolution of parameter χ⋆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The comparison in (b) is with respect to the spectral derivates of test data according to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It is no surprise that the automatic differentiation incurs greater errors in estimating the higher order derivatives involved in the antiplane wave motion compared to the second-order derivatives of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In addition, the PINN v2⋆ = v2⋆(ξ, f, x|γ, χ⋆ 2) with seven hidden layers of respectively 40, 40, 120, 120, 80, 40, and 40 neurons is created to reconstruct (i) particle velocity field v2 in the layered specimen Π exp 2 , and (ii) distribution of the PDE parameter χ2 in the sampling area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The latter is defined as an unknown parameter of the network with dimension 40×120, and the scaler weight γ is set to 1 h(2πf2)2 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='84 for the low-frequency reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this setting, the input/output dimension for v2⋆ reads 4800×1×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 22 provides a comparative analysis between the experimental and PINN-predicted maps of velocity and PDE parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' The associated statistics are provided in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' It is evident from the waveform in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 22 (a) that the most pronounced errors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 22 (d) occur at the loci of vanishing particle velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Similar to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2, this could be potentially addressed by enriching the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Conclusions The ML-based direct inversion and physics-informed neural networks are investigated for full-field ultra- sonic characterization of layered components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Direct inversion makes use of signal processing tools to directly compute the field derivatives from dense datasets furnished by laser-based ultrasonic experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This allows for a simplified and controlled learning process that specifically recovers the sought-for physical fields through minimizing a single-objective loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' PINNs are by design more versatile and particularly advantageous with limited test data where waveform completion is desired (or required) for mechanical characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' PINNs multi-objective learning from ultrasonic data may be more complex but can be accomplished via carefully gauged loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In direct inversion, Tikhonov regularization is critical for stable reconstruction of distributed PDE param- eters from limited or multi-fidelity experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In this vein, deep learning offers a unique opportunity to simultaneously recover the regularization parameter as an auxiliary field which proved to be particularly insightful in inversion of experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In training PINNs, two strategies were remarkably helpful: (1) identifying the reference length scale by the dominant wavelength in an effort to control the norm of spatial derivatives – which turned out to be crucial in the case of flexural waves in thin plates with the higher order PDE, and (2) estimating the Lagrange multiplier by taking advantage of the inertia term in the governing PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 19 Figure 21: PINN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' experimental maps of particle velocity and its derivatives in Π exp 1 : (a) PINN estimates for {v1⋆, v1⋆ ,1111, v1⋆ ,2222, v1⋆ ,1122} wherein the derivatives are obtained by automatic differentiation, (b) normalized LDV-captured par- ticle velocity field v1 and its corresponding spectral derivatives, (c) normal misfit 8 between (a) and (b), (d) PINN-reconstructed PDE parameter χ⋆ 1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' the number of epochs Ne, and (e) total loss Lϖ and its components (the PDE residue and data misfit) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Ne in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Laboratory implementations at multiple frequencies exposed that verification and validation are indis- pensable for predictive data-driven modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' More specifically, both direct inversion and PINNs recover the unknown “physical” quantities that best fit the data to specific equations (with often unspecified range of va- lidity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This may lead to mathematically decent but physically incompatible reconstructions especially when the perceived physical laws are near their limits such that the discrepancy in capturing the actual physics is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' In which case, the inversion algorithms try to compensate for this discrepancy by adjusting the PDE parameters which leads to non-physical reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Thus, it is paramount to conduct comple- mentary experiments to (a) establish the applicability of prescribed PDEs, and (b) validate the predictive capabilities of the reconstructed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Authors’ contributions Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' investigation, methodology, data curation, software, visualization, writing – original draft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' con- ceptualization, methodology, funding acquisition, supervision, writing – original draft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' experimental data curation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' experimental data curation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 20 1× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='1* 1 * 1* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 (a) 0 0 0 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 v,1111 V,2222 v,1122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 (b) 0 0 0 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 1 8 三(v,1111) 3 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='L 1 * 6 6 6 4 4 (c) 4 2 2 2 ×10-3 ×10-1 ×10-1 ×10-1 1 x1 PDE loss 2 data loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 total loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 (d) (e) 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 MA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 6 Ne Ne 0 ×105 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='8 1Figure 22: Low-frequency PINN reconstruction in Π exp 2 using test data from a single source at f2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='17: (a) PINN-predicted distri- bution of particle velocity v2⋆, (b) normalized LDV-captured particle velocity v2, (c) normal misfit between (a) and (b), (d) PINN- predicted distribution of the PDE parameter χ⋆ 2, and (e) total loss Lϖ and its components (the PDE residue and data misfit) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' the number of epochs Ne in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' Table 9: Mean and standard deviation of the PINN-reconstructed distributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 22 from a single-source, low- frequency test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' β 2Ti 2St 2Br χβ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='91 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='51 ⟨χ⋆ β⟩Πexp β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='790 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='890 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='414 σ(χ⋆ β|Πexp β ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content='134 Acknowledgments This study was funded by the National Science Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' 1944812) and the University of Colorado Boulder through FP’s startup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE0T4oBgHgl3EQfdgCu/content/2301.02378v1.pdf'} +page_content=' This work utilized 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sha256:2e229130f0a2830929f13145af656bae7d6b98bef08dfeeb167c708a34c4d75e +size 122815 diff --git a/6NE5T4oBgHgl3EQfPg5N/content/tmp_files/2301.05505v1.pdf.txt b/6NE5T4oBgHgl3EQfPg5N/content/tmp_files/2301.05505v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2213e33ffe21886f8177fc0f25127e5f0a5bfdcd --- /dev/null +++ b/6NE5T4oBgHgl3EQfPg5N/content/tmp_files/2301.05505v1.pdf.txt @@ -0,0 +1,1019 @@ +Astronomy & Astrophysics manuscript no. dust_filtering +©ESO 2023 +January 16, 2023 +Leaky Dust Traps: How Fragmentation impacts Dust Filtering by +Planets +Sebastian Markus Stammler1, Tim Lichtenberg2, Joanna Dr˛a˙zkowska3, and Tilman Birnstiel1, 4 +1 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679, Munich, Germany +2 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands +3 Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany +4 Exzellenzcluster ORIGINS, Boltzmannstr. 2, D-85748 Garching, Germany +January 16, 2023 +ABSTRACT +The nucleosynthetic isotope dichotomy between carbonaceous and non-carbonaceous meteorites has been interpreted as evidence for +spatial separation and coexistence of two distinct planet-forming reservoirs for several million years in the solar protoplanetary disk. +Rapid formation of Jupiter’s core within one million years after CAIs has been suggested as a potential mechanism for spatial and +temporal separation. In this scenario, Jupiter’s core would open a gap in the disk and trap inwards-drifting dust grains in the pressure +bump at the outer edge of the gap, separating the inner and outer disk materials from each other. We performed simulations of dust +particles in a protoplanetary disk with a gap opened by an early formed Jupiter core, including dust growth and fragmentation as well +as dust transport using the dust evolution software DustPy. Our numerical experiments indicate that particles trapped in the outer +edge of the gap rapidly fragment and are transported through the gap, contaminating the inner disk with outer disk materials on a +timescale that is inconsistent with the meteoritic record. This suggests that other processes must have initiated or at least contributed +to the isotopic separation between the inner and outer Solar System. +Key words. Meteorites, meteors, meteoroids — Methods: numerical — Protoplanetary disks — Planets and satellites: formation – +Planets and satellites: composition +1. Introduction +Recent high-precision isotopic measurements reveal a di- +chotomy between carbonaceous and non-carbonaceous mete- +orites indicating that both have been formed in separate reser- +voirs within the early Solar System (Trinquier et al. 2007, 2009; +Leya et al. 2009; Warren 2011; Mezger et al. 2020; Kleine et al. +2020). Kruijer et al. (2017) and Desch et al. (2018) argued that +these reservoirs must have been well separated for at least two +million years without interchanging solid material, proposing the +rapid formation of Jupiter’s core opening a gap in the protoplan- +etary disk as possible mechanism to prevent the mixing of both +reservoirs. The physical origin of the isotopic separation is a po- +tential critical clue to the timescales of planet formation in both +the inner and outer Solar System (Nimmo et al. 2018), and thus +ultimately the origin of the chemical abundances in the terres- +trial planets and similar exoplanets (Krijt et al. 2022; Lichten- +berg et al. 2022). +This concept of a gap-opening Jupiter preventing dust reser- +voir mixing, however, intimately depends on the evolution of the +dust flux during the evolution of the protoplanetary disk. Dust +particles in protoplanetary disks are subject to gas drag and drift +(Whipple 1973; Weidenschilling 1977; Takeuchi & Lin 2002). +The radial dust velocity is given by: +vd = vg +1 +St2 + 1 ++ 2vP +St +St2 + 1 +. +(1) +The Stokes number St is an aerodynamic measure and propor- +tional to the particle size. Small particles with small Stokes num- +bers are dragged along with the gas with velocity vg as can be +seen by Equation 1. The gas is, in contrast to the dust, pressure +supported and orbits the star with sub-Keplerian velocities in a +typical smooth disk with inward pointing pressure gradient. The +dust particles, on the other hand, are not pressure supported, ex- +change angular momentum with the gas and drift in direction of +pressure gradients. Intermediate particle sizes are most affected +by this effect. Small particles are well coupled to the gas, while +large particles are completely decoupled. From Equation 1 it can +be seen that particles with Stokes number of unity will experi- +ence maximum drift in direction of the pressure gradient with +velocity vP, which is given by: +vP = 1 +2 +c2 +s +vK +∂ log P +∂ log r , +(2) +with the sound speed cs, pressure P, and the Keplerian velocity +vK. Particles typically grow to maximum sizes with Stokes num- +bers between 10−2 to 10−1 (see Birnstiel et al. 2012), depending +on the disk parameters, and are therefore affected by radial drift. +Growing planets can perturb the pressure structure in the disk +by opening a gap in the gas (Paardekooper & Mellema 2006; +Rice et al. 2006). At the outer edge of the gap the pressure gradi- +ent reverses and is pointing outward. If the pressure pertubation +is large enough, large dust pebbles that are affected by drift can +be prevented from crossing the gap. The planetary mass at which +the pressure pertubation is large enough to stop particle dift is +called pebble isolation mass (see Lambrechts et al. 2014; Bitsch +et al. 2018) and is given by Drazkowska et al. (2022) as: +Miso ≃ 25 M⊕ +�HP/r +0.05 +�3 M⋆ +M⊙ +. +(3) +Article number, page 1 of 8 +arXiv:2301.05505v1 [astro-ph.EP] 13 Jan 2023 + +A&A proofs: manuscript no. dust_filtering +From NASA’s Juno mission Jupiter’s core is estimated to +have a mass of up to 25 M⊕ (Wahl et al. 2017) and would have +therefore been able to open a gap and stop the flux of dust peb- +bles in the disk. A rapid formation of Jupiter’s core could there- +fore explain two isolated dust reservoirs with the dust in the outer +disk forming the carbonaceous and the dust in the inner disk the +non-carbonaceous bodies in the Solar System. +Dr˛a˙zkowska et al. (2019), however, showed in two- +dimensional hydrodynamic simulations of gas and dust includ- +ing collisional dust evolution, that the pressure bump at the outer +edge of planetary gaps does not only show an accumulation of +large dust pebbles, but also of small dust particles. But in con- +trast to large pebbles, these small particles are not trapped by +the pressure bump, they are produced in situ by collisions of +large particles leading to fragmentation. These small fragments +can escape the pressure bump due to diffusion and gas drag. The +equations of motion of the dust particles are given by: +∂ +∂tΣd + 1 +r +∂ +∂r +� +rΣdvd − rDΣg +∂ +∂r +�Σd +Σg +�� += 0, +(4) +with the dust diffusivity given by Youdin & Lithwick (2007) as +D = δrc2 +s +ΩK +1 +St2 + 1 +. +(5) +with δr being a free parameter that defines the strength of radial +dust diffusion. Small particles are therefore most affected by dif- +fusion. If the diffusivity is high enough, these small particles can +diffuse out of the pressure maximum and are dragged with the +gas through the gap. If this is the case the inner disk would be +contaminated with dust from the outer disk negating the idea of +two distinct dust reservoirs separated by an early formed Jupiter +core. +In this letter we test this hypothesis. In section 2 we present +a toy model which initially has dust placed only outside of the +planet to show as a proof of concept, that solid material can pen- +etrate planetary gaps if the dust is subject to fragmentation and +diffusion. In section 3 we investigate the influence of the plan- +etary mass and the dust diffusivity on the dust permeability of +planetary gaps. In section 4 we present models with a realis- +tic evolution of the planetary mass, as it has been suggested for +Jupiter, for models with both fragmentation and bouncing. Fi- +nally, in section 5 we discuss our results, before we conclude in +section 6. +2. Toy Model +To investigate the influence of a planet on the dust flux in the +inner disk, we model dust coagulation and transport in a proto- +planetary disk with a planet opening a gap at 5 AU using the dust +evolution software DustPy1 (Stammler & Birnstiel 2022). In a +first simplified toy model, we initialize the disk only with dust +outside of a Saturn mass planet. Therefore, any dust flux mea- +sured inside the planet must have crossed the gap. We use this +simplified model to investigate different scenarios: dust growth +limited by fragmentation, dust growth limited by bouncing, and +unlimited dust growth. Furthermore, we compare the toy model +to a model without a gap. +We initialize the gas surface density with the self similar so- +lution of Lynden-Bell & Pringle (1974): +Σg (r) = Mdisk +2πr2c +� r +rc +�−1 +exp +� +− r +rc +� +(6) +1 DustPy v1.0.1 has been used for the simulations presented in this +work. +with a cutoff radius of rc = 30 AU and an initial disk mass of +Mdisk = 0.05 M⊙. We impose a gap onto this gas surface density +profile originating from a Saturn mass planet located at 5 AU, for +which we use the gap profile fits provided by Kanagawa et al. +(2017). To maintain this gap profile F (r) throughout the sim- +ulation we impose the inverse of this profile onto the turbulent +viscosity parameter α, since the product of gas surface density +and viscosity is constant in quasi steady-state: +α (r) = +α0 +F (r). +(7) +In the default setup, we use α0 = δr = 10−3. Please note, that this +change in α (r) does not affect the turbulent diffusion of the dust +particles, since δr is a constant in our models. +We initialize the dust surface density with a constant gas-to- +dust ratio of 100 and the dust size distribution according Mathis +et al. (1977) as n (a) = a−3.5 with a maximum initial particle size +of 1 µm. In the toy model we initially have dust only outside of +15 AU. +DustPy simulates dust growth by solving the Smoluchowski +equation of a dust mass distribution. Dust transport is simulated +by solving Equation 4 for every dust size individually. +The gas surface density is evolved by solving the viscous +advection-diffusion equation +∂ +∂tΣg + 1 +r +∂ +∂r +� +rΣgvg +� += 0 +(8) +with the gas velocity given by Lynden-Bell & Pringle (1974) as +vg = − +3 +Σg +√r +∂ +∂r +� +Σgν √r +� +(9) +and the kinematic viscosity given by +ν = αcsHP +(10) +with the sound speed cs = +� +kBT/µ, the pressure scale height +HP = cs/ΩK, and the viscosity parameter α given by Equation 7. +We run five different flavors of the toy model: one with a +fragmentation velocity of vfrag = 10 m/s (fiducial), one with no +fragmentation at all, one with a fragmentation velocity of 1 m/s, +one with bouncing as described by Windmark et al. (2012), and +one without a gap, i.e. F (r) = 1. In the default collision model +used by DustPy particles fragment once their relative collision +velocities exceed the fragmentation velocity. Fragmenting colli- +sions of equal size particles lead to catastrophic fragmentation +of both collision partners. If the target particle is significantly +larger, only the projectile particle fragments entirely while erod- +ing mass off the target particle (Schräpler et al. 2018; Hasegawa +et al. 2021). The transition between pure sticking and fragmenta- +tion is smooth, since DustPy is assuming a velocity distribution +of possible collision velocities. For details on the collision model +we refer to Stammler & Birnstiel (2022). +Panel A of Figure 1 shows the initial dust distribution with +dust located outside of 15 AU with particles sizes up to 1 µm. +The white lines are contour lines of Stokes numbers St = +� +10−3, 10−2, 10−1, 100� +with the bold white line corresponding to +St = 1. Panel B shows the fiducial simulation with a Saturn mass +planet at 5 AU and the fragmentation velocity vfrag = 10 m/s +after 1 Myr. Particles trapped in the pressure bump outside the +planetary gap can reach sizes with Stokes numbers of up to +St = 10−1 corresponding to particle sizes of a few centimeters. +It can be seen that even small particles are accumulated in the +pressure bump, even though their Stokes numbers are too small +Article number, page 2 of 8 + +Sebastian Markus Stammler et al.: Leaky Dust Traps: How Fragmentation impacts Dust Filtering by Planets +101 +102 +Distance from star [AU] +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Particle size [cm] +A: initial +101 +102 +Distance from star [AU] +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Particle size [cm] +B: fiducial +101 +102 +Distance from star [AU] +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Particle size [cm] +C: without planet +101 +102 +Distance from star [AU] +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Particle size [cm] +D: no fragmentation +101 +102 +Distance from star [AU] +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Particle size [cm] +E: vfrag = 1 m/s +101 +102 +Distance from star [AU] +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Particle size [cm] +F: bouncing +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +dust [g/cm²] +Fig. 1. Panel A: Initial dust distribution. The white lines correspond to Stokes numbers of St = +� +10−3, 10−2, 10−1, 100� +with the bold white line +corresponding to St = 1. All other panels show snapshots of models at 1 Myr. Panel B: The fiducial toy model with a Saturn mass planet at +5 AU and a fragmentation velocity of 10 m/s. Panel C: Model without a planet. The vertical dashed lines are the location at which the dust flux +is measured in Figure 2. Panel D: Model without fragmentation. Panel E: Model with a reduced fragmentation velocity of 1 m/s. Bottom right: +Model with bouncing instead of fragmentation. +to be affected by drift. These small dust particles are produced +by collisional fragmentation of larger particles trapped in the +bump. They diffuse out of the bump and are dragged with the +gas contaminating the inner disk with outer disk material. It can +be seen that particles with Stokes numbers of about St = 10−2, +corresponding to particle sizes of a few millimeter, can diffuse +through the gap into the inner disk. Particles in the inner disk +can again grow to centimeter sizes and can contribute to phe- +nomena like the streaming instability or pebble accretion. Panel +C shows a simulation with identical initial conditions but with- +out a planet opening a gap. The vertical yellow and green dashed +lines in panels B and C are the locations at which the dust fluxes +shown in Figure 2 are measured. +The dust fluxes at the outer disk are identical in both sim- +ulations with the solid and dashed green lines overlapping in +Figure 2. The fluxes in the inner disk, however, differ in both +simulations. The onset of dust flux in the inner disk in the sim- +ulation with a planet is delayed by about 20 000 yr compared to +the simulations without a planet. Without a planet, the large dust +particles can freely drift into the inner disk. With a planet, how- +ever, they are first trapped in the pressure bump at the outer edge +of the gap, fragment down to smaller sizes, and diffuse out of the +pressure bump before the gas can drag them into the inner disk +where they grow to larger particles again. Due to this delayed +processing the maximum dust flux is reduced by about one or- +der of magnitude. The duration, however, is prolonged such that +Article number, page 3 of 8 + +A&A proofs: manuscript no. dust_filtering +103 +104 +105 +106 +107 +Time [yrs] +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +Dust flux [M +/yr] +r = 2 AU +r = 15 AU +with planet +w/o planet +103 +104 +105 +106 +107 +Time [yrs] +10 +2 +10 +1 +100 +Fraction of total dust mass accreted +Fig. 2. Top: Comparison of the dust flux in the inner disk (at 2 AU) +and outer disk (at 15 AU) in the toy model with a Saturn mass planet +at 5 AU (panel B in Figure 1) and a model without a planet (panel C +in Figure 1). Both green 15 AU lines overlap. Bottom: Total dust mass +accreted through the inner disk over time. +the total mass of dust flowing through the inner disk is identical +after 10 Myr as can be seen in the bottom panel of Figure 2. The +Saturn mass planet did not separate the inner from outer disk +material, but only delayed the material transport. +Panel D of Figure 1 shows a simulation without fragmen- +tation. In this scenario, particles sizes are limited only by the +radial drift, consistent with the model presented by Kobayashi & +Tanaka (2021). In the center of the pressure bump, the pressure +gradient is zero and the growth is in principle unlimited until the +particles accumulate at the upper end of the simulation grid. This +scenario most closely represents the separation of inner and outer +dust reservoirs with only very few particles being able to diffuse +through the gap, because they were not able to grow to large par- +ticles quickly enough. It is, however, rather unlikely that the dust +particles do not fragment or get eroded at some point given the +relative velocities they typically experience (see Blum & Münch +1993; Wada et al. 2009; Schräpler et al. 2018). +Panel E of Figure 1 shows a model with a fragmentation ve- +locity of 1 m/s as indicated by recent experiments (see e.g. Blum +2018; Gundlach et al. 2018; Musiolik & Wurm 2019). In this +case, the particles cannot reach particles sizes large enough to be +efficiently trapped in the pressure bump. +The objective is therefore to halt particle growth without pro- +ducing small particles. This can be achieved if the growth is lim- +ited by bouncing, when particles simply bounce of each other +without growing or fragmenting. Panel F of Figure 1 shows a +simulation with the bouncing barrier implemented as described +by Windmark et al. (2012). In this model, bouncing starts when +103 +104 +105 +106 +107 +Time [yrs] +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +Pebble flux [M +/yr] +103 +104 +105 +106 +107 +Time [yrs] +10 +4 +10 +3 +10 +2 +10 +1 +100 +Fraction of dust mass accreted +no planet +30 M +50 M +Msat, +r = 10 +2 +Msat, +r = 10 +3 +Msat, +r = 10 +4 +Msat, +r = 10 +5 +200 M +Mjup +Fig. 3. Top: Dust flux through the planetary gap in models with different +planet masses. The blue line is for a model without a planet. The dashed, +dotted, and dash-dotted red lines show additional simulations with a +Saturn mass planet for different radial dust diffusivity parameters δr. +Bottom: Total fraction of outer dust mass accreted through the planetary +gap. +the relative velocity reaches a few centimeters per second. In this +case, however, the particles only reach sizes of a few 100 µm +corresponding to Stokes numbers lower than 10−3, which is too +small to be efficiently trapped in the pressure bump created by +the planet. The particles can diffuse through the gap and contam- +inate the inner disk. +3. Full Disk Models +The toy model in section 2 served as a proof of concept that +planets do not prevent dust flux if particles are subject to frag- +mentation. In this section we discuss full disk models with dif- +ferent planet masses in which dust is initialized in the entire disk +to investigate the dust permeability of the gap. The top panel of +Figure 3 shows the dust flux through the planetary gap for differ- +ent planet masses from 30 Earth masses to one Jupiter mass. In +the case of a Saturn mass planet we additionally performed sim- +ulations with different dust diffusivity parameters δr (see Equa- +tion 5). The bottom panel of Figure 3 shows the total fraction of +outer disk dust material that has been accreted through the gap +over time. In all models the planets have their respective masses +already from the beginning of the simulations. +The smallest planetary mass considered here is 30 M⊕, which +is already higher than the upper estimate of Jupiter’s core mass. +The largest mass considered is 1 Mjup. The smallest planetary +mass is not capable of efficiently suppressing the dust flux +through the gap. After about 300 000 yr almost the entire dust +Article number, page 4 of 8 + +Sebastian Markus Stammler et al.: Leaky Dust Traps: How Fragmentation impacts Dust Filtering by Planets +mass (horizontal line in bottom panel) of the outer disk has been +accreted through the gap. Increasing the planetary mass simply +delays the accretion time, but is not able to prevent accretion. +The maximum delay of accretion seems to be achieved already +with a 200 M⊕ planet. Increasing the planet mass further to a +Jupiter mass planet does not significantly change the accretion +history. At the end of the simulation at 10 Myr about 80 % of the +dust mass has been accreted through the gap. +The dust diffusivity δr has a more significant influence on the +accretion. Increasing the diffusivity by a factor of 10 to δr = 10−2 +in the Saturn mass simulation has the same effect as reducing the +planet mass by a factor of about 2, mimicking the accretion his- +tory of a 40 M⊕ planet with diffusivity of δr = 10−3. Note that we +only changed δr, while keeping α0 = 10−3 and therefore keeping +the shape of planetary gap. The relative collision velocities of +the dust particles are not affected by this change in δr. Decreas- +ing δr by a factor of 10 is more efficient in retaining the dust +than having a Jupiter mass planet with the standard diffusivity. +In this case only about 10 % of the dust mass has been accreted at +the end of the simulation after 10 Myr. Lowering the diffusivity +even further to δr = 10−5 reduces the dust permeability further +to a about 5 % of the outer disk mass after 10 Myr. It is however +noted that the fraction of outer disk material present in the inner +disk is usually significantly larger, since the inner disk material +is accreted onto the star on short timescales and only re-supplied +with outer disk material. +4. Time-dependent planet mass +In the previous models we assumed that the planets are fully +formed from the beginning of the simulation and the planet mass +does not evolve over time. Kruijer et al. (2017) argue that the +two dust reservoirs have been separated from about 1 Myr to +3−4 Myr after CAI formation. They therefore claim that Jupiter’s +core must have been massive enough to open a gap at 1 Myr +and must have reached a mass of about 50 M⊕ after 4 Myr to be +able to scatter planetesimals from the outer disk to the inner disk +where they are observed today in the asteroid belt. We there- +fore performed simulations with a time-dependent planet mass +as shown in the top left panel of Figure 4. The solid blue line +shows an evolutionary track where the planet reaches 30 M⊕ af- +ter 1 Myr, 50 M⊕ after 4 Myr and a final mass of Mjup at the end +of the simulation after 10 Myr. +The bottom left panel of Figure 4 shows the fraction of mass +accreted through the planetary gap normalized to the dust mass +in the outer disk at 1 Myr when the planet was massive enough +to open a gap. We performed simulations with different values of +the dust diffusivity δr between 10−5 and 10−3. In the standard run +with δr = 10−3 about 80 % of the dust mass has been accreted +through the gap after 4 Myr (vertical solid line) when the assem- +bly of the meteorite parent bodies has been completed. Even in +the low diffusivity run with δr = 10−5 about 60 % of the mass has +been accreted though the gap between 1 Myr and 4 Myr, strongly +contaminating the inner disk with dust from the outer reservoir +on a system-wide scale. Lowering the dust diffusivity to very low +values does not help keeping both reservoirs separated, since the +planet mass is too low in this scenario. +The bottom right panel of Figure 4 shows a model with +bouncing instead of fragmentation. The solid blue line shows a +model with radial dust diffusivity δr = 10−3. As already shown in +section 2, this is not sufficient to stop dust accretion through the +gap. Only after 7 Myr when the planet already reached a mass +of about 200 M⊕ the gap is deep enough and accretion is halted. +Allowing the planet to reach these masses at earlier times would, +however, not change the dust redistribution, since these massive +planets are able to scatter planetesimals from the outer disk into +the inner disk, which is inconsistent with observations from the +meteoritic record at these early times (Deienno et al. 2022). +The green solid line shows a model with δr = δt = δz = 10−5. +The parameters δt and δz are similar to δr and parametrize the +strength of turbulent motion and vertical settling of the particles +(see Stammler & Birnstiel 2022; Pinilla et al. 2021, for details). +In that way the relative velocities between the particles are re- +duced, allowing them to grow to larger sizes before being lim- +ited by bouncing. They can therefore be trapped by gaps created +by smaller mass planets. However, even in that case accretion is +only halted after abut 3 Myr, when the planet reached a mass of +about 40 M⊕. +The dashed green line shows a model where the planet +reaches a mass of 40 M⊕ already after 1 Myr (dashed line in top +left panel of Figure 4). In this case accretion of dust through the +gap is efficiently stopped at 1 Myr. The top right panel of Fig- +ure 4 shows a snapshot of this simulation after 4 Myr. The inner +disk is heavily depleted in dust, all of which has been accreted +onto the star. The dust mass in the inner disk at this stage was +all supplied from the outer disk. Meteoritic bodies formed in the +inner disk would therefore be entirely made out of outer disk +material. +5. Discussion +Isotopic measurements of meteoritic material indicate that me- +teorites must have formed in two dust reservoirs, that coexisted +spatially separated for several million years. The early formation +of Jupiter’s core has been proposed as natural explanation for +the observed separation. A planet exceeding the pebble isolation +mass opens a gap in the gas disk creating a pressure bump at the +outer edge of the gap, which can trap large dust particles. Two- +dimensional hydrodynamical simulations by Dr˛a˙zkowska et al. +(2019) including dust coagulation and fragmentation showed an +overabundance of small dust particles at the location of the pres- +sure bump, which should be too small to be efficiently trapped. +These particles were created in fragmenting collision of large +dust pebbles that have been trapped in the pressure bump. These +small dust fragments can diffuse out of the bump and can be +dragged by the gas through the gap. +Our simulations in this work suggest that collisional frag- +mentation of dust pebbles in pressure bumps and subsequent dif- +fusion of small fragments can act as a leak for dust traps. As +can be seen by Figure 3, gaps opened by planets can only de- +lay but not fully prevent dust accretion if particles are subject to +fragmentation. To act as an efficient dust barrier, particles need +to grow to large pebbles that can be trapped without producing +small particles as shown in the panel D of Figure 1. +We investigated different planet masses and showed in Fig- +ure 3 that no planet mass was able to completely isolate the inner +disk from outer dust material on timescales that are relevant for +the assumed reservoir separation. Even an initial gap formed by +a fully-grown Jupiter mass planet would leak 20 % of the outer +disk material into the inner disk within 1 Myr. Smaller proto- +Jupiter masses typically lead to complete homogenization within +∼ 105 to at maximum a few 106 yr. This presents a problem +for the suggestion that the age differences in carbonaceus and +non-carbonaceous meteorites may be used as a tracer to track +the growth timescale of proto-Jupiter within the disk (Kruijer +et al. 2017; Alibert et al. 2018): the initial spatial distribution of +nucleosynthetic isotopes at the end of disk infall is degenerate +Article number, page 5 of 8 + +A&A proofs: manuscript no. dust_filtering +0 +2 +4 +6 +8 +10 +Time [Myr] +0 +50 +100 +150 +200 +250 +300 +Planet mass [M +] +default model +rapid early growth +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Time [Myr] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Fraction of dust mass accreted +Fragmentation +r = 10 +3 +r = 10 +4 +r = 10 +5 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Time [Myr] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Fraction of dust mass accreted +Bouncing +i = 10 +3 +i = 10 +5 +i = 10 +5 +101 +102 +Distance from star [AU] +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Particle size [cm] +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +dust [g/cm²] +Fig. 4. Top left: Evolution of the planetary mass in the time-dependent model. The solid line shows the default model where the planet reaches +20 M⊕ at 1 Myrs. The dashed line shows the evolution in a model with rapid early growth in which the planet reaches 40 M⊕ at 1 Myr. Bottom +left: Fraction of outer disk dust mass accreted through the gap after 1 Myr in the default planetary mass evolution model for different values of +dust diffusivity δr with fragmentation limited growth. Bottom right: The solid lines show the fraction of outer disk material accreted through the +gap after 1 Myr for bouncing limited growth for different values of the δi parameters in the default planetary growth model. The dashed green line +shows a model of bouncing limited growth with δi = 10−5 and rapid early growth of the planet (dashed line in top left panel). The vertical lines +mark 4 Myr until which both reservoirs need to be separated. Top right: Snapshot of the dust distribution at 4 Myr for the model with bouncing +limited growth and δi = 10−5 (dashed green line in bottom right panel). The inner disk is depleted in dust and only supplied with small amounts of +outer disk material. +with different Jupiter growth tracks in the Jupiter barrier hypoth- +esis. Only significantly lowering the dust diffusivity to a value +of δr = 10−5 could decrease the dust permeability such that the +inner disk is only contaminated with a few percent of outer disk +material. However, isolating the inner disk from dust flux would +quickly drain the inner disk from solids that got accreted onto +the star, which was also previously noted by Liu et al. (2022). At +later stages the dust in the inner disk then consists to large parts +of outer disk material that has been slowly diffused through the +gap, which is inconsistent with the meteoritic record. +The situation gets worse when using a more realistic evo- +lution of the planetary mass, assuming Jupiter’s core reached a +mass of 20 M⊕ after 1 Myr and 50 M⊕ after 4 Myr. These masses +are not large enough to isolate the inner disk even in models +with very low diffusivity. Even in the most optimistic cases at +least 60 % of the outer disk dust has been accreted through the +planetary gap after 4 Myr as can be seen by Figure 4. However, +increasing the core mass even more and earlier would enable +Jupiter to scatter outer disk planetesimals into the inner disk pol- +luting the inner dust reservoir, which has not been accounted +for in this simple model. Only in models with bouncing lim- +ited growth without small particles, early planetary growth and +reduced relative particle collision velocities, the inner disk can +be efficiently isolated from the inner disk as seen by Figure 4. +In these cases, however, the inner disk is quickly depleted from +dust and only re-supplied from small amount of outer disk ma- +terial. Meteoritic bodies formed in the inner disk after this point +would therefore consist almost entirely of outer disk material. +Dr˛a˙zkowska et al. (2019) noted that the shape of planetary +gaps in two-dimensional simulations is not axisymmetric, which +is ignored in the simple one-dimensional model in this publi- +cation. They further noted, however, that the asymmetry at the +planet location would increase the dust flux through the gap. +Weber et al. (2018) compared one- and two-dimensional simula- +tions of dust transport through planetary gaps and indeed found +that gaps in two-dimensional simulations are more permeable to +dust particles. Our one-dimensional simulations, therefore, need +to be considered more conservative. If it is not possible to sepa- +rate two reservoirs in one-dimensional models, it is less likely to +do so in higher dimensions. +We furthermore assumed a dust fragmentation velocity of +10 m/s, which may be rather high even for icy particles as in- +dicated by recent laboratory experiments which are suggesting +values of 1 m/s (see Blum 2018; Gundlach et al. 2018; Musiolik +& Wurm 2019). Lowering the fragmentation velocity, however, +generally decreases the particle sizes making them even less +likely to be trapped in pressure bumps (see panel E in Figure 1). +Other experiments indicate a significantly higher fragmentation +Article number, page 6 of 8 + +Sebastian Markus Stammler et al.: Leaky Dust Traps: How Fragmentation impacts Dust Filtering by Planets +velocity (e.g. Kimura et al. 2020) than the 10 m/s used in this +work. The exact value of the fragmentation velocity, however, +does not significantly influence the problem of inner disk con- +tamination. Either the fragmentation velocity is exceeded, which +will lead to pollution of the inner disk with outer disk material +(see panel B of Figure 1). Or the fragmentation velocity is larger +than the maximum collision velocity of dust particles in the disk, +in which case the particles will efficiently grow to larger parti- +cles, that are being trapped in the outer edge of the disk, which +will quickly deplete the inner disk (see panel D in Figure 1). +Similarily, the porosity evolution may have an effect on the +collisional physics of dust particles (e.g. Suyama et al. 2008; +Krijt et al. 2015; Kobayashi & Tanaka 2021). However, as for +the fragmentation velocity the details of the collision model do +not have a strong effect on the outcome of the simulation. Either +the particles fragment and the inner disk is polluted with outer +disk material, or the particles grow unhindered to large particles +that are trapped in the pressure bump, which is quickly depleting +the inner disk. +We furthermore did not consider the formation of planetes- +imals in the pressure bump in this work. Previous publications +have shown that the conditions in pressure maxima at the outer +edges of gaps can facilitate planetesimal formation (Stammler +et al. 2019; Miller et al. 2021) or even the formation of planets +(Lau et al. 2022; Jiang & Ormel 2023). One could conceive that +small dust fragments could not penetrate the inner disk because +they are quickly converted into planetesimals before they could +transverse the gap. This would, however, require a nearly per- +fect planetesimal formation efficiency to efficiently isolate both +dust reservoirs, which has not been observed in previous simu- +lations. Additionally, planetesimals formed at gap edges quickly +have been shown in simulations to quickly ablate (Eriksson et al. +2021). Enstatite and ordinary chondrites would thus have to be +explained by planetesimal formation where the dust is replen- +ished by, for instance, late-stage planetesimal collisions in the +NC reservoir (Dullemond et al. 2014; Lichtenberg et al. 2018; +Bernabò et al. 2022). +This suggests that it is unlikely that the formation of Jupiter +could have solely separated both dust reservoirs in the Solar Sys- +tem if the dust particles were subject to fragmentation. This does +not only apply to gaps created by planets, but also to other sub- +structures of non-planetary origin where particles are trapped +in pressure maxima as described in Brasser & Mojzsis (2020). +Other suggested mechanisms to explain the observations include +a temporal change in the isotopic content of inward-streaming +dust grains (Schiller et al. 2018), and the formation of multi- +ple distinct planetesimal populations in the inner and outer disk +(Lichtenberg et al. 2021; Morbidelli et al. 2022; Izidoro et al. +2021; Liu et al. 2022). How these physical mechanisms are con- +nected to the structures and gaps seen in ALMA disks (Miotello +et al. 2022) and the underlying mechanisms of protoplanet for- +mation (Drazkowska et al. 2022) and differentiation (Lichten- +berg et al. 2022) remain to be explored. +6. Conclusions +Protoplanet-induced gaps in circumstellar disks are not able to +efficiently separate dust in the inner disk from dust in the outer +disk on million-year timescales if the particles are subject to +fragmentation. Particles limited by bouncing without producing +small fragments are usually too small to be trapped by pressure +bumps. Only significantly reducing the relative collision veloci- +ties allows particles to be efficiently trapped in pressure bumps +within 1 Myr, if the planet grew to 40 M⊕. In this case, however, +the inner disk is quickly depleted from dust making it difficult to +form meteoritic bodies in situ. Our simulations suggest that other +physical mechanism must have initiated or at least substantially +contributed to the large-scale separation of nucleosynthetic iso- +topes observed in the planetary materials of the inner and outer +Solar System. +Acknowledgements. This project has received funding from the European Re- +search Council (ERC) under the European Union’s Horizon 2020 research and +innovation programme under grant agreement No 714769. This project has re- +ceived funding by the Deutsche Forschungsgemeinschaft (DFG, German Re- +search Foundation) through grants FOR 2634/1 and 361140270. This research +was supported by the Munich Institute for Astro-, Particle and BioPhysics +(MIAPbP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, +German Research Foundation) under Germany’s Excellence Strategy – EXC- +2094 – 390783311. JD was funded by the European Union under the Euro- +pean Union’s Horizon Europe Research & Innovation Programme 101040037 +(PLANETOIDS). Views and opinions expressed are however those of the au- +thors only and do not necessarily reflect those of the European Union or the Eu- +ropean Research Council. Neither the European Union nor the granting authority +can be held responsible for them. 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N. & Lithwick, Y. 2007, Icarus, 192, 588 +Article number, page 8 of 8 + diff --git a/6NE5T4oBgHgl3EQfPg5N/content/tmp_files/load_file.txt b/6NE5T4oBgHgl3EQfPg5N/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..be235587e71f4e21e8e2c1a20b788499657a5739 --- /dev/null +++ b/6NE5T4oBgHgl3EQfPg5N/content/tmp_files/load_file.txt @@ -0,0 +1,754 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf,len=753 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' dust_filtering ©ESO 2023 January 16, 2023 Leaky Dust Traps: How Fragmentation impacts Dust Filtering by Planets Sebastian Markus Stammler1, Tim Lichtenberg2, Joanna Dr˛a˙zkowska3, and Tilman Birnstiel1, 4 1 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 1, 81679, Munich, Germany 2 Kapteyn Astronomical Institute, University of Groningen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Box 800, 9700 AV Groningen, The Netherlands 3 Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany 4 Exzellenzcluster ORIGINS, Boltzmannstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2, D-85748 Garching, Germany January 16, 2023 ABSTRACT The nucleosynthetic isotope dichotomy between carbonaceous and non-carbonaceous meteorites has been interpreted as evidence for spatial separation and coexistence of two distinct planet-forming reservoirs for several million years in the solar protoplanetary disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Rapid formation of Jupiter’s core within one million years after CAIs has been suggested as a potential mechanism for spatial and temporal separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In this scenario, Jupiter’s core would open a gap in the disk and trap inwards-drifting dust grains in the pressure bump at the outer edge of the gap, separating the inner and outer disk materials from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' We performed simulations of dust particles in a protoplanetary disk with a gap opened by an early formed Jupiter core, including dust growth and fragmentation as well as dust transport using the dust evolution software DustPy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Our numerical experiments indicate that particles trapped in the outer edge of the gap rapidly fragment and are transported through the gap, contaminating the inner disk with outer disk materials on a timescale that is inconsistent with the meteoritic record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' This suggests that other processes must have initiated or at least contributed to the isotopic separation between the inner and outer Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Meteorites, meteors, meteoroids — Methods: numerical — Protoplanetary disks — Planets and satellites: formation – Planets and satellites: composition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Introduction Recent high-precision isotopic measurements reveal a di- chotomy between carbonaceous and non-carbonaceous mete- orites indicating that both have been formed in separate reser- voirs within the early Solar System (Trinquier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2007, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Leya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Warren 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Mezger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Kleine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Kruijer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (2017) and Desch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (2018) argued that these reservoirs must have been well separated for at least two million years without interchanging solid material, proposing the rapid formation of Jupiter’s core opening a gap in the protoplan- etary disk as possible mechanism to prevent the mixing of both reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The physical origin of the isotopic separation is a po- tential critical clue to the timescales of planet formation in both the inner and outer Solar System (Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2018), and thus ultimately the origin of the chemical abundances in the terres- trial planets and similar exoplanets (Krijt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Lichten- berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' This concept of a gap-opening Jupiter preventing dust reser- voir mixing, however, intimately depends on the evolution of the dust flux during the evolution of the protoplanetary disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Dust particles in protoplanetary disks are subject to gas drag and drift (Whipple 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Weidenschilling 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Takeuchi & Lin 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The radial dust velocity is given by: vd = vg 1 St2 + 1 + 2vP St St2 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (1) The Stokes number St is an aerodynamic measure and propor- tional to the particle size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Small particles with small Stokes num- bers are dragged along with the gas with velocity vg as can be seen by Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The gas is, in contrast to the dust, pressure supported and orbits the star with sub-Keplerian velocities in a typical smooth disk with inward pointing pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The dust particles, on the other hand, are not pressure supported, ex- change angular momentum with the gas and drift in direction of pressure gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Intermediate particle sizes are most affected by this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Small particles are well coupled to the gas, while large particles are completely decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' From Equation 1 it can be seen that particles with Stokes number of unity will experi- ence maximum drift in direction of the pressure gradient with velocity vP, which is given by: vP = 1 2 c2 s vK ∂ log P ∂ log r , (2) with the sound speed cs, pressure P, and the Keplerian velocity vK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Particles typically grow to maximum sizes with Stokes num- bers between 10−2 to 10−1 (see Birnstiel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2012), depending on the disk parameters, and are therefore affected by radial drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Growing planets can perturb the pressure structure in the disk by opening a gap in the gas (Paardekooper & Mellema 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Rice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' At the outer edge of the gap the pressure gradi- ent reverses and is pointing outward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' If the pressure pertubation is large enough, large dust pebbles that are affected by drift can be prevented from crossing the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The planetary mass at which the pressure pertubation is large enough to stop particle dift is called pebble isolation mass (see Lambrechts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Bitsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2018) and is given by Drazkowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (2022) as: Miso ≃ 25 M⊕ �HP/r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='05 �3 M⋆ M⊙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (3) Article number, page 1 of 8 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='05505v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='EP] 13 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' dust_filtering From NASA’s Juno mission Jupiter’s core is estimated to have a mass of up to 25 M⊕ (Wahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2017) and would have therefore been able to open a gap and stop the flux of dust peb- bles in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' A rapid formation of Jupiter’s core could there- fore explain two isolated dust reservoirs with the dust in the outer disk forming the carbonaceous and the dust in the inner disk the non-carbonaceous bodies in the Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Dr˛a˙zkowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (2019), however, showed in two- dimensional hydrodynamic simulations of gas and dust includ- ing collisional dust evolution, that the pressure bump at the outer edge of planetary gaps does not only show an accumulation of large dust pebbles, but also of small dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' But in con- trast to large pebbles, these small particles are not trapped by the pressure bump, they are produced in situ by collisions of large particles leading to fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' These small fragments can escape the pressure bump due to diffusion and gas drag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The equations of motion of the dust particles are given by: ∂ ∂tΣd + 1 r ∂ ∂r � rΣdvd − rDΣg ∂ ∂r �Σd Σg �� = 0, (4) with the dust diffusivity given by Youdin & Lithwick (2007) as D = δrc2 s ΩK 1 St2 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (5) with δr being a free parameter that defines the strength of radial dust diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Small particles are therefore most affected by dif- fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' If the diffusivity is high enough, these small particles can diffuse out of the pressure maximum and are dragged with the gas through the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' If this is the case the inner disk would be contaminated with dust from the outer disk negating the idea of two distinct dust reservoirs separated by an early formed Jupiter core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In this letter we test this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In section 2 we present a toy model which initially has dust placed only outside of the planet to show as a proof of concept, that solid material can pen- etrate planetary gaps if the dust is subject to fragmentation and diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In section 3 we investigate the influence of the plan- etary mass and the dust diffusivity on the dust permeability of planetary gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In section 4 we present models with a realis- tic evolution of the planetary mass, as it has been suggested for Jupiter, for models with both fragmentation and bouncing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Fi- nally, in section 5 we discuss our results, before we conclude in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Toy Model To investigate the influence of a planet on the dust flux in the inner disk, we model dust coagulation and transport in a proto- planetary disk with a planet opening a gap at 5 AU using the dust evolution software DustPy1 (Stammler & Birnstiel 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In a first simplified toy model, we initialize the disk only with dust outside of a Saturn mass planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Therefore, any dust flux mea- sured inside the planet must have crossed the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' We use this simplified model to investigate different scenarios: dust growth limited by fragmentation, dust growth limited by bouncing, and unlimited dust growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Furthermore, we compare the toy model to a model without a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' We initialize the gas surface density with the self similar so- lution of Lynden-Bell & Pringle (1974): Σg (r) = Mdisk 2πr2c � r rc �−1 exp � − r rc � (6) 1 DustPy v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='1 has been used for the simulations presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' with a cutoff radius of rc = 30 AU and an initial disk mass of Mdisk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='05 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' We impose a gap onto this gas surface density profile originating from a Saturn mass planet located at 5 AU, for which we use the gap profile fits provided by Kanagawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' To maintain this gap profile F (r) throughout the sim- ulation we impose the inverse of this profile onto the turbulent viscosity parameter α, since the product of gas surface density and viscosity is constant in quasi steady-state: α (r) = α0 F (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (7) In the default setup, we use α0 = δr = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Please note, that this change in α (r) does not affect the turbulent diffusion of the dust particles, since δr is a constant in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' We initialize the dust surface density with a constant gas-to- dust ratio of 100 and the dust size distribution according Mathis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (1977) as n (a) = a−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='5 with a maximum initial particle size of 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In the toy model we initially have dust only outside of 15 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' DustPy simulates dust growth by solving the Smoluchowski equation of a dust mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Dust transport is simulated by solving Equation 4 for every dust size individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The gas surface density is evolved by solving the viscous advection-diffusion equation ∂ ∂tΣg + 1 r ∂ ∂r � rΣgvg � = 0 (8) with the gas velocity given by Lynden-Bell & Pringle (1974) as vg = − 3 Σg √r ∂ ∂r � Σgν √r � (9) and the kinematic viscosity given by ν = αcsHP (10) with the sound speed cs = � kBT/µ, the pressure scale height HP = cs/ΩK, and the viscosity parameter α given by Equation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' We run five different flavors of the toy model: one with a fragmentation velocity of vfrag = 10 m/s (fiducial), one with no fragmentation at all, one with a fragmentation velocity of 1 m/s, one with bouncing as described by Windmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (2012), and one without a gap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' F (r) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In the default collision model used by DustPy particles fragment once their relative collision velocities exceed the fragmentation velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Fragmenting colli- sions of equal size particles lead to catastrophic fragmentation of both collision partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' If the target particle is significantly larger, only the projectile particle fragments entirely while erod- ing mass off the target particle (Schräpler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Hasegawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The transition between pure sticking and fragmenta- tion is smooth, since DustPy is assuming a velocity distribution of possible collision velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' For details on the collision model we refer to Stammler & Birnstiel (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Panel A of Figure 1 shows the initial dust distribution with dust located outside of 15 AU with particles sizes up to 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The white lines are contour lines of Stokes numbers St = � 10−3, 10−2, 10−1, 100� with the bold white line corresponding to St = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Panel B shows the fiducial simulation with a Saturn mass planet at 5 AU and the fragmentation velocity vfrag = 10 m/s after 1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Particles trapped in the pressure bump outside the planetary gap can reach sizes with Stokes numbers of up to St = 10−1 corresponding to particle sizes of a few centimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' It can be seen that even small particles are accumulated in the pressure bump, even though their Stokes numbers are too small Article number, page 2 of 8 Sebastian Markus Stammler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' : Leaky Dust Traps: How Fragmentation impacts Dust Filtering by Planets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Distance from star [AU] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Particle size [cm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='A: initial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Distance from star [AU] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Particle size [cm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='B: fiducial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Distance from star [AU] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Particle size [cm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='C: without planet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Distance from star [AU] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Particle size [cm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='D: no fragmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Distance from star [AU] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Particle size [cm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='E: vfrag = 1 m/s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Distance from star [AU] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Particle size [cm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='F: bouncing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='dust [g/cm²] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Panel A: Initial dust distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The white lines correspond to Stokes numbers of St = � 10−3, 10−2, 10−1, 100� with the bold white line corresponding to St = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' All other panels show snapshots of models at 1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Panel B: The fiducial toy model with a Saturn mass planet at 5 AU and a fragmentation velocity of 10 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Panel C: Model without a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The vertical dashed lines are the location at which the dust flux is measured in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Panel D: Model without fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Panel E: Model with a reduced fragmentation velocity of 1 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Bottom right: Model with bouncing instead of fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' to be affected by drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' These small dust particles are produced by collisional fragmentation of larger particles trapped in the bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' They diffuse out of the bump and are dragged with the gas contaminating the inner disk with outer disk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' It can be seen that particles with Stokes numbers of about St = 10−2, corresponding to particle sizes of a few millimeter, can diffuse through the gap into the inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Particles in the inner disk can again grow to centimeter sizes and can contribute to phe- nomena like the streaming instability or pebble accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Panel C shows a simulation with identical initial conditions but with- out a planet opening a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The vertical yellow and green dashed lines in panels B and C are the locations at which the dust fluxes shown in Figure 2 are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The dust fluxes at the outer disk are identical in both sim- ulations with the solid and dashed green lines overlapping in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The fluxes in the inner disk, however, differ in both simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The onset of dust flux in the inner disk in the sim- ulation with a planet is delayed by about 20 000 yr compared to the simulations without a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Without a planet, the large dust particles can freely drift into the inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' With a planet, how- ever, they are first trapped in the pressure bump at the outer edge of the gap, fragment down to smaller sizes, and diffuse out of the pressure bump before the gas can drag them into the inner disk where they grow to larger particles again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Due to this delayed processing the maximum dust flux is reduced by about one or- der of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The duration, however, is prolonged such that Article number, page 3 of 8 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' dust_filtering 103 104 105 106 107 Time [yrs] 10 7 10 6 10 5 10 4 10 3 10 2 Dust flux [M /yr] r = 2 AU r = 15 AU with planet w/o planet 103 104 105 106 107 Time [yrs] 10 2 10 1 100 Fraction of total dust mass accreted Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Top: Comparison of the dust flux in the inner disk (at 2 AU) and outer disk (at 15 AU) in the toy model with a Saturn mass planet at 5 AU (panel B in Figure 1) and a model without a planet (panel C in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Both green 15 AU lines overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Bottom: Total dust mass accreted through the inner disk over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' the total mass of dust flowing through the inner disk is identical after 10 Myr as can be seen in the bottom panel of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The Saturn mass planet did not separate the inner from outer disk material, but only delayed the material transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Panel D of Figure 1 shows a simulation without fragmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In this scenario, particles sizes are limited only by the radial drift, consistent with the model presented by Kobayashi & Tanaka (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In the center of the pressure bump, the pressure gradient is zero and the growth is in principle unlimited until the particles accumulate at the upper end of the simulation grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' This scenario most closely represents the separation of inner and outer dust reservoirs with only very few particles being able to diffuse through the gap, because they were not able to grow to large par- ticles quickly enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' It is, however, rather unlikely that the dust particles do not fragment or get eroded at some point given the relative velocities they typically experience (see Blum & Münch 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Wada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Schräpler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Panel E of Figure 1 shows a model with a fragmentation ve- locity of 1 m/s as indicated by recent experiments (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Blum 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Gundlach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Musiolik & Wurm 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In this case, the particles cannot reach particles sizes large enough to be efficiently trapped in the pressure bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The objective is therefore to halt particle growth without pro- ducing small particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' This can be achieved if the growth is lim- ited by bouncing, when particles simply bounce of each other without growing or fragmenting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Panel F of Figure 1 shows a simulation with the bouncing barrier implemented as described by Windmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In this model, bouncing starts when 103 104 105 106 107 Time [yrs] 10 7 10 6 10 5 10 4 10 3 Pebble flux [M /yr] 103 104 105 106 107 Time [yrs] 10 4 10 3 10 2 10 1 100 Fraction of dust mass accreted no planet 30 M 50 M Msat, r = 10 2 Msat, r = 10 3 Msat, r = 10 4 Msat, r = 10 5 200 M Mjup Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Top: Dust flux through the planetary gap in models with different planet masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The blue line is for a model without a planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The dashed, dotted, and dash-dotted red lines show additional simulations with a Saturn mass planet for different radial dust diffusivity parameters δr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Bottom: Total fraction of outer dust mass accreted through the planetary gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' the relative velocity reaches a few centimeters per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In this case, however, the particles only reach sizes of a few 100 µm corresponding to Stokes numbers lower than 10−3, which is too small to be efficiently trapped in the pressure bump created by the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The particles can diffuse through the gap and contam- inate the inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Full Disk Models The toy model in section 2 served as a proof of concept that planets do not prevent dust flux if particles are subject to frag- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In this section we discuss full disk models with dif- ferent planet masses in which dust is initialized in the entire disk to investigate the dust permeability of the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The top panel of Figure 3 shows the dust flux through the planetary gap for differ- ent planet masses from 30 Earth masses to one Jupiter mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In the case of a Saturn mass planet we additionally performed sim- ulations with different dust diffusivity parameters δr (see Equa- tion 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The bottom panel of Figure 3 shows the total fraction of outer disk dust material that has been accreted through the gap over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In all models the planets have their respective masses already from the beginning of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The smallest planetary mass considered here is 30 M⊕, which is already higher than the upper estimate of Jupiter’s core mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The largest mass considered is 1 Mjup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The smallest planetary mass is not capable of efficiently suppressing the dust flux through the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' After about 300 000 yr almost the entire dust Article number, page 4 of 8 Sebastian Markus Stammler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' : Leaky Dust Traps: How Fragmentation impacts Dust Filtering by Planets mass (horizontal line in bottom panel) of the outer disk has been accreted through the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Increasing the planetary mass simply delays the accretion time, but is not able to prevent accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The maximum delay of accretion seems to be achieved already with a 200 M⊕ planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Increasing the planet mass further to a Jupiter mass planet does not significantly change the accretion history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' At the end of the simulation at 10 Myr about 80 % of the dust mass has been accreted through the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The dust diffusivity δr has a more significant influence on the accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Increasing the diffusivity by a factor of 10 to δr = 10−2 in the Saturn mass simulation has the same effect as reducing the planet mass by a factor of about 2, mimicking the accretion his- tory of a 40 M⊕ planet with diffusivity of δr = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Note that we only changed δr, while keeping α0 = 10−3 and therefore keeping the shape of planetary gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The relative collision velocities of the dust particles are not affected by this change in δr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Decreas- ing δr by a factor of 10 is more efficient in retaining the dust than having a Jupiter mass planet with the standard diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In this case only about 10 % of the dust mass has been accreted at the end of the simulation after 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Lowering the diffusivity even further to δr = 10−5 reduces the dust permeability further to a about 5 % of the outer disk mass after 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' It is however noted that the fraction of outer disk material present in the inner disk is usually significantly larger, since the inner disk material is accreted onto the star on short timescales and only re-supplied with outer disk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Time-dependent planet mass In the previous models we assumed that the planets are fully formed from the beginning of the simulation and the planet mass does not evolve over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Kruijer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (2017) argue that the two dust reservoirs have been separated from about 1 Myr to 3−4 Myr after CAI formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' They therefore claim that Jupiter’s core must have been massive enough to open a gap at 1 Myr and must have reached a mass of about 50 M⊕ after 4 Myr to be able to scatter planetesimals from the outer disk to the inner disk where they are observed today in the asteroid belt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' We there- fore performed simulations with a time-dependent planet mass as shown in the top left panel of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The solid blue line shows an evolutionary track where the planet reaches 30 M⊕ af- ter 1 Myr, 50 M⊕ after 4 Myr and a final mass of Mjup at the end of the simulation after 10 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The bottom left panel of Figure 4 shows the fraction of mass accreted through the planetary gap normalized to the dust mass in the outer disk at 1 Myr when the planet was massive enough to open a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' We performed simulations with different values of the dust diffusivity δr between 10−5 and 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In the standard run with δr = 10−3 about 80 % of the dust mass has been accreted through the gap after 4 Myr (vertical solid line) when the assem- bly of the meteorite parent bodies has been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Even in the low diffusivity run with δr = 10−5 about 60 % of the mass has been accreted though the gap between 1 Myr and 4 Myr, strongly contaminating the inner disk with dust from the outer reservoir on a system-wide scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Lowering the dust diffusivity to very low values does not help keeping both reservoirs separated, since the planet mass is too low in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The bottom right panel of Figure 4 shows a model with bouncing instead of fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The solid blue line shows a model with radial dust diffusivity δr = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' As already shown in section 2, this is not sufficient to stop dust accretion through the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Only after 7 Myr when the planet already reached a mass of about 200 M⊕ the gap is deep enough and accretion is halted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Allowing the planet to reach these masses at earlier times would, however, not change the dust redistribution, since these massive planets are able to scatter planetesimals from the outer disk into the inner disk, which is inconsistent with observations from the meteoritic record at these early times (Deienno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The green solid line shows a model with δr = δt = δz = 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The parameters δt and δz are similar to δr and parametrize the strength of turbulent motion and vertical settling of the particles (see Stammler & Birnstiel 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Pinilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2021, for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In that way the relative velocities between the particles are re- duced, allowing them to grow to larger sizes before being lim- ited by bouncing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' They can therefore be trapped by gaps created by smaller mass planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' However, even in that case accretion is only halted after abut 3 Myr, when the planet reached a mass of about 40 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The dashed green line shows a model where the planet reaches a mass of 40 M⊕ already after 1 Myr (dashed line in top left panel of Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In this case accretion of dust through the gap is efficiently stopped at 1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The top right panel of Fig- ure 4 shows a snapshot of this simulation after 4 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The inner disk is heavily depleted in dust, all of which has been accreted onto the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The dust mass in the inner disk at this stage was all supplied from the outer disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Meteoritic bodies formed in the inner disk would therefore be entirely made out of outer disk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Discussion Isotopic measurements of meteoritic material indicate that me- teorites must have formed in two dust reservoirs, that coexisted spatially separated for several million years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The early formation of Jupiter’s core has been proposed as natural explanation for the observed separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' A planet exceeding the pebble isolation mass opens a gap in the gas disk creating a pressure bump at the outer edge of the gap, which can trap large dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Two- dimensional hydrodynamical simulations by Dr˛a˙zkowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (2019) including dust coagulation and fragmentation showed an overabundance of small dust particles at the location of the pres- sure bump, which should be too small to be efficiently trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' These particles were created in fragmenting collision of large dust pebbles that have been trapped in the pressure bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' These small dust fragments can diffuse out of the bump and can be dragged by the gas through the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Our simulations in this work suggest that collisional frag- mentation of dust pebbles in pressure bumps and subsequent dif- fusion of small fragments can act as a leak for dust traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' As can be seen by Figure 3, gaps opened by planets can only de- lay but not fully prevent dust accretion if particles are subject to fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' To act as an efficient dust barrier, particles need to grow to large pebbles that can be trapped without producing small particles as shown in the panel D of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' We investigated different planet masses and showed in Fig- ure 3 that no planet mass was able to completely isolate the inner disk from outer dust material on timescales that are relevant for the assumed reservoir separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Even an initial gap formed by a fully-grown Jupiter mass planet would leak 20 % of the outer disk material into the inner disk within 1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Smaller proto- Jupiter masses typically lead to complete homogenization within ∼ 105 to at maximum a few 106 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' This presents a problem for the suggestion that the age differences in carbonaceus and non-carbonaceous meteorites may be used as a tracer to track the growth timescale of proto-Jupiter within the disk (Kruijer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Alibert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2018): the initial spatial distribution of nucleosynthetic isotopes at the end of disk infall is degenerate Article number, page 5 of 8 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' dust_filtering 0 2 4 6 8 10 Time [Myr] 0 50 100 150 200 250 300 Planet mass [M ] default model rapid early growth 1 2 3 4 5 6 7 8 9 10 Time [Myr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='0 Fraction of dust mass accreted Fragmentation r = 10 3 r = 10 4 r = 10 5 1 2 3 4 5 6 7 8 9 10 Time [Myr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='0 Fraction of dust mass accreted Bouncing i = 10 3 i = 10 5 i = 10 5 101 102 Distance from star [AU] 10 4 10 3 10 2 10 1 100 101 102 Particle size [cm] 10 5 10 4 10 3 10 2 10 1 100 101 dust [g/cm²] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Top left: Evolution of the planetary mass in the time-dependent model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The solid line shows the default model where the planet reaches 20 M⊕ at 1 Myrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The dashed line shows the evolution in a model with rapid early growth in which the planet reaches 40 M⊕ at 1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Bottom left: Fraction of outer disk dust mass accreted through the gap after 1 Myr in the default planetary mass evolution model for different values of dust diffusivity δr with fragmentation limited growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Bottom right: The solid lines show the fraction of outer disk material accreted through the gap after 1 Myr for bouncing limited growth for different values of the δi parameters in the default planetary growth model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The dashed green line shows a model of bouncing limited growth with δi = 10−5 and rapid early growth of the planet (dashed line in top left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The vertical lines mark 4 Myr until which both reservoirs need to be separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Top right: Snapshot of the dust distribution at 4 Myr for the model with bouncing limited growth and δi = 10−5 (dashed green line in bottom right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The inner disk is depleted in dust and only supplied with small amounts of outer disk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' with different Jupiter growth tracks in the Jupiter barrier hypoth- esis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Only significantly lowering the dust diffusivity to a value of δr = 10−5 could decrease the dust permeability such that the inner disk is only contaminated with a few percent of outer disk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' However, isolating the inner disk from dust flux would quickly drain the inner disk from solids that got accreted onto the star, which was also previously noted by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' At later stages the dust in the inner disk then consists to large parts of outer disk material that has been slowly diffused through the gap, which is inconsistent with the meteoritic record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The situation gets worse when using a more realistic evo- lution of the planetary mass, assuming Jupiter’s core reached a mass of 20 M⊕ after 1 Myr and 50 M⊕ after 4 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' These masses are not large enough to isolate the inner disk even in models with very low diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Even in the most optimistic cases at least 60 % of the outer disk dust has been accreted through the planetary gap after 4 Myr as can be seen by Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' However, increasing the core mass even more and earlier would enable Jupiter to scatter outer disk planetesimals into the inner disk pol- luting the inner dust reservoir, which has not been accounted for in this simple model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Only in models with bouncing lim- ited growth without small particles, early planetary growth and reduced relative particle collision velocities, the inner disk can be efficiently isolated from the inner disk as seen by Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In these cases, however, the inner disk is quickly depleted from dust and only re-supplied from small amount of outer disk ma- terial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Meteoritic bodies formed in the inner disk after this point would therefore consist almost entirely of outer disk material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Dr˛a˙zkowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (2019) noted that the shape of planetary gaps in two-dimensional simulations is not axisymmetric, which is ignored in the simple one-dimensional model in this publi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' They further noted, however, that the asymmetry at the planet location would increase the dust flux through the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' (2018) compared one- and two-dimensional simula- tions of dust transport through planetary gaps and indeed found that gaps in two-dimensional simulations are more permeable to dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Our one-dimensional simulations, therefore, need to be considered more conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' If it is not possible to sepa- rate two reservoirs in one-dimensional models, it is less likely to do so in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' We furthermore assumed a dust fragmentation velocity of 10 m/s, which may be rather high even for icy particles as in- dicated by recent laboratory experiments which are suggesting values of 1 m/s (see Blum 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Gundlach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Musiolik & Wurm 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Lowering the fragmentation velocity, however, generally decreases the particle sizes making them even less likely to be trapped in pressure bumps (see panel E in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Other experiments indicate a significantly higher fragmentation Article number, page 6 of 8 Sebastian Markus Stammler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' : Leaky Dust Traps: How Fragmentation impacts Dust Filtering by Planets velocity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Kimura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2020) than the 10 m/s used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' The exact value of the fragmentation velocity, however, does not significantly influence the problem of inner disk con- tamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Either the fragmentation velocity is exceeded, which will lead to pollution of the inner disk with outer disk material (see panel B of Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Or the fragmentation velocity is larger than the maximum collision velocity of dust particles in the disk, in which case the particles will efficiently grow to larger parti- cles, that are being trapped in the outer edge of the disk, which will quickly deplete the inner disk (see panel D in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Similarily, the porosity evolution may have an effect on the collisional physics of dust particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Suyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Krijt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Kobayashi & Tanaka 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' However, as for the fragmentation velocity the details of the collision model do not have a strong effect on the outcome of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Either the particles fragment and the inner disk is polluted with outer disk material, or the particles grow unhindered to large particles that are trapped in the pressure bump, which is quickly depleting the inner disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' We furthermore did not consider the formation of planetes- imals in the pressure bump in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Previous publications have shown that the conditions in pressure maxima at the outer edges of gaps can facilitate planetesimal formation (Stammler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2021) or even the formation of planets (Lau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Jiang & Ormel 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' One could conceive that small dust fragments could not penetrate the inner disk because they are quickly converted into planetesimals before they could transverse the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' This would, however, require a nearly per- fect planetesimal formation efficiency to efficiently isolate both dust reservoirs, which has not been observed in previous simu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Additionally, planetesimals formed at gap edges quickly have been shown in simulations to quickly ablate (Eriksson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Enstatite and ordinary chondrites would thus have to be explained by planetesimal formation where the dust is replen- ished by, for instance, late-stage planetesimal collisions in the NC reservoir (Dullemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Lichtenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Bernabò et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' This suggests that it is unlikely that the formation of Jupiter could have solely separated both dust reservoirs in the Solar Sys- tem if the dust particles were subject to fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' This does not only apply to gaps created by planets, but also to other sub- structures of non-planetary origin where particles are trapped in pressure maxima as described in Brasser & Mojzsis (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Other suggested mechanisms to explain the observations include a temporal change in the isotopic content of inward-streaming dust grains (Schiller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2018), and the formation of multi- ple distinct planetesimal populations in the inner and outer disk (Lichtenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Morbidelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Izidoro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' How these physical mechanisms are con- nected to the structures and gaps seen in ALMA disks (Miotello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2022) and the underlying mechanisms of protoplanet for- mation (Drazkowska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2022) and differentiation (Lichten- berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2022) remain to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Conclusions Protoplanet-induced gaps in circumstellar disks are not able to efficiently separate dust in the inner disk from dust in the outer disk on million-year timescales if the particles are subject to fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Particles limited by bouncing without producing small fragments are usually too small to be trapped by pressure bumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Only significantly reducing the relative collision veloci- ties allows particles to be efficiently trapped in pressure bumps within 1 Myr, if the planet grew to 40 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' In this case, however, the inner disk is quickly depleted from dust making it difficult to form meteoritic bodies in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Our simulations suggest that other physical mechanism must have initiated or at least substantially contributed to the large-scale separation of nucleosynthetic iso- topes observed in the planetary materials of the inner and outer Solar System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' This project has received funding from the European Re- search Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 714769.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' This project has re- ceived funding by the Deutsche Forschungsgemeinschaft (DFG, German Re- search Foundation) through grants FOR 2634/1 and 361140270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' This research was supported by the Munich Institute for Astro-, Particle and BioPhysics (MIAPbP) which is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC- 2094 – 390783311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' JD was funded by the European Union under the Euro- pean Union’s Horizon Europe Research & Innovation Programme 101040037 (PLANETOIDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Views and opinions expressed are however those of the au- thors only and do not necessarily reflect those of the European Union or the Eu- ropean Research Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' Neither the European Union nor the granting authority can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' TL was supported by grants from the Simons Foundation (SCOL Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 611576) and the Branco Weiss Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' References Alibert, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=', Venturini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=', Helled, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=', et al.' metadata={'source': 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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' & Lithwick, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} +page_content=' 2007, Icarus, 192, 588 Article number, page 8 of 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE5T4oBgHgl3EQfPg5N/content/2301.05505v1.pdf'} diff --git a/99AzT4oBgHgl3EQfg_xe/content/tmp_files/2301.01477v1.pdf.txt b/99AzT4oBgHgl3EQfg_xe/content/tmp_files/2301.01477v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..caa7ab7da6d9a37efd78074cd9dba08397d2d627 --- /dev/null +++ b/99AzT4oBgHgl3EQfg_xe/content/tmp_files/2301.01477v1.pdf.txt @@ -0,0 +1,3666 @@ +arXiv:2301.01477v1 [stat.ME] 4 Jan 2023 +Reliability Analysis of Load-sharing Systems using a +Flexible Model with Piecewise Linear Functions +Shilpi Biswas ∗, Ayon Ganguly †, and Debanjan Mitra ‡ +Abstract +Aiming for accurate estimation of system reliability of load-sharing systems, a flex- +ible model for such systems is constructed by approximating the cumulative hazard +functions of component lifetimes using piecewise linear functions. The advantages of +the resulting model are that it is data-driven and it does not use prohibitive assump- +tions on the underlying component lifetimes. Due to its flexible nature, the model is +capable of providing a good fit to data obtained from load-sharing systems in general, +thus resulting in an accurate estimation of important reliability characteristics. Es- +timates of reliability at a mission time, quantile function, mean time to failure, and +mean residual time for load-sharing systems are developed under the proposed model +involving piecewise linear functions. Maximum likelihood estimation and construction +of confidence intervals for the proposed model are discussed in detail. The performance +of the proposed model is observed to be quite satisfactory through a detailed Monte +Carlo simulation study. Analysis of a load-sharing data pertaining to the lives of a +two-motor load-sharing system is provided as an illustrative example. In summary, +this article presents a comprehensive discussion on a flexible model that can be used +for load-sharing systems under minimal assumptions. +Keywords: Load-sharing systems; Cumulative hazard function; Baseline hazard; Piecewise +linear approximation; Maximum likelihood estimation; Fisher information; Bootstrap; Con- +fidence interval; Quantile function; Mean time to failure; Reliability at a mission time; Mean +residual time. +1 +Introduction +1.1 +Background +Dynamic models are suitable for reliability systems where failure or degradation of one or +more components affects the performance of the surviving or operating components. Load- +sharing systems are appropriate examples where such models can be used. The total load +on a load-sharing system is shared between its components; when a component fails within +∗Indian Institute of Technology Guwahati, Assam 781039, India; Email: shilpi.biswas@iitg.ac.in +†Indian Institute of Technology Guwahati, Assam 781039, India; Email: aganguly@iitg.ac.in +‡Indian Institute of Management Udaipur, Rajasthan 313001, India; Email: debanjan.mitra@iimu.ac.in +1 + +the system, the total load gets redistributed over the remaining operating components. As a +result of a higher stress due to this extra load, the failure rates of the operating components +increase. +Common examples of load-sharing systems are those where components are connected +in parallel, such as central processing units (CPUs) of multi-processor computers, cables +of a suspension bridge, valves or pumps in hydraulic systems, electrical generator systems +etc. Load-sharing systems are found in other spheres as well, such as the kidney system in +humans. When one of the kidneys fails or deteriorates, the other kidney experiences elevated +stress and has an increased chance of failure. +The load-share rule among the operating components depends on the physical charac- +teristics of the system involved. In an equal load-share rule, the extra load caused by the +failed components is shared equally by the operating components. On the other hand, a +local load-share rule implies that the extra load is shared by the neighboring components of +the failed ones. A monotone load-sharing rule more generally assumes that the load on the +operating components is non-decreasing with respect to the failure of other components in +the system [18]. +1.2 +Literature review +One of the early major contributions to the literature on load-sharing systems was by +Daniels [9], describing the increasing stress on yarn fibres with successive breakings of indi- +vidual fibres within a bundle. In the same context of the textile industry, the early-period +literature saw developments by Coleman [4, 5], Rosen [29], and Harlow and Phoenix [13, 14], +among others. In general, the topic attracted the attentions of several researchers, and sig- +nificant theoretical contributions were made, for example, by Birnbaum and Saunders [6], +Freund [12], Ross [30], Schechner [31], Lee et al. [19], Hollander and Pena [15], and Lynch [22]. +While most studies on load-sharing systems in the early-period were based on a known +load-share rule, Kim and Kvam [16] presented a statistical methodology for multicompo- +nent load-sharing systems with an unknown load-share rule. In fact, the work of Kim and +Kvam [16] was also important for another reason: they used the hypothetical latent variable +approach for modelling the component lifetimes. The latent variable approach was later +adapted by Park [27, 28] for developing an inferential framework for load-sharing systems +assuming the component lifetimes to be exponential, Weibull, and lognormally distributed +random variables. +The use of parametric models has a long history in the literature on load-sharing models. +Exponential distribution has been extensively used for modelling lifetimes of components of +load-sharing systems [32, 20, 24]. However, the property of a constant hazard rate of the +exponential distribution is not practical for most applications. The tampered failure rate +model for load-sharing systems, proposed by Suprasad et al. [33], was thus developed to +accommodate a wide range of failure-time distributions for the components. In this connec- +tion, the use of accelerated life testing models for load-sharing systems may be mentioned; +see Mettas and Vassiliou [23], Amari and Bergman [1], and Kong and Ye [17]. A family +of parametric distributions was used for modelling the lives of two-component load-sharing +systems by Deshpande et al. [10]. Asha et al. [2] used a frailty-based model to this effect. A +recent contribution in this direction is by Franco et al. [11] who used generalized Freund’s +2 + +bivariate exponential model for two-component load-sharing systems. See also the references +cited in these articles. +Recently, several authors have explored diverse areas concerning load-sharing systems. +The damage accumulation of load-sharing systems was modelled by M¨uller and Meyer [25]. +Luo et al.[21] developed a model for correlated lifetimes in dynamic environments incorpo- +rating the load-sharing criterion. Brown et al. [7] explored a spatial model for load-sharing +where the extra load due to failure of a component is shared more by the operating com- +ponents that are in close proximity of the failed component than those that are distant. +Nezakati and Ramzakh [26], and Zhao et al. [36] connected degradation of components to +load-sharing phenomena. +In an interesting development, Che et al. [8] considered man- +machine units (MMUs) as units of analysis where load-sharing was possible due to machine +issues as well as human issues. They studied the load-sharing of the MMUs, attempting to +capture the complex dependence between machines and their operators. A general model, +called the load-strength model, was studied by Zhang et al. [35]. It is to be noted that most +of the studies on load-sharing systems have used parametric models for analysis so far, thus +heavily relying on the modelling assumptions for suitability of their analyses. +1.3 +Aim and Motivation +Our aim in this paper is to develop an appropriate estimate for the system reliability or +reliability at mission time (RMT) of load-sharing systems. The aim, also, is to accurately +estimate quantile function of the underlying system lifetime distribution, mean time to failure +(MTTF), and mean residual time (MRT) of load-sharing systems. +These quantities are +important to fully understand the characteristics of a load-sharing system; also, they are of +practical importance for making various strategies and plans. +Naturally, the quality of estimation of RMT, quantile function, MTTF, and MRT of a +load-sharing system depends on the suitability of the model that is fitted to the lifetimes +of its components capturing the load-share rule. +To this effect, we develop a model for +the component lifetimes involving piecewise linear approximations (PLAs) of the cumulative +hazard functions, capturing the unknown load-share rule at each of the successive stages of +component failures. The model is data-driven, and does not require prohibitive parametric +assumptions for component lifetime distributions. Due to this flexibility, the PLA-based +model is capable of providing a good fit to load-sharing data. An example, elaborated in a +later section, is as follows. +Data pertaining to a load-sharing system where each system was a parallel combination +of two motors were analysed by Asha et al. [2] and Franco et al. [11]. +Asha et al. [2] +assumed Weibull distributions for the component lifetimes, although data for one of the two +component motors showed clear empirical evidence that the assumption was not satisfied. A +generalized bivariate Freund distribution was assumed for the component lifetimes by Franco +et al. [11]. To this data, we have fitted our proposed PLA-based model, and have observed +according to the Akaike’s information criterion (AIC) for model selection, the PLA-based +model is a much better fit compared to the Weibull model of Asha et al. [2] and generalized +bivariate Freund model of Franco et al. [11]. The immediate and obvious result of this is +a much more accurate estimation of the RMT, quantile function, MTTF, and MRT of the +system lifetimes. The details of this analysis are given in a later section. +3 + +The main contributions of this paper are as follows: +• We develop a flexible, data-driven model based on PLA for modelling component +lifetimes of a load-sharing system. The model does not require prohibitive parametric +assumptions on the underlying component lifetimes. +• We develop inference for the proposed PLA-based model based on data from multi- +component load-sharing systems. +• Under the proposed PLA-based model, we develop methods to accurately estimate im- +portant reliability characteristics such as system reliability or RMT, quantile function, +MTTF, and MRT of load-sharing systems. +The rest of this article is structured as follows. In Section 2, the proposed PLA-based +model for load-sharing systems is presented. +Section 3 contains likelihood inference for +the model based on data from multi-component load-sharing systems, including relevant +details of derivation of MLEs, construction of confidence intervals, and a general guidance +on selection of cut-points for the piecewise linear functions. Estimation of system reliability, +quantile function, MTTF, and MRT of load-sharing systems in this setting are given in +Section 4. Based on component lifetime data from a two-component load-sharing system, +an illustrative example of application of the PLA-based model and estimation of various +important reliability characteristics are presented in Section 5. In Section 6, results of a +detailed Monte Carlo simulation experiment investigating the efficacy and robustness of the +PLA-based model are presented. +Finally, the paper is concluded with some remarks in +Section 7. +2 +The Piecewise Linear Approximation Model for +Cumulative Hazard +In general, a PLA is a helpful tool for modelling data, avoiding strong parametric assump- +tions. In survival analysis, piecewise linear functions are used extensively. Recently, Bal- +akrishnan et al. [3] proposed a PLA-based model for the hazard rate of a population with +a cured proportion; see also the references therein. In this article, we develop a PLA-based +model for load-sharing systems with unknown load-share rules. Specifically, we model the +cumulative hazard functions of the component lifetime distributions using PLAs. At each +of the successive stages of component failures, as the lifetime distributions of the remaining +operating components change, a new PLA for the cumulative hazard is used. The model +can be suitably tuned by choosing the number of linear pieces for the PLA at each stage of +failure. The principal advantage of the proposed PLA-based modelling approach is that it +uses minimal model assumptions. +Consider a J-component load-sharing system. Here, a J-component load-sharing system +means a load-sharing system with J components that are connected in parallel. Assume that +the failed components of the system are not replaced or repaired. When the components fail +one by one, after each failure the total load on the system gets redistributed over the remain- +ing operational components. As a result the operational components experience a higher load +4 + +than before. At the beginning when all components are operational, let U(0) +1 , U(0) +2 , . . . , U(0) +J +denote the latent lifetimes of the components, and Y (0) denote the system lifetime till the +first component failure. Obviously, +Y (0) = min +� +U(0) +1 , U(0) +2 , . . . , U(0) +J +� +. +Similarly, for j = 1, 2, . . . , J − 1, let Y (j) denote the system lifetime between j-th and +(j + 1)-st component failures. Then, +Y (j) = min +� +U(j) +1 , U(j) +2 , . . . , U(j) +J−j +� +, +where U(j) +1 , U(j) +2 , . . . U(j) +J−j denote the latent lifetimes of the operational components after the +j-th component failure, j = 1, 2, . . . , J − 1. For all values of j, U(j) +1 , . . . , U(j) +J−j are assumed +to be independent and identically distributed random variables. It is further assumed that +� +U(j) +ℓ , ℓ = 1, 2, . . . , J − j; j = 0, 1, . . . , J − 1 +� +are independent random variables. +Let h(j)(·) and H(j)(·) denote the hazard rate (HR) and cumulative hazard function +(CHF), respectively, of the distribution of U(j) +1 , j = 0, 1, 2, . . . , J −1. Here, we assume that +the HR h(j) (·) is a non-decreasing function for all j. For y > 0, the survival function (SF) +of Y (j) is given by +P +� +Y (j) > y +� += P +� +min +� +U(j) +1 , U(j) +2 , . . . , U(j) +J−j +� +> y +� += e−(J−j)H(j)(y). +Hence, for y > 0, the cumulative distribution function (CDF) and probability density func- +tion (PDF) of Y (j) are given by +F (j)(y) = 1 − e−(J−j)H(j)(y) +and +f (j)(y) = (J − j)h(j)(y) e−(J−j)H(j)(y), +respectively. +Now, suppose there are n J-component load-sharing systems, and let Y (j) +i +denote the +system lifetime between j-th and (j + 1)-st component failures for the i-th system, i = +1, 2, . . . , n, j = 0, 1, . . . , J − 1. Suppose the observed values of Y (j) +1 , Y (j) +2 , . . . , Y (j) +n +are +y(j) +1 , y(j) +2 , . . . , y(j) +n , respectively. Let, for j = 0, 1, . . . , J − 1, ξ(j) = +� +τ (j) +0 , τ (j) +1 , . . . , τ (j) +N +� +denote a set of N + 1 cut-points over the time scale y(j) +1 , . . . , y(j) +n , with the restrictions that +τ (j) +0 +< τ (j) +1 +< τ (j) +2 +< . . . < τ (j) +N , +τ (j) +0 +≤ min +� +y(j) +1 , . . . , y(j) +n +� +and τ (j) +N ≥ max +� +y(j) +1 , . . . , y(j) +n +� +. +Initially, ξ(j) is taken to be fixed and known. We discuss how to choose ξ(j) in a later section. +The proposed model approximates the CHF H(j)(·) by a piecewise linear function defined +over intervals [τ (j) +k−1, τ (j) +k ), k = 1, 2, . . . , N, constructed by the consecutive cut points in ξ(j). +Therefore, over the range [τ (0) +0 , τ (0) +N ), the CHF H(0)(·) is approximated by Λ(0)(·), where +Λ(0)(t) = +N +� +k=1 +(ak + bkt) 1[τ (0) +k−1, τ (0) +k +)(t), +(2.1) +5 + +with ak’s and bk’s as real constants and +1A(t) = +� +1 +if t ∈ A +0 +if t ̸∈ A. +One of the possible ways to extend the PLA beyond τ (0) +N +would be to extend the last line +segment aN + bNt to [τ (0) +N , ∞). Therefore, the CHF corresponding to PLA over the range +[τ (0) +0 , ∞) is +Λ(0)(t) = +N +� +k=1 +(ak + bkt) 1[τ (0) +k−1, τ (0) +k +)(t) + (aN + bNt)1[τ (0) +N , ∞)(t), +with Λ(0)(τ (0) +0 ) = 0. We also assume that Λ(0)(·) is a continuous function. As Λ(0)(τ (0) +0 ) = 0, +using the assumption of continuity, ai’s can be expressed in terms of bi’s as follows: +a1 = −b1τ (0) +0 +and +ak = +k−1 +� +ℓ=1 +(bℓ − bℓ+1) τ (0) +ℓ ++ a1 = +k−1 +� +ℓ=1 +bℓ +� +τ (0) +ℓ +− τ (0) +ℓ−1 +� +− bkτ (0) +k−1, +for k = 1, 2, 3, . . . , N. +Note that the above model can be equivalently described in terms of HRs. +In this +approach, h(0)(·) over the range [τ (0) +0 , τ (0) +N ) is approximated by a piecewise constant function +λ(0)(·), where +λ(0)(t) = +N +� +i=1 +bk1[τ (0) +k−1, τ (0) +k +) (t) . +(2.2) +After failure of one or more components within the system, the direct impact of the +increased load will be an increased HR for the operational components. To incorporate this +information, after the failure of j components of the system, we approximate h(j)(·) over +[τ (j) +0 , τ (j) +N ), j = 1, 2, . . . , J − 1, using the piecewise constant function λ(j)(·), where +λ(j)(t) = γj +N +� +k=1 +bk1[τ (j) +k−1, τ (j) +k +) (t) , +(2.3) +with +1 < γ1 < γ2 < . . . < γJ−1. +The PLAs to the CHFs, corresponding to the PLAs of the HRs given in Eq.(2.3) are given +by +Λ(j)(t) = γj +N +� +k=1 +�k−1 +� +ℓ=1 +bℓ +� +τ (j) +ℓ +− τ (j) +ℓ−1 +� ++ bk +� +t − τ (j) +k−1 +�� +1[τ (j) +k−1, τ (j) +k +) (t) . +(2.4) +To meet the non-decreasing nature of the HR, we assume that 0 < b1 < b2 < . . . < bN. Note +that the parameters γ1, γ2, . . . , γJ−1 reflect the load-share rule of increased HRs. We treat +γ1, γ2, . . . , γJ−1 as unknown parameters, and estimate them from component failure data. +It may be mentioned here that the PLA model can be interpreted as an approximation +of the underlying lifetime distribution by several exponential models (with different rate +parameters) over the ranges specified by the cut-points. +6 + +3 +Likelihood Inference +The parameters involved in the PLA-based model are estimated from the component failure +data obtained from a set of load-sharing systems. The available data on component failures +from n J-component load-sharing systems is of the form +Data = +� +y(j) +i +: i = 1, 2, . . . , n; j = 0, 1, . . . , J − 1 +� +, +where y(j) +i +is the observed system lifetime between j-th and (j + 1)-st component failures for +the i-th system. For j = 0, 1, 2, . . . , J − 1, and k = 1, 2, . . . , N, define +I(j) +k += +� +i : y(j) +i +∈ +� +τ (j) +k−1, τ (j) +k +�� +and +n(j) +k += |I(j) +k |. +Obviously, �N +k=1 n(j) +k += n. The likelihood function for the PLA model is then given by +L (θ) = +n +� +i=1 +J−1 +� +j=0 +� +(J − j)γj +N +� +k=1 +bk1[τ (j) +k−1, τ (j) +k +) +� +y(j) +i +� +e +−(J−j)γj +��k−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� ++bk +� +y(j) +i +−τ (j) +k−1 +��� +, +(3.1) +where γ0 = 1 and θ = (γ1, γ2, . . . , γJ−1, b1, b2, . . . , bN)′ is the vector of parameters. The +corresponding log-likelihood function, ignoring additive constant, can be expressed as +l (θ) = +N +� +k=1 +��J−1 +� +j=0 +n(j) +k +� +ln bk − +�J−1 +� +j=0 +(J − j)γjT (j) +k +� +bk +� ++ n +J−1 +� +j=0 +ln γj, +(3.2) +where +T (j) +k += +� +i∈I(j) +k +� +y(j) +i +− τ (j) +k−1 +� ++ +� +n − +k +� +ℓ=1 +n(j) +ℓ +� � +τ (j) +k +− τ (j) +k−1 +� +, +for k = 1, 2, . . . , N; j = 0, 1, . . . , J − 1. Equating partial derivative of the log-likelihood +function in Eq.(3.2) with respect to bk to zero, we can express bk in terms of the load-share +parameters γ = (γ1, γ2, . . . , γJ−1) as +bk = bk (γ) = +J−1 +� +j=0 +n(j) +k +J−1 +� +j=0 +(J − j)γjT (j) +k +, +k = 1, ..., N. +(3.3) +Substituting bk(γ) from Eq.(3.3) in Eq.(3.2), the profile log-likelihood in γ, ignoring additive +constant, is obtained as +˜l (γ) = +N +� +k=1 +��J−1 +� +j=0 +n(j) +k +� � +ln +�J−1 +� +j=0 +n(j) +k +� +− ln +�J−1 +� +j=0 +(J − j)γjT (j) +k +��� ++ n +J−1 +� +j=0 +ln γj. +(3.4) +7 + +For optimizing the profile log-likelihood ˜l (γ) in γ, any routine maximizer of a standard +statistical software may be used. Once the MLEs �γ1, �γ2, . . . , �γJ−1 of γ1, γ2, . . . , γJ−1 are +obtained by numerical optimization of ˜l (γ), they can be plugged into Eq.(3.3) to get MLEs +of bk as +�bk = bk (�γ1, . . . , �γJ−1) , +k = 1, 2, . . . , N. +3.1 +A special case: two-component load-sharing systems +For analysing data from two-component load-sharing systems, if two linear pieces are used +in the PLA-based model, MLEs can be derived analytically and explicitly. Consider the case +when J = 2 and N = 2. In this case, the log-likelihood function simplifies to +l (θ) = +2 +� +k=1 +�� +1 +� +j=0 +n(j) +k +� +ln bk − +� +1 +� +j=0 +(2 − j)γjT (j) +k +� +bk +� ++ n +1 +� +j=0 +ln γj, +(3.5) +with +T (j) +k += +� +i∈I(j) +k +� +y(j) +i +− τ (j) +k−1 +� ++ +� +n − +k +� +ℓ=1 +n(j) +ℓ +� � +τ (j) +k +− τ (j) +k−1 +� +, +for k = 1, 2, j = 0, 1 and γ0 = 1. Here, θ = (γ1, b1, b2). +Equating ∂l(θ) +∂b1 and ∂l(θ) +∂b2 to zero, we get +b1 = +n(0) +1 ++ n(1) +1 +2T (0) +1 ++ γ1T (1) +1 +(3.6) +b2 = +n(0) +2 ++ n(1) +2 +2T (0) +2 ++ γ1T (1) +2 +. +(3.7) +Equating ∂l(θ) +∂γ1 to zero gives +γ1 = T (1) +1 b1 + T (1) +2 b2, +(3.8) +in which, substituting b1 and b2 from Eqs.(3.6) and (3.7), a quadratic equation in γ1 is +obtained as follows +Q(γ1) = nγ2 +1B0,12 + 2γ1 +�� +n(0) +1 ++ n(1) +1 +− n +� +B2,1 + +� +n(0) +2 ++ n(1) +2 +− n +� +B1,2 +� +− 4nB12,0 = 0, +(3.9) +with B0,12 = T (1) +1 T (1) +2 , B1,2 = T (0) +1 T (1) +2 , B2,1 = T (0) +2 T (1) +1 +and B12,0 = T (0) +1 T (0) +2 . Solving Q(γ1) = +0, we have two values of γ1 from which we choose the suitable one, and then from equations +(3.6) and (3.7) we get the MLEs of b1 and b2, respectively. +8 + +3.2 +Confidence Intervals +As discussed above, the MLEs for the parameters of the PLA-based model are not available +in explicit form in general, except for the special case of two-component load-sharing systems +considered in Section 3.1. As a result, exact confidence intervals for the model parameters +cannot be obtained. Asymptotic confidence intervals may be constructed in two possible +ways: by using the Fisher information matrix, and by applying a bootstrap-based technique. +3.2.1 +CIs using Fisher information matrix +Using the asymptotic properties of the MLEs, it can be shown that for large sample size +n, the distribution of √n(�θ − θ) is approximated by a multi-variate normal distribution +N(0, I−1(�θ)), where the dimension of the multi-variate normal distribution is same as that +of the parameter vector θ, and the asymptotic variance-covariance matrix I−1(θ) is the in- +verse of the Fisher information matrix I(θ), evaluated at the MLE �θ. The Fisher information +matrix I(θ) is defined as the expected value of the observed information matrix J(θ) which +is calculated from the negative of the second-order derivatives of the log-likelihood function. +That is, I(θ) = E(J(θ)), where J(θ) = −∇2(log L(θ)). In situations where analytical calcu- +lation of the Fisher information is difficult or intractable, it may be either replaced by the +observed information matrix, or may be calculated by simulations. +From the asymptotic variance-covariance matrix I−1(θ), individual asymptotic variances +of the MLEs can be pulled out, and asymptotic confidence intervals can be constructed. For +example, corresponding to the MLE �γ1 using the asymptotic variance +� +V ar(ˆγ1) obtained from +I−1(θ), asymptotic confidence intervals for γ1 can be constructed as: +� +�γ1 − zα/2 +� +� +V ar(ˆγ1), �γ1 + zα/2 +� +� +V ar(ˆγ1) +� +, +where zα is the 100(1 − α)% point of the standard normal distribution. +Special case: two-component load-sharing systems +For the special case of two-component load-sharing systems considered in Section 3.1, the +Fisher information matrix can be worked out explicitly. In this case, +J(θ) = − + + + + + + + + +∂2l(θ) +∂γ2 +1 +∂2l(θ) +∂γ1∂b1 +∂2l(θ) +∂γ1∂b2 +∂2l(θ) +∂b1∂γ1 +∂2l(θ) +∂b2 +1 +∂2l(θ) +∂b1∂b2 +∂2l(θ) +∂b2∂γ1 +∂2l(θ) +∂b2∂b1 +∂2l(θ) +∂b2 +2 + + + + + + + + += − + + + + +− n +γ2 +1 +−T (1) +1 +−T (1) +2 +−T (1) +1 +−n(0) +1 +n(1) +1 +b12 +0 +−T (1) +2 +0 +−n(0) +2 +n(1) +2 +b22 + + + + . +9 + +Hence, the Fisher information matrix is +I(θ) = + + + + + + + + + + + + + + + + + + + + + +n +γ2 +1 +E + + + +� +i∈I(1) +1 +Y (1) +i + + + + E +� +N(1) +2 +� +τ (1) +1 +E + + + +� +i∈I(1) +1 +Y (1) +i + + + − E +� +N(1) +2 +� +τ (1) +1 +E + + + +� +i∈I(1) +1 +Y (1) +i + + + + E +� +N(1) +2 +� +τ (1) +1 +E +� +N(0) +1 +� ++E +� +N(1) +1 +� +b2 +1 +0 +E + + + +� +i∈I(1) +1 +Y (1) +i + + + − E +� +N(1) +2 +� +τ (1) +1 +0 +E +� +N(0) +2 +� ++E +� +N(1) +2 +� +b2 +2 + + + + + + + + + + + + + + + + + + + + + +, +where N(j) +k +is the number of Y (j) +i +in [τ (j) +k−1, τ (j) +k ), k = 1, 2, j = 0, 1, i = 1, ..., n. An outline +of calculations of the relevant expectations for the Fisher information matrix is given in +Appendix A. The inverse of the Fisher information matrix is obtained as +� +I−1(θ) +� += +1 +|I(θ)| + + +A11(θ) +−A12(θ) +A13(θ) +−A21(θ) +A22(θ) +−A23(θ) +A31(θ) +−A32(θ) +A33(θ) + + , +where the determinant of I(θ) is +|I(θ)| += +n +� +2 − +� +e−2b1τ (0) +1 ++ e−γ1b1τ (1) +1 +�� � +e−2b1τ (0) +1 ++ e−γ1b1τ (1) +1 +� +γ2 +1b2 +1b2 +2 +− +e−2γ1b1τ (1) +1 +� +1 +γ1b2 +�2 � +2 − +� +e−2b1τ (0) +1 ++ e−γ1b1τ (1) +1 +�� +b2 +1 +− +� +1 +γ1b1 +� +1 − (1 + γ1b1τ (1) +1 )e−γ1b1τ (1) +1 +� ++ τ (1) +1 e−γ1b1τ (1) +1 +�2 � +e−2b1τ (0) +1 ++ e−γ1b1τ (1) +1 +� +b2 +2 +, +A11(θ) = +� +2 − +� +e−2b1τ (0) +1 ++ e−γ1b1τ (1) +1 +�� � +e−2b1τ (0) +1 ++ e−γ1b1τ (1) +1 +� +b2 +1b2 +2 +, +A22(θ) = +n +� +e−2b1τ (0) +1 ++ e−γ1b1τ (1) +1 +� +γ2 +1b2 +2 +− e−2γ1b1τ (1) +1 +� 1 +γ1b2 +�2 +, +A33(θ) = +n +� +2 − +� +e−2b1τ (0) +1 ++ e−γ1b1τ (1) +1 +�� +γ2 +1b2 +1 +− +� 1 +γ1b1 +� +1 − (1 + γ1b1τ (1) +1 )e−γ1b1τ (1) +1 +� ++ τ (1) +1 e−γ1b1τ (1) +1 +�2 +, +A12(θ) = A21(θ) = +� +1 +γ1b1 +� +1 − (1 + γ1b1τ (1) +1 )e−γ1b1τ (1) +1 +� ++ τ (1) +1 e−γ1b1τ (1) +1 +� � +e−2b1τ (0) +1 ++ e−γ1b1τ (1) +1 +� +b2 +2 +, +10 + +A13(θ) = A31(θ) = − +e−γ1b1τ (1) +1 +� +1 +γ1b2 +� � +2 − +� +e−2b1τ (0) +1 ++ e−γ1b1τ (1) +1 +�� +b2 +1 +, +A23(θ) = A32(θ) = − +�� +1 − (1 + γ1b1τ (1) +1 )e−γ1b1τ (1) +1 +� ++ γ1b1τ (1) +1 e−γ1b1τ (1) +1 +� +e−γ1b1τ (1) +1 +γ2 +1b1b2 +. +Evaluating I−1(θ) at the MLE �θ, the asymptotic variance-covariance matrix of the MLEs +is obtained. +Hence, 100(1 − α)% asymptotic confidence intervals for γ1, b1, and b2 are +obtained as +� +�γ1 −zα/2 +� +A11(ˆθ) +|I(ˆθ)| , �γ1 +zα/2 +� +A11(ˆθ) +|I(ˆθ)| +� +, +� +�b1 −zα/2 +� +A22(ˆθ) +|I(ˆθ)| , �b1 +zα/2 +� +A22(ˆθ) +|I(ˆθ)| +� +, and +� +�b2 − zα/2 +� +A33(ˆθ) +|I(ˆθ)| , �b2 + zα/2 +� +A33(ˆθ) +|I(ˆθ)| +� +, respectively. +3.2.2 +Bootstrap confidence intervals +Using the MLE �θ, B bootstrap samples can be obtained in the same sampling framework; +let �θ +∗ +s = +� +�γ∗ +1s,�b∗ +1s,�b∗ +2s +� +denote the bootstrap estimates, s = 1, ..., B. Bootstrap bias and +standard error are defined as +biasb(�γ1) = �γ∗ +1 − �γ1, +biasb(�b1) = �b∗ +1 −�b1, +biasb(�b2) = �b∗ +2 −�b2 +and +SEb(�γ1) = +� +� +� +� +1 +B − 1 +B +� +s=1 +� +�γ∗ +1s − � +γ∗ +1 +�2 +, SEb(�b1) = +� +� +� +� +1 +B − 1 +B +� +s=1 +� +�b∗ +1s − �b∗ +1 +�2 +, SEb(�b2) = +� +� +� +� +1 +B − 1 +B +� +s=1 +� +�b∗ +2s − �b∗ +2 +�2 +, +where +�γ∗ +1 = 1 +B +B +� +s=1 +�γ∗ +1s, +�b∗ +1 = 1 +B +B +� +s=1 +�b∗ +1s, +�b∗ +2 = 1 +B +B +� +s=1 +�b∗ +2s. +Finally, a 100(1 − α)% bootstrap confidence interval for γ1 can be calculated as +� +�γ1 − biasb(�γ1) − zα/2SEb(�γ1), �γ1 − biasb(�γ1) + zα/2SEb(�γ1) +� +. +Bootstrap confidence intervals for b1 and b2 can be calculated similarly. +For percentile bootstrap confidence intervals for, say γ1, the bootstrap estimates of �γ1 +are first ordered in terms of magnitude: +�γ∗ +1(1) < �γ∗ +1(2) < ... < �γ∗ +1(B). +Then, a 100(1−α)% percentile bootstrap confidence interval for γ1 is +� +�γ∗ +1([ αB +2 ]), �γ∗ +1([(1− α +2 )B]) +� +. +Similarly, percentile bootstrap confidence intervals can be calculated for b1 and b2. +11 + +3.3 +Choice of Cut Points +The number and position of the cut-points for constructing the PLA-based model need to +be suitably chosen, so that the model can closely approximate the underlying CHF, but +avoid overfitting. A large number of cut points would provide a close local approximation +to the underlying CHF. However, apart from being computationally expensive, a close local +approximation may also lead to overfitting in which case it would be difficult to use the +PLA-based model to predict future failures of components or systems. +One of the possible ways to choose the number and position of the cut-points is by looking +at the plot of the nonparametric estimator of CHF. From such a plot, observing the areas +where the nonparametric estimate changes significantly, one can determine the positions and +number of cut-points. +More objectively, one can choose the positions of a given number of cut-points by max- +imizing the log-likelihood function. For example, for three cut-points (N = 2), the natural +choice for τ (j) +0 +is min +� +y(j) +1 , . . . , y(j) +n +� +and τ (j) +2 +is max +� +y(j) +1 , . . . , y(j) +n +� +. Now to choose the +position of τ (j) +1 , one may take τ (j) +1 +equal to different sample quantiles of +� +y(j) +1 , . . . , y(j) +n +� +and +choose one that provides the maximum value of log-likelihood function evaluated at MLE. +This process can be expressed as an algorithm as follows. +Algorithm: +• Step 1: Fix 0 < p1 < p2 < 1. +• Step 2: Find the number of y(j) +1 , . . . , y(j) +n +that are between p1-th and p2-th sample +quantiles of +� +y(j) +1 , . . . , y(j) +n +� +. Denote this number by l. Note that l does not depend +on j = 0, 1, . . . , J − 1. +• Step 3: Set aj1 = p1-th quantile of +� +y(j) +1 , . . . , y(j) +n +� +, j = 0, 1, . . . , J − 1. +• Step 4: Set LL1= the value of log-likelihood function evaluated at MLE taking τ (j) +1 += +aj1, j = 0, 1, . . . , J − 1. +• Step 5: Set aj2 = min +� +y(j) +i +> aj1; i = 1, 2, . . . , n +� +, j = 0, 1, . . . , J − 1. +• Step 6: Set LL2= the value of log-likelihood function evaluated at MLE taking τ (j) +1 += +aj2, j = 0, 1, . . . , J − 1. +• Step 7: Repeat the steps 5 and 6 to obtain LL1, LL2, . . . , LLl. +• Step 8: Set k∗ = arg max +1≤k≤l +LLk. +• Step 9: The final cut points are τ (j) +1 += ajk∗, j = 0, 1, . . . , J − 1. +12 + +4 +Estimation of various reliability characteristics +The final goal of fitting a model to load-sharing data, naturally, is accurate estimation of +reliability characteristics of load-sharing systems. As the PLA-based model provides a good +fit to load-sharing data due to the model’s flexible nature, it is natural that the important +reliability characteristics of load-sharing systems can also be estimated quite accurately +under this model. In this section, we develop estimates of reliability characteristics such as +the quantile function, MTTF, RMT, and MRT of load-sharing systems under the PLA-based +model. Details of these derivations are given in Appendix B for interested readers. +Under the PLA-based model, the quantile function of Y (j) which is the system lifetime +between the j-th and (j + 1)-st component failures, j = 0, ..., J − 1, is given by +η(p) = inf +� +y ∈ R : G(j)(y) ≥ p +� +, +0 < p < 1, +where G(j)(y) = 1 − e−(J−j)Λ(j)(y). Using the expression of Λ(j)(y) given in Section 2, it is +possible to work out an explicit formula for the quantile function η(p), as follows: +η(p) = + + + + + + + + + + + + + + + + + +τ (j) +k−1 − +log(1−p) +(J−j)γjbk − 1 +bk · +k−1 +� +ℓ=1 +bℓ(τ (j) +ℓ +− τ (j) +ℓ−1), if p ∈ +� +G(j)(τ (j) +k−1), G(j)(τ (j) +k ) +� +, +for k = 1, 2, . . . , N. +τ (j) +N−1 − +log(1−p) +(J−j)γjbN − +1 +bN · +N−1 +� +ℓ=1 +bℓ(τ (j) +ℓ +− τ (j) +ℓ−1), if p ∈ +� +G(j)(τ (j) +N ), 1 +� +. +The mean time to failure or MTTF of a load-sharing system is the expected time the +system operates till its failure. Let T denote the system failure time; then, T = �J−1 +j=0 Y (j). +The MTTF of a load-sharing system under the PLA-based model is given by +E(T) = +J−1 +� +j=0 +N +� +s=1 +�e−κj,s−1 − e−κj,s +(J − j)γjbℓ +� +, +where +κj,s = (J − j)γj +s +� +ℓ=1 +bℓ +� +τ (j) +ℓ +− τ (j) +ℓ−1 +� +. +Reliability at a mission time or RMT of a system is the probability that the system will +operate till a desired time t0; it is calculated as the survival probability of the system at +time t0, i.e., S(t0) = P(T > t0) = P +�J−1 +� +j=0 +Y (j) > t0 +� +. An explicit expression for RMT may +be derived by using the distribution of the system lifetime T. +However, as Y (j)s, j = 0, ..., J − 1 are independent but not identically distributed, it is +difficult to obtain an explicit expression for the distribution of the system lifetime T, where +T = �J−1 +j=0 Y (j). It is evident from the moment generating function φT(t) of T, which, under +the PLA-based model, is given by +φT(t) = +J−1 +� +j=0 +N +� +s=1 +(J − j)bsγj +(J − j)bsγj − t +� +etτ (j) +s−1−κj,s−1 − etτ (j) +s +−κj,s +� +if t < γ1bN, +13 + +where +κj,s = (J − j)γj +s +� +ℓ=1 +bℓ +� +τ (j) +ℓ +− τ (j) +ℓ−1 +� +. +From here, it is clear that it is difficult to find the RMT analytically under this model. +However, for this model, RMT can be estimated using Monte Carlo simulations. +For a +Monte Carlo estimate of the RMT at a pre-specified time t0, one needs to generate R data +points ti, i = 1, 2, . . . , R, as realisations of the system lifetime T, and find R(t0) +R , where R(t0) +is the number of realisations of the system lifetime that exceed t0. For a reasonably good +estimate of RMT, a large value of R should be used. +The mean residual time or MRT of a system is the expected additional time the system +will survive if it has already survived a given time t. That is, +MRT(t) = E(T − t|T > t) = +� ∞ +t +sfT|T>t(s)ds − t. +Therefore, analytical derivation of MRT requires the truncated distribution of the system +lifetime T, and it is difficult to obtain the truncated distribution of T in this case. Instead, +an estimate of the MRT can be given using Monte Carlo simulations. We generate R data +points t∗ +i , i = 1, 2, . . . , R, as realisations of the truncated lifetime T|T > t, and a Monte +Carlo estimate of the MRT for load-sharing systems under the PLA-based model is then +given by +� +MRT(t) = +R +� +i=1 +t∗ +i +R +− t. +5 +Data Analysis +In this section, we present an illustrative example using data from load-sharing systems +comprising of two components. Very recently, this data have been analysed by Sutar and +Naik-Nimbalkar [34], Asha et al. [2] and Franco et al. [11]. The data consist of information +on component lifetimes of 18 two-component load-sharing systems. Each system is a parallel +combination of two motors - “A” and “B”. When both motors A and B are in working +condition, the total load on the system is shared between them. When one of the motors +fails, the entire load goes to the operational motor. +Sutar and Naik-Nimbalkar [34] observed that the load-sharing phenomenon existed for +the systems considered in this dataset. Asha et al. [2] assumed Weibull lifetimes for the +components. From the Weibull Q-Q plots for the lifetimes of motor A and B reported in +Asha et al. [2], it was observed that although the Weibull model assumption for the lifetimes +of motor B was reasonable, the lifetimes of motor A did not follow a Weibull distribution. +This motivated us to consider the PLA-based modelling approach for the lifetimes of the +load-sharing systems in this case. +The dataset is reproduced in Table 1 for ready reference of the readers. The average +and standard deviation of first component failure times are 178.61 and 62.75, respectively, +14 + +0 +100 +200 +300 +0 +100 +200 +300 +Sample quantile +Population quantile +(a) Q-Q plot for Y (0) +0 +25 +50 +75 +100 +125 +0 +25 +50 +75 +100 +125 +Sample quantile +Population quantile +(b) Q-Q plot for Y (1) +Figure 1: Q-Q plots +0.25 +0.50 +0.75 +1.00 +100 +150 +200 +250 +300 +Time +SF +(a) Plot of SF for Y (0) +0.00 +0.25 +0.50 +0.75 +1.00 +0 +40 +80 +120 +Time +SF +(b) Plot of SF for Y (1) +Figure 2: Plots of SFs +while those of the lifetime between first and second component failures are 49.72 and 29.45, +respectively. We consider three cut points for the PLA-based model (i.e., N = 2). The +estimates of the model parameters are reported in Table 2. The Q-Q plots for Y (0) and Y (1) +are given in Figures 1a and 1b, respectively. The plots of the estimated SF and CHF are +given in Figures 2 and 3, respectively. These figures indicate that the PLA-based model fits +the data quite adequately. +15 + +0.0 +0.5 +1.0 +1.5 +100 +150 +200 +250 +300 +Time +CHF +(a) Plot of CHF for Y (0) +0 +2 +4 +6 +0 +40 +80 +120 +Time +CHF +(b) Plot of CHF for Y (1) +Figure 3: Plots of CHFs +Table 1: Time to failure (in days) data set for two motors in a load-sharing configuration +System +Time to failure of motor A +Time to failure of motor B +Event description +1 +102 +65 +B failed first +2 +84 +148 +A failed first +3 +88 +202 +A failed first +4 +156 +121 +B failed first +5 +148 +123 +B failed first +6 +139 +150 +A failed first +7 +245 +156 +B failed first +8 +235 +172 +B failed first +9 +220 +192 +B failed first +10 +207 +214 +A failed first +11 +250 +212 +B failed first +12 +212 +220 +A failed first +13 +213 +265 +A failed first +14 +220 +275 +A failed first +15 +243 +300 +A failed first +16 +300 +248 +B failed first +17 +257 +330 +A failed first +18 +263 +350 +A failed first +A Kolmogorov-Smirnov type test has been performed to test the following hypotheses: +H0 : True model is specified by Eqs. (2.2) and (2.3) +against +H1 : True model is not specified by Eqs. (2.2) and (2.3) +16 + +Table 2: Point and interval estimates of parameters of the PLA-based model when applied +to the two-motor load-sharing data +Parameter +MLE +Std. Error +Asymptotic +Percentile bootstrap +Bootstrap +γ1 +4.2712 +1.1901 +(1.9386, 6.6038) +(3.0754, 8.0279) +(0.8456, 5.8172) +b1 +0.0034 +0.0008 +(0.0019, 0.0048) +(0.0021, 0.0062) +(0.0008, 0.0052) +b2 +0.0134 +0.0039 +(0.0056, 0.0212) +(0.0061, 0.0209) +(0.0083, 0.0232) +Table 3: Mean residual time and reliability in mission time +t0 +MRTt0 +RMTt0 +102.00 +124.223 +0.963 +167.50 +88.678 +0.706 +227.50 +60.646 +0.466 +272.50 +42.794 +0.271 +350.00 +36.919 +0.044 +based on the test statistics +Tn = max +1≤i≤n +���� �G(0) � +Y (0) +i:n +� +− i +n +���� + max +1≤i≤n +���� �G(1) � +Y (1) +i:n +� +− i +n +���� , +where �G(j)(·) is the estimated cumulative distribution function corresponding to PLA-based +model, and Y (j) +i:n is the i-th order statistics corresponding to Y (j) +i +, j = 0, 1, i = 1, 2, . . . , n. +The observed value of the test statistics Tn is found to be 0.414 based on this data. The +Monte Carlo estimate of the corresponding p-value is 0.71. Therefore, the null hypothesis +cannot be rejected at significance level 0.05, and we conclude that it is quite reasonable to +use the PLA-based model for this data. +It may also be noted here that for this data, the value of the Akaike’s information +criterion (AIC) for the model considered by Asha et al. [2] is 480.50, and that for the best +model considered by Franco et al. [11] is 409.65. In contrast, the AIC value for the PLA- +based model turns out to be 369.34, implying that the PLA-based model is more suitable +for the two-motor load-sharing systems data considered here. +For the PLA-based model, the estimated value of γ1 is 4.2712, which empirically implies +that the load-sharing model is quite appropriate in this case. The same comment can also +be made from the plots, by noting that the plot of the SF of the distribution of time between +first and second failure component times diminishes to zero more quickly compared to that +of first component failure times in Figure 2. +The reliability characteristics of the two-motor load-sharing systems are also estimated +by using the expressions and techniques described in Section 4. The MTTF is calculated +to be 221.36 days. Monte Carlo estimates of the MRT and RMT are calculated at different +sample percentile points of the system failure times and are presented in Table 3. +17 + +6 +Simulation Study +The accuracy of the proposed PLA-based model in fitting data from load-sharing systems is +of utmost importance as the subsequent estimation of reliability characteristics depends on +the PLA-based model. In this section, we present results of a Monte Carlo simulation study +that examines the performance of the proposed PLA-based model in two directions. First, +based on samples generated from a parent process with piecewise linear CHF, we assess the +performance of the proposed estimation method that is presented in Section 3. Then, the +efficacy of the PLA-based model in fitting data generated from a parent process represented +by some parametric models is also assessed. The simulations are carried out by using R +software. For the simulations, we consider two-component load-sharing systems. +6.1 +Assessing performance of the estimation method +To assess the performance of the estimation methods, we consider an underlying cumulative +hazard that is made up of two linear pieces. To this effect, we generate samples from the +model specified by Eqs.(2.2) and (2.3) with J = 2 and N = 2. The true parameter values +are taken to be b1 = 0.01, 0.05; b2 = 0.1, 0.5; γ1 = 5; τ (0) +1 += ln 2 +2b1 ; τ (1) +1 += +ln 2 +γ1b1. The estimation +is performed based on samples of size n = 100 and 200. The average estimates (AE), mean +square errors (MSE), variance (VAR) of the MLEs based on 5000 Monte Carlo replications +are reported in Tables 4, 5, and 6. The coverage percentage (CP) and average lengths (AL) +of 95% confidence intervals are also reported in the same tables. +From the Tables 4, 5 and 6, we observe that the average estimates of γ1, b1 and b2 are +very close to the true values, and the MSEs as well as VARs are quite small as desired. It is +also noticed that the performance of all the constructed confidence intervals is satisfactory. +These results demonstrate that the proposed inferential techniques can accurately estimate +the parameters of the PLA-based model. +Table 4: Performance measures for estimates of γ1 +n +b1 +b2 +AE +MSE +VAR +Asymptotic +Percentile bootstrap +Bootstrap +CP +AL +CP +AL +CP +AL +0.01 0.1 +5.0231 +0.3388811 +0.3384155 +94.38 +2.2566 +99.94 +2.2012 +83.58 +2.2156 +0.5 +5.0178 +0.2880872 +0.2878297 +95.84 +2.2509 +99.94 +2.1278 +86.68 +2.1993 +100 +0.05 0.1 +5.0209 +0.3556721 +0.3553026 +93.98 +2.2711 +98.52 +2.3093 +88.60 +2.3226 +0.5 +5.0231 +0.3388811 +0.3384155 +94.38 +2.2566 +99.94 +2.2012 +83.58 +2.2156 +0.01 0.1 +5.0144 +0.1472563 +0.1470773 +96.20 +1.5963 +99.90 +1.5000 +85.06 +1.5093 +0.5 +5.0127 +0.1373738 +0.1372388 +96.64 +1.5949 +99.86 +1.4395 +84.42 +1.4476 +200 +0.05 0.1 +5.0145 +0.1698002 +0.1696241 +94.84 +1.6037 +98.56 +1.6165 +89.56 +1.6246 +0.5 +5.0144 +0.1472563 +0.1470773 +96.20 +1.5963 +99.90 +1.5000 +85.06 +1.5093 +18 + +Table 5: Performance measures for estimates of b1 +n +b1 +b2 +AE +MSE +VAR +Asymptotic +Percentile bootstrap +Bootstrap +CP +AL +CP +AL +CP +AL +0.01 0.1 +0.0108 +0.0000022 +0.0000016 +87.24 +0.0041 +81.66 +0.0052 +92.88 +0.0052 +0.5 +0.0110 +0.0000025 +0.0000016 +84.56 +0.0042 +68.70 +0.0053 +93.96 +0.0054 +100 +0.05 0.1 +0.0513 +0.0000389 +0.0000371 +90.10 +0.0201 +95.96 +0.0245 +93.70 +0.0246 +0.5 +0.0528 +0.0000457 +0.0000376 +89.18 +0.0203 +89.88 +0.0252 +92.70 +0.0253 +0.01 0.1 +0.0105 +0.0000010 +0.0000008 +86.42 +0.0029 +81.74 +0.0035 +93.48 +0.0036 +0.5 +0.0106 +0.0000011 +0.0000008 +84.74 +0.0029 +74.08 +0.0036 +94.56 +0.0036 +200 +0.05 0.1 +0.0508 +0.0000186 +0.0000180 +90.06 +0.0140 +95.14 +0.0169 +93.08 +0.0169 +0.5 +0.0525 +0.0000255 +0.0000193 +86.42 +0.0143 +81.74 +0.0177 +93.48 +0.0178 +Table 6: Performance measures for estimates of b2 +n +b1 +b2 +AE +MSE +VAR +Asymptotic +Percentile bootstrap +Bootstrap +CP +AL +CP +AL +CP +AL +0.01 0.1 +0.1006 +0.0001691 +0.0001688 +92.70 +0.0464 +96.60 +0.0534 +94.00 +0.0530 +0.5 +0.5067 +0.0038339 +0.0037895 +94.52 +0.2344 +97.12 +0.2675 +95.12 +0.2676 +100 +0.05 0.1 +0.1030 +0.0001794 +0.0001705 +93.76 +0.0472 +95.46 +0.0529 +94.70 +0.0532 +0.5 +0.5028 +0.0042337 +0.0042268 +92.68 +0.2315 +96.34 +0.2650 +94.00 +0.2633 +0.01 0.1 +0.1002 +0.0000725 +0.0000725 +94.22 +0.0325 +96.72 +0.0343 +93.76 +0.0345 +0.5 +0.5030 +0.0017348 +0.0017261 +95.06 +0.1632 +96.52 +0.1665 +93.60 +0.1674 +200 +0.05 0.1 +0.1011 +0.0000780 +0.0000768 +93.84 +0.0326 +96.10 +0.0351 +94.08 +0.0352 +0.5 +0.5010 +0.0018134 +0.0018128 +94.22 +0.1623 +96.72 +0.1717 +93.76 +0.1725 +6.2 +Assessing efficacy of the PLA-based model in fitting data +from other models +Now, we examine the robustness of the PLA-based model in the following manner. +We +generate load-sharing data from parametric models, and then fit the PLA-based model to +the data. +The model fit is then assessed with respect to an integrated measure that is +suitably defined to reflect the quality of approximation provided by the PLA-based model. +The measure, which we call the Absolute Integrated Error (AIE), is as follows. For j = 0, 1, +let S(j) +TGP(·) and H(j) +TGP(·) denote the SF and CHF of the lifetimes between j-th and (j + 1)- +st failures. Also, assume that the estimated SF and CHF based on PLA-based model are +denoted by �S(j) +P LA(·) and �H(j) +P LA(·), respectively. Then the AIE, based on the SF and CHF, +respectively, are defined as +AIE(j) +SF = 1 +R +R +� +k=1 +1 +y(j) +max − y(j) +min +� y(j) +max +y(j) +min +���S(j) +TGP(t) − �S(j) +P CA(t) +��� dt, +19 + +Table 7: AIE based on SF and CHF for Weibull distribution with k = 3, β = 1. +n +α +AIE(0) +SF +AIE(1) +SF +AIE(0) +CHF +AIE(1) +CHF +50 +1.0 +0.0379 +0.0291 +0.1503 +0.2981 +1.5 +0.0436 +0.0434 +0.1329 +0.2633 +100 +1.0 +0.0266 +0.0183 +0.1282 +0.2541 +1.5 +0.0326 +0.0301 +0.1231 +0.2440 +Table 8: AIE of the survival and cumulative hazard function of quadratic distribution for +κ1 = 0.5, ˜κ1 = 2κ1 = 1, ˜κ2 > 2κ2. +n +κ2 +˜κ2 +AIE(0) +SF +AIE(1) +SF +AIE(0) +CHF +AIE(1) +CHF +50 +0.50 +1.50 +0.0380 +0.0368 +0.1262 +0.2536 +2.00 +0.0380 +0.0389 +0.1261 +0.2555 +0.70 +1.50 +0.0389 +0.0363 +0.1261 +0.2524 +2.00 +0.0388 +0.0383 +0.1258 +0.2539 +100 +0.50 +1.50 +0.0289 +0.0262 +0.1185 +0.2506 +2.00 +0.0289 +0.0281 +0.1178 +0.2575 +0.70 +1.50 +0.0301 +0.0257 +0.1217 +0.2465 +2.00 +0.0299 +0.0274 +0.1206 +0.2528 +AIE(j) +CHF = 1 +R +R +� +k=1 +1 +y(j) +max − y(j) +min +� y(j) +max +y(j) +min +���H(j) +TGP(t) − �H(j) +P CA(t) +��� dt, +where y(j) +min = min +� +y(j) +1 , y(j) +2 , . . . , y(j) +n +� +, y(j) +max = max +� +y(j) +1 , y(j) +2 , . . . , y(j) +n +� +, j = 0, 1. +For generating load-sharing data from parametric models, two scenarios are considered: +(a) Case - 1: It is assumed that the lifetimes of each components of a two-component load- +sharing system are independent and identically distributed as Weibull distribution with shape +parameter α and scale parameter β when both the components are working. After the first +failure, the lifetime of the surviving component is assumed to follow a Weibull distribution +with same shape parameter α, but a different scale parameter kβ, where k > 2 is to ensure +the increase of load on the surviving component. For β = 1, k = 3, we take α = 1 and 1.5. +(b) Case - 2: In the second scenario, the component lifetimes are assumed to be independent +and identically distributed according to a distribution with quadratic CHF κ1t + κ2t2 when +both components are working. After the first failure, the lifetime of the surviving component +is assumed to follow a quadratic CHF with different parameters ˜κ1 and ˜κ2. We take several +values of the parameters κ1, κ2, ˜κ1, and ˜κ2 ensuring the fact that the CHF increases after +one component fails in the system. +The numerical results are reported in Tables 7, 8, and 9. For all cases, it is observed +that the values of AIE based on SF and CHF are reasonably small, indicating that the +20 + +Table 9: AIE of the survival and cumulative hazard function of quadratic distribution for +˜κ1 > 2κ1, κ2 = 0.5, ˜κ2 = 2κ2 = 1. +n +κ1 +˜κ1 +AIE(0) +SF +AIE(1) +SF +AIE(0) +CHF +AIE(1) +CHF +50 +0.50 +1.50 +0.0388 +0.0313 +0.1283 +0.2570 +2.00 +0.0397 +0.0307 +0.1309 +0.2672 +0.70 +1.50 +0.0372 +0.0314 +0.1284 +0.2567 +2.00 +0.0377 +0.0304 +0.1301 +0.2644 +100 +0.50 +1.50 +0.0306 +0.0210 +0.1265 +0.2290 +2.00 +0.0319 +0.0206 +0.1325 +0.2285 +0.70 +1.50 +0.0278 +0.0210 +0.1184 +0.2331 +2.00 +0.0285 +0.0198 +0.1227 +0.2271 +PLA-based model provides quite a satisfactory approximation to the data generated from +different parent populations. +7 +Concluding Remarks +In this article, a PLA-based model for the CHF is proposed for data from load-sharing sys- +tems and then important reliability characteristics such as quantile function, RMT, MTTF, +and MRT of load-sharing systems are estimated under the proposed model. The principal +advantages of the model are that it is data-driven, and does not use strong parametric as- +sumptions for the underlying lifetime variable. Likelihood inference for the proposed model is +discussed in detail. It is observed that for two-component load-sharing systems, it is possible +to obtain explicit expressions for the MLEs of parameters of the PLA-based model. Construc- +tion of confidence intervals using the Fisher information matrix and bootstrap approaches +are also discussed. Derivations of the important reliability characteristics are provided in +this setting. +A Monte Carlo simulation study is performed to examine (a) the performance of the +methods of inference, and (b) the efficacy of the PLA-based model to fit load-sharing data +in general. +It is shown that the PLA-based model performs quite satisfactorily in both +cases. +Analysis of data pertaining to components lifetimes of a two-motor load-sharing +system is provided as an illustration. It is illustrated that the PLA-based model is supe- +rior to the models that have been considered for this data in the literature of load-sharing +systems. In summary, in this paper, an efficient PLA-based modelling framework using mini- +mal assumptions for load-sharing systems is discussed, and estimates of important reliability +characteristics for load-sharing systems in this setting are developed. +21 + +Funding information +• The research of Ayon Ganguly is supported by the Mathematical Research Impact Cen- +tric Support (File no. MTR/2017/000700) from the Science and Engineering Research +Board, Department of Science and Technology, Government of India. +• The research of Debanjan Mitra is supported by the Mathematical Research Impact +Centric Support (File no. MTR/2021/000533) from the Science and Engineering Re- +search Board, Department of Science and Technology, Government of India. +References +[1] S.V. Amari and R. Bergman. 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IEEE Transactions on Reliability, 67:1096– +1110, 2018. +Appendix A: Calculation of Fisher information ma- +trix for two-component load sharing systems +For calculating I(θ), the required expectations are E +� +N(0) +1 +� +, E +� +N(0) +2 +� +, E +� +N(1) +1 +� +, E +� +N(1) +2 +� +, +E + + + +� +i∈I(1) +1 +Y (1) +i + + + and E + + + +� +i∈I(1) +2 +Y (1) +i + + +. +Note that +N(0) +k +∼ Bin(n, p(0) +k ), +N(1) +k +∼ Bin(n, p(1) +k ), +with +p(0) +k += P +� +Y (0) +i +∈ [τ (0) +k−1, τ (0) +k ) +� +, +p(1) +k += P +� +Y (1) +i +∈ [τ (1) +k−1, τ (1) +k ) +� +, +k = 1, 2. +24 + +In case of a two-component load-sharing system, PDF of Y (j) +i +, j = 1, 2, is given by +gY (j) +i (y) = (2 − j)λ(j)(y)e−(2−j) +� y +0 λ(j)(u)du. +Hence, +p(0) +1 += +� τ (0) +1 +0 +gY (0) +i +(y)dy = 1 − e−2b1τ (0) +1 , +p(1) +1 += +� τ (1) +1 +0 +gY (1) +i +(y)dy = 1 − e−γ1b1τ (1) +1 . +Then, p(0) +2 += 1 − p(0) +1 += e−2b1τ (0) +1 +and p(1) +2 += 1 − p(1) +1 += e−γ1b1τ (1) +1 . Therefore, +E(N(0) +1 ) = 1−e−2b1τ (0) +1 , +E(N(0) +2 ) = e−2b1τ (0) +1 , +E(N(1) +1 ) = 1−e−γ1b1τ (1) +1 , +E(N(1) +2 ) = e−γ1b1τ (1) +1 . +Now, +E + + + +� +i∈I(1) +1 +Y (1) +i + + + = E + + +E + + + +� +i∈I(1) +1 +Y (1) +i +|N(1) +1 += n(1) +1 + + + + + + . +For i ∈ I(1) +1 , Y (1) +i +follows a right truncated exponential distribution with PDF +γ1b1e−γ1b1y +1−e−γ1b1τ(1) +1 +for +0 < y < τ (1) +1 . Hence, for i ∈ I(1) +1 , +E +� +Y (1) +i +� += +� τ (1) +1 +0 +y γ1b1e−γ1b1y +1 − e−γ1b1τ (1) +1 +dy = +1 +γ1b1 +� +1 − (1 + γ1b1τ (1) +1 )e−γ1b1τ (1) +1 +1 − e−γ1b1τ (1) +1 +� +. +Therefore, +E + + + +� +i∈I(1) +1 +Y (1) +i + + + = +1 +γ1b1 +� +1 − (1 + γ1b1τ (1) +1 )e−γ1b1τ (1) +1 +1 − e−γ1b1τ (1) +1 +� +E(N(1) +1 ) += +1 +γ1b1 +� +1 − (1 + γ1b1τ (1) +1 )e−γ1b1τ (1) +1 +1 − e−γ1b1τ (1) +1 +� � +1 − e−γ1b1τ (1) +1 +� += +1 +γ1b1 +� +1 − (1 + γ1b1τ (1) +1 )e−γ1b1τ (1) +1 +� +. +Similarly, +E + + + +� +i∈I(1) +2 +Y (1) +i + + + = E + + +E + + + +� +i∈I(1) +2 +Y (1) +i +|N(1) +2 += n(1) +2 + + + + + + . +For i ∈ I(1) +2 , Y (1) +i +follows a left truncated exponential distribution with PDF γ1b2e−γ1b2y +e−γ1b2τ(1) +1 +for +y > τ (1) +1 . Hence, +E +� +Y (1) +i +� += +� ∞ +τ (1) +1 +yγ1b2e−γ1b2y +e−γ1b2τ (1) +1 +dy = +1 +γ1b2 ++ τ (1) +1 . +Therefore, +E + + + +� +i∈I(1) +2 +Y (1) +i + + + = +� 1 +γ1b2 ++ τ (1) +1 +� +E(N(1) +2 ) = +� 1 +γ1b2 ++ τ ′ +1 +� +e−γ1b1τ (1) +1 . +25 + +Appendix B: Calculations of some important relia- +bility characteristics +Derivation of the quantile function: +Denote p = G(j)(y) for y ∈ +� +τ (j) +k−1, τ (j) +k +� +; then, y = η(p) for p ∈ +� +G(j)(τ (j) +k−1), G(j)(τ (j) +k ) +� +, +k = 1, 2, . . . , N. Now, +p = 1 − e +−(J−j)γj +��k−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� ++bk +� +y−τ (j) +k−1 +�� +=⇒ bk +� +y − τ (j) +k−1 +� += −log(1 − p) +(J − j)γj +− +k−1 +� +ℓ=1 +bℓ +� +τ (j) +ℓ +− τ (j) +ℓ−1 +� +=⇒ y = τ (j) +k−1 − log(1 − p) +(J − j)γjbk +− 1 +bk +k−1 +� +ℓ=1 +bℓ +� +τ (j) +ℓ +− τ (j) +ℓ−1 +� +, if p ∈ +� +G(j)(τ (j) +k−1), G(j)(τ (j) +k ) +� +, +k = 1, 2, . . . , N. +If y ∈ +� +τ (j) +N , ∞ +� +, then y = η(p) for p ∈ +� +G(j)(τ (j) +N ), 1 +� +. +Therefore, +p = 1 − e +−(J−j)γj +��N−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� ++bN +� +y−τ (j) +N−1 +�� +=⇒ y = τ (j) +N−1 − log(1 − p) +(J − j)γjbN +− 1 +bN +N−1 +� +ℓ=1 +bℓ +� +τ (j) +ℓ +− τ (j) +ℓ−1 +� +, if p ∈ +� +G(j)(τ (j) +N ), 1 +� +. +Derivation of MTTF: +MTTF of the system lifetime T is given by E(T) = E +�J−1 +� +j=0 +Y (j) +� += +J−1 +� +j=0 +E(Y (j)), where +E(Y (j)) = +� ∞ +0 +P(Y (j) > y)dy = +� τ (j) +N−1 +0 +e−(J−j)Λ(j)(y)dy+ +� ∞ +τ (j) +N−1 +e−(J−j)Λ(j)(y)dy = I1+I2 (say). +Here, +I1 = +� τ (j) +N−1 +0 +e +−(J−j)γj +�N +k=1 +��k−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� ++bk +� +y−τ (j) +k−1 +�� +1 +[τ(0) +k−1, τ(0) +k +)(y) +dy += +N−1 +� +s=1 +� τ (j) +s +τ (j) +s−1 +e +−(J−j)γj +��s−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� ++bs +� +y−τ (j) +s−1 +�� +dy += +N−1 +� +s=1 +� +e +−(J−j)γj +�s−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� � τ (j) +s +τ (j) +s−1 +e +−(J−j)γjbs +� +y−τ (j) +s−1 +� +dy +� += +N−1 +� +s=1 + + +e +−(J−j)γj +�s−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +�  +1 − e +−(J−j)γjbs +� +τ (j) +s +−τ (j) +s−1 +� +(J − j)γjbs + + + + + +26 + += +N−1 +� +s=1 + + + +e +−(J−j)γj +�s−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� +− e +−(J−j)γj +�s +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� +(J − j)γjbs + + + +and +I2 = +� ∞ +τ (j) +N−1 +e +−(J−j)γj +�N +k=1 +��k−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� ++bk +� +y−τ (j) +k−1 +�� +1 +[τ(0) +k−1, τ(0) +k +)(y) +dy += +� ∞ +τ (j) +N−1 +e +−(J−j)γj +��N−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� ++bN +� +y−τ (j) +N−1 +�� +dy += e +−(J−j)γj +�N−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� � ∞ +τ (j) +N−1 +e +−(J−j)γjbN +� +y−τ (j) +N−1 +� +dy += e +−(J−j)γj +�N−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� � +1 +(J − j)γjbN +� += e +−(J−j)γj +�N−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� +(J − j)γjbN +. +Therefore, +E(Y (j)) = +N +� +s=1 + + + +e +−(J−j)γj +�s−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� +− e +−(J−j)γj +�s +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� +(J − j)γjbs + + + . +From here, the results follows immediately. +Derivation of moment generating function of system lifetime: +Note that the system lifetime MGF of T is T = +J−1 +� +j=0 +Y (j), where Y (j)’s are independent +for j = 0, 1, . . . , (J − 1). Therefore, the MGF of T is φT(t) = +J−1 +� +j=0 +φY (j)(t). Now, +φY (j)(t) += E(etY (j)) = +� ∞ +0 +etygY (j)(y)dy += +� τ (j) +N−1 +0 +ety(J − j)λ(j)(y)e−(J−j)Λ(j)(y)dy + +� ∞ +τ (j) +N−1 +ety(J − j)λ(j)(y)e−(J−j)Λ(j)(y)dy += I1 + I2 (say) , +where gY (j)(y) = (J − j)λ(j)(y)e−(J−j)Λ(j)(y). For t ∈ R, +I1 = +� τ (j) +N−1 +0 +ety(J − j)γj +N +� +k=1 +bk1[τ (j) +k−1, τ (j) +k +) (y) e +−(J−j)γj +�N +k=1 +��k−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� ++bk +� +y−τ (j) +k−1 +�� +dy += +N−1 +� +s=1 +(J − j)bsγj +� τ (j) +s +τ (j) +s−1 +e +− +� +(J−j)γj +��s−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� ++bs +� +y−τ (j) +s−1 +�� +−ty +� +dy +27 + += +N−1 +� +s=1 +� +(J − j)bsγje +−(J−j)γj +�s−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� � τ (j) +s +τ (j) +s−1 +e +− +� +(J−j)γjbs +� +y−τ (j) +s−1 +� +−ty +� +dy +� += +N−1 +� +s=1 + + +(J − j)bsγje +−(J−j)γj +�s−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +�  +etτ (j) +s−1 − e +− +� +(J−j)γjbs +� +τ (j) +s +−τ (j) +s−1 +� +−tτ (j) +s +� +(J − j)γjbs − t + + + + + += +N−1 +� +s=1 +(J − j)bsγj +(J − j)bsγj − t +� +e +− +� +(J−j)γj +�s−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� +−tτ (j) +s−1 +� +−e +− +� +(J−j)γj +�s +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� +−tτ (j) +s +�� +. +For t < (J − j)γjbN, +I2 = +� ∞ +τ (j) +N−1 +ety(J − j)γj +N +� +k=1 +bk1[τ (j) +k−1, τ (j) +k +) (y) e +−(J−j)γj +�N +k=1 +��k−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� ++bk +� +y−τ (j) +k−1 +�� +dy += (J − j)bNγj +� ∞ +τ (j) +N−1 +e +ty−(J−j)γj +��N−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� ++bN +� +y−τ (j) +N−1 +�� +dy += (J − j)bNγje +−(J−j)γj +�N−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� � ∞ +τ (j) +N−1 +e +− +� +(J−j)γjbN +� +y−τ (j) +N−1 +� +−ty +� +dy += (J − j)bNγje +−(J−j)γj +�N−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� � +etτ (j) +N−1 +(J − j)γjbN − t +� += (J − j)bNγj · e +tτ (j) +N−1−(J−j)γj +�N−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� +(J − j)bNγj − t +. +Therefore, for t < (J − j)γjbN, +φY (j)(t) = +N +� +s=1 +(J − j)bsγj +(J − j)bsγj − t +� +e +− +� +(J−j)γj +�s−1 +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� +−tτ (j) +s−1 +� +−e +− +� +(J−j)γj +�s +ℓ=1 bℓ +� +τ (j) +ℓ +−τ (j) +ℓ−1 +� +−tτ (j) +s +�� +. +From here the result follows immediately. +28 + diff --git a/99AzT4oBgHgl3EQfg_xe/content/tmp_files/load_file.txt b/99AzT4oBgHgl3EQfg_xe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4a7fe7b1353b473200df187ece04b75da48b7e6b --- /dev/null +++ b/99AzT4oBgHgl3EQfg_xe/content/tmp_files/load_file.txt @@ -0,0 +1,1910 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf,len=1909 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='01477v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ME] 4 Jan 2023 Reliability Analysis of Load-sharing Systems using a Flexible Model with Piecewise Linear Functions Shilpi Biswas ∗, Ayon Ganguly †, and Debanjan Mitra ‡ Abstract Aiming for accurate estimation of system reliability of load-sharing systems, a flex- ible model for such systems is constructed by approximating the cumulative hazard functions of component lifetimes using piecewise linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The advantages of the resulting model are that it is data-driven and it does not use prohibitive assump- tions on the underlying component lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Due to its flexible nature, the model is capable of providing a good fit to data obtained from load-sharing systems in general, thus resulting in an accurate estimation of important reliability characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Es- timates of reliability at a mission time, quantile function, mean time to failure, and mean residual time for load-sharing systems are developed under the proposed model involving piecewise linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Maximum likelihood estimation and construction of confidence intervals for the proposed model are discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The performance of the proposed model is observed to be quite satisfactory through a detailed Monte Carlo simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Analysis of a load-sharing data pertaining to the lives of a two-motor load-sharing system is provided as an illustrative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In summary, this article presents a comprehensive discussion on a flexible model that can be used for load-sharing systems under minimal assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Keywords: Load-sharing systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Cumulative hazard function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Baseline hazard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Piecewise linear approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Maximum likelihood estimation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Fisher information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Bootstrap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Con- fidence interval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Quantile function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Mean time to failure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Reliability at a mission time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Mean residual time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1 Background Dynamic models are suitable for reliability systems where failure or degradation of one or more components affects the performance of the surviving or operating components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Load- sharing systems are appropriate examples where such models can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The total load on a load-sharing system is shared between its components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' when a component fails within ∗Indian Institute of Technology Guwahati, Assam 781039, India;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Email: shilpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='biswas@iitg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='in †Indian Institute of Technology Guwahati, Assam 781039, India;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Email: aganguly@iitg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='in ‡Indian Institute of Management Udaipur, Rajasthan 313001, India;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Email: debanjan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='mitra@iimu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='in 1 the system, the total load gets redistributed over the remaining operating components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' As a result of a higher stress due to this extra load, the failure rates of the operating components increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Common examples of load-sharing systems are those where components are connected in parallel, such as central processing units (CPUs) of multi-processor computers, cables of a suspension bridge, valves or pumps in hydraulic systems, electrical generator systems etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Load-sharing systems are found in other spheres as well, such as the kidney system in humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' When one of the kidneys fails or deteriorates, the other kidney experiences elevated stress and has an increased chance of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The load-share rule among the operating components depends on the physical charac- teristics of the system involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In an equal load-share rule, the extra load caused by the failed components is shared equally by the operating components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' On the other hand, a local load-share rule implies that the extra load is shared by the neighboring components of the failed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' A monotone load-sharing rule more generally assumes that the load on the operating components is non-decreasing with respect to the failure of other components in the system [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2 Literature review One of the early major contributions to the literature on load-sharing systems was by Daniels [9], describing the increasing stress on yarn fibres with successive breakings of indi- vidual fibres within a bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In the same context of the textile industry, the early-period literature saw developments by Coleman [4, 5], Rosen [29], and Harlow and Phoenix [13, 14], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In general, the topic attracted the attentions of several researchers, and sig- nificant theoretical contributions were made, for example, by Birnbaum and Saunders [6], Freund [12], Ross [30], Schechner [31], Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [19], Hollander and Pena [15], and Lynch [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' While most studies on load-sharing systems in the early-period were based on a known load-share rule, Kim and Kvam [16] presented a statistical methodology for multicompo- nent load-sharing systems with an unknown load-share rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In fact, the work of Kim and Kvam [16] was also important for another reason: they used the hypothetical latent variable approach for modelling the component lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The latent variable approach was later adapted by Park [27, 28] for developing an inferential framework for load-sharing systems assuming the component lifetimes to be exponential, Weibull, and lognormally distributed random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The use of parametric models has a long history in the literature on load-sharing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Exponential distribution has been extensively used for modelling lifetimes of components of load-sharing systems [32, 20, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' However, the property of a constant hazard rate of the exponential distribution is not practical for most applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The tampered failure rate model for load-sharing systems, proposed by Suprasad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [33], was thus developed to accommodate a wide range of failure-time distributions for the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In this connec- tion, the use of accelerated life testing models for load-sharing systems may be mentioned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' see Mettas and Vassiliou [23], Amari and Bergman [1], and Kong and Ye [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' A family of parametric distributions was used for modelling the lives of two-component load-sharing systems by Deshpande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Asha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [2] used a frailty-based model to this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' A recent contribution in this direction is by Franco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [11] who used generalized Freund’s 2 bivariate exponential model for two-component load-sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' See also the references cited in these articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Recently, several authors have explored diverse areas concerning load-sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The damage accumulation of load-sharing systems was modelled by M¨uller and Meyer [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [21] developed a model for correlated lifetimes in dynamic environments incorpo- rating the load-sharing criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [7] explored a spatial model for load-sharing where the extra load due to failure of a component is shared more by the operating com- ponents that are in close proximity of the failed component than those that are distant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Nezakati and Ramzakh [26], and Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [36] connected degradation of components to load-sharing phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In an interesting development, Che et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [8] considered man- machine units (MMUs) as units of analysis where load-sharing was possible due to machine issues as well as human issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' They studied the load-sharing of the MMUs, attempting to capture the complex dependence between machines and their operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' A general model, called the load-strength model, was studied by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' It is to be noted that most of the studies on load-sharing systems have used parametric models for analysis so far, thus heavily relying on the modelling assumptions for suitability of their analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='3 Aim and Motivation Our aim in this paper is to develop an appropriate estimate for the system reliability or reliability at mission time (RMT) of load-sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The aim, also, is to accurately estimate quantile function of the underlying system lifetime distribution, mean time to failure (MTTF), and mean residual time (MRT) of load-sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' These quantities are important to fully understand the characteristics of a load-sharing system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' also, they are of practical importance for making various strategies and plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Naturally, the quality of estimation of RMT, quantile function, MTTF, and MRT of a load-sharing system depends on the suitability of the model that is fitted to the lifetimes of its components capturing the load-share rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' To this effect, we develop a model for the component lifetimes involving piecewise linear approximations (PLAs) of the cumulative hazard functions, capturing the unknown load-share rule at each of the successive stages of component failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The model is data-driven, and does not require prohibitive parametric assumptions for component lifetime distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Due to this flexibility, the PLA-based model is capable of providing a good fit to load-sharing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' An example, elaborated in a later section, is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Data pertaining to a load-sharing system where each system was a parallel combination of two motors were analysed by Asha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [2] and Franco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Asha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [2] assumed Weibull distributions for the component lifetimes, although data for one of the two component motors showed clear empirical evidence that the assumption was not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' A generalized bivariate Freund distribution was assumed for the component lifetimes by Franco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' To this data, we have fitted our proposed PLA-based model, and have observed according to the Akaike’s information criterion (AIC) for model selection, the PLA-based model is a much better fit compared to the Weibull model of Asha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [2] and generalized bivariate Freund model of Franco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The immediate and obvious result of this is a much more accurate estimation of the RMT, quantile function, MTTF, and MRT of the system lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The details of this analysis are given in a later section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 3 The main contributions of this paper are as follows: We develop a flexible, data-driven model based on PLA for modelling component lifetimes of a load-sharing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The model does not require prohibitive parametric assumptions on the underlying component lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' We develop inference for the proposed PLA-based model based on data from multi- component load-sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Under the proposed PLA-based model, we develop methods to accurately estimate im- portant reliability characteristics such as system reliability or RMT, quantile function, MTTF, and MRT of load-sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The rest of this article is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In Section 2, the proposed PLA-based model for load-sharing systems is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Section 3 contains likelihood inference for the model based on data from multi-component load-sharing systems, including relevant details of derivation of MLEs, construction of confidence intervals, and a general guidance on selection of cut-points for the piecewise linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Estimation of system reliability, quantile function, MTTF, and MRT of load-sharing systems in this setting are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Based on component lifetime data from a two-component load-sharing system, an illustrative example of application of the PLA-based model and estimation of various important reliability characteristics are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In Section 6, results of a detailed Monte Carlo simulation experiment investigating the efficacy and robustness of the PLA-based model are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Finally, the paper is concluded with some remarks in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 2 The Piecewise Linear Approximation Model for Cumulative Hazard In general, a PLA is a helpful tool for modelling data, avoiding strong parametric assump- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In survival analysis, piecewise linear functions are used extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Recently, Bal- akrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [3] proposed a PLA-based model for the hazard rate of a population with a cured proportion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' see also the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In this article, we develop a PLA-based model for load-sharing systems with unknown load-share rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Specifically, we model the cumulative hazard functions of the component lifetime distributions using PLAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' At each of the successive stages of component failures, as the lifetime distributions of the remaining operating components change, a new PLA for the cumulative hazard is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The model can be suitably tuned by choosing the number of linear pieces for the PLA at each stage of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The principal advantage of the proposed PLA-based modelling approach is that it uses minimal model assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Consider a J-component load-sharing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Here, a J-component load-sharing system means a load-sharing system with J components that are connected in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Assume that the failed components of the system are not replaced or repaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' When the components fail one by one, after each failure the total load on the system gets redistributed over the remain- ing operational components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' As a result the operational components experience a higher load 4 than before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' At the beginning when all components are operational, let U(0) 1 , U(0) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , U(0) J denote the latent lifetimes of the components, and Y (0) denote the system lifetime till the first component failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Obviously, Y (0) = min � U(0) 1 , U(0) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , U(0) J � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Similarly, for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1, let Y (j) denote the system lifetime between j-th and (j + 1)-st component failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Then, Y (j) = min � U(j) 1 , U(j) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , U(j) J−j � , where U(j) 1 , U(j) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' U(j) J−j denote the latent lifetimes of the operational components after the j-th component failure, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For all values of j, U(j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , U(j) J−j are assumed to be independent and identically distributed random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' It is further assumed that � U(j) ℓ , ℓ = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1 � are independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Let h(j)(·) and H(j)(·) denote the hazard rate (HR) and cumulative hazard function (CHF), respectively, of the distribution of U(j) 1 , j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Here, we assume that the HR h(j) (·) is a non-decreasing function for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For y > 0, the survival function (SF) of Y (j) is given by P � Y (j) > y � = P � min � U(j) 1 , U(j) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , U(j) J−j � > y � = e−(J−j)H(j)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Hence, for y > 0, the cumulative distribution function (CDF) and probability density func- tion (PDF) of Y (j) are given by F (j)(y) = 1 − e−(J−j)H(j)(y) and f (j)(y) = (J − j)h(j)(y) e−(J−j)H(j)(y), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Now, suppose there are n J-component load-sharing systems, and let Y (j) i denote the system lifetime between j-th and (j + 1)-st component failures for the i-th system, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , n, j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Suppose the observed values of Y (j) 1 , Y (j) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , Y (j) n are y(j) 1 , y(j) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , y(j) n , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Let, for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1, ξ(j) = � τ (j) 0 , τ (j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , τ (j) N � denote a set of N + 1 cut-points over the time scale y(j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , y(j) n , with the restrictions that τ (j) 0 < τ (j) 1 < τ (j) 2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' < τ (j) N , τ (j) 0 ≤ min � y(j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , y(j) n � and τ (j) N ≥ max � y(j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , y(j) n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Initially, ξ(j) is taken to be fixed and known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' We discuss how to choose ξ(j) in a later section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The proposed model approximates the CHF H(j)(·) by a piecewise linear function defined over intervals [τ (j) k−1, τ (j) k ), k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , N, constructed by the consecutive cut points in ξ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Therefore, over the range [τ (0) 0 , τ (0) N ), the CHF H(0)(·) is approximated by Λ(0)(·), where Λ(0)(t) = N � k=1 (ak + bkt) 1[τ (0) k−1, τ (0) k )(t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1) 5 with ak’s and bk’s as real constants and 1A(t) = � 1 if t ∈ A 0 if t ̸∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' One of the possible ways to extend the PLA beyond τ (0) N would be to extend the last line segment aN + bNt to [τ (0) N , ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Therefore, the CHF corresponding to PLA over the range [τ (0) 0 , ∞) is Λ(0)(t) = N � k=1 (ak + bkt) 1[τ (0) k−1, τ (0) k )(t) + (aN + bNt)1[τ (0) N , ∞)(t), with Λ(0)(τ (0) 0 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' We also assume that Λ(0)(·) is a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' As Λ(0)(τ (0) 0 ) = 0, using the assumption of continuity, ai’s can be expressed in terms of bi’s as follows: a1 = −b1τ (0) 0 and ak = k−1 � ℓ=1 (bℓ − bℓ+1) τ (0) ℓ + a1 = k−1 � ℓ=1 bℓ � τ (0) ℓ − τ (0) ℓ−1 � − bkτ (0) k−1, for k = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Note that the above model can be equivalently described in terms of HRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In this approach, h(0)(·) over the range [τ (0) 0 , τ (0) N ) is approximated by a piecewise constant function λ(0)(·), where λ(0)(t) = N � i=1 bk1[τ (0) k−1, τ (0) k ) (t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2) After failure of one or more components within the system, the direct impact of the increased load will be an increased HR for the operational components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' To incorporate this information, after the failure of j components of the system, we approximate h(j)(·) over [τ (j) 0 , τ (j) N ), j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1, using the piecewise constant function λ(j)(·), where λ(j)(t) = γj N � k=1 bk1[τ (j) k−1, τ (j) k ) (t) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='3) with 1 < γ1 < γ2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' < γJ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The PLAs to the CHFs, corresponding to the PLAs of the HRs given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='3) are given by Λ(j)(t) = γj N � k=1 �k−1 � ℓ=1 bℓ � τ (j) ℓ − τ (j) ℓ−1 � + bk � t − τ (j) k−1 �� 1[τ (j) k−1, τ (j) k ) (t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='4) To meet the non-decreasing nature of the HR, we assume that 0 < b1 < b2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' < bN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Note that the parameters γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , γJ−1 reflect the load-share rule of increased HRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' We treat γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , γJ−1 as unknown parameters, and estimate them from component failure data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' It may be mentioned here that the PLA model can be interpreted as an approximation of the underlying lifetime distribution by several exponential models (with different rate parameters) over the ranges specified by the cut-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 6 3 Likelihood Inference The parameters involved in the PLA-based model are estimated from the component failure data obtained from a set of load-sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The available data on component failures from n J-component load-sharing systems is of the form Data = � y(j) i : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1 � , where y(j) i is the observed system lifetime between j-th and (j + 1)-st component failures for the i-th system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1, and k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , N, define I(j) k = � i : y(j) i ∈ � τ (j) k−1, τ (j) k �� and n(j) k = |I(j) k |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Obviously, �N k=1 n(j) k = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The likelihood function for the PLA model is then given by L (θ) = n � i=1 J−1 � j=0 � (J − j)γj N � k=1 bk1[τ (j) k−1, τ (j) k ) � y(j) i � e −(J−j)γj ��k−1 ℓ=1 bℓ � τ (j) ℓ −τ (j) ℓ−1 � +bk � y(j) i −τ (j) k−1 ��� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1) where γ0 = 1 and θ = (γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , γJ−1, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , bN)′ is the vector of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The corresponding log-likelihood function, ignoring additive constant, can be expressed as l (θ) = N � k=1 ��J−1 � j=0 n(j) k � ln bk − �J−1 � j=0 (J − j)γjT (j) k � bk � + n J−1 � j=0 ln γj, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2) where T (j) k = � i∈I(j) k � y(j) i − τ (j) k−1 � + � n − k � ℓ=1 n(j) ℓ � � τ (j) k − τ (j) k−1 � , for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Equating partial derivative of the log-likelihood function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2) with respect to bk to zero, we can express bk in terms of the load-share parameters γ = (γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , γJ−1) as bk = bk (γ) = J−1 � j=0 n(j) k J−1 � j=0 (J − j)γjT (j) k , k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='3) Substituting bk(γ) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='3) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2), the profile log-likelihood in γ, ignoring additive constant, is obtained as ˜l (γ) = N � k=1 ��J−1 � j=0 n(j) k � � ln �J−1 � j=0 n(j) k � − ln �J−1 � j=0 (J − j)γjT (j) k ��� + n J−1 � j=0 ln γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='4) 7 For optimizing the profile log-likelihood ˜l (γ) in γ, any routine maximizer of a standard statistical software may be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Once the MLEs �γ1, �γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , �γJ−1 of γ1, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , γJ−1 are obtained by numerical optimization of ˜l (γ), they can be plugged into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='3) to get MLEs of bk as �bk = bk (�γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , �γJ−1) , k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1 A special case: two-component load-sharing systems For analysing data from two-component load-sharing systems, if two linear pieces are used in the PLA-based model, MLEs can be derived analytically and explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Consider the case when J = 2 and N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In this case, the log-likelihood function simplifies to l (θ) = 2 � k=1 �� 1 � j=0 n(j) k � ln bk − � 1 � j=0 (2 − j)γjT (j) k � bk � + n 1 � j=0 ln γj, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='5) with T (j) k = � i∈I(j) k � y(j) i − τ (j) k−1 � + � n − k � ℓ=1 n(j) ℓ � � τ (j) k − τ (j) k−1 � , for k = 1, 2, j = 0, 1 and γ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Here, θ = (γ1, b1, b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Equating ∂l(θ) ∂b1 and ∂l(θ) ∂b2 to zero, we get b1 = n(0) 1 + n(1) 1 2T (0) 1 + γ1T (1) 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='6) b2 = n(0) 2 + n(1) 2 2T (0) 2 + γ1T (1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='7) Equating ∂l(θ) ∂γ1 to zero gives γ1 = T (1) 1 b1 + T (1) 2 b2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='8) in which, substituting b1 and b2 from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='7), a quadratic equation in γ1 is obtained as follows Q(γ1) = nγ2 1B0,12 + 2γ1 �� n(0) 1 + n(1) 1 − n � B2,1 + � n(0) 2 + n(1) 2 − n � B1,2 � − 4nB12,0 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='9) with B0,12 = T (1) 1 T (1) 2 , B1,2 = T (0) 1 T (1) 2 , B2,1 = T (0) 2 T (1) 1 and B12,0 = T (0) 1 T (0) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Solving Q(γ1) = 0, we have two values of γ1 from which we choose the suitable one, and then from equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='7) we get the MLEs of b1 and b2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2 Confidence Intervals As discussed above, the MLEs for the parameters of the PLA-based model are not available in explicit form in general, except for the special case of two-component load-sharing systems considered in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' As a result, exact confidence intervals for the model parameters cannot be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Asymptotic confidence intervals may be constructed in two possible ways: by using the Fisher information matrix, and by applying a bootstrap-based technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1 CIs using Fisher information matrix Using the asymptotic properties of the MLEs, it can be shown that for large sample size n, the distribution of √n(�θ − θ) is approximated by a multi-variate normal distribution N(0, I−1(�θ)), where the dimension of the multi-variate normal distribution is same as that of the parameter vector θ, and the asymptotic variance-covariance matrix I−1(θ) is the in- verse of the Fisher information matrix I(θ), evaluated at the MLE �θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The Fisher information matrix I(θ) is defined as the expected value of the observed information matrix J(θ) which is calculated from the negative of the second-order derivatives of the log-likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' That is, I(θ) = E(J(θ)), where J(θ) = −∇2(log L(θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In situations where analytical calcu- lation of the Fisher information is difficult or intractable, it may be either replaced by the observed information matrix, or may be calculated by simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' From the asymptotic variance-covariance matrix I−1(θ), individual asymptotic variances of the MLEs can be pulled out, and asymptotic confidence intervals can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For example, corresponding to the MLE �γ1 using the asymptotic variance � V ar(ˆγ1) obtained from I−1(θ), asymptotic confidence intervals for γ1 can be constructed as: � �γ1 − zα/2 � � V ar(ˆγ1), �γ1 + zα/2 � � V ar(ˆγ1) � , where zα is the 100(1 − α)% point of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Special case: two-component load-sharing systems For the special case of two-component load-sharing systems considered in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1, the Fisher information matrix can be worked out explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In this case, J(θ) = − \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed ∂2l(θ) ∂γ2 1 ∂2l(θ) ∂γ1∂b1 ∂2l(θ) ∂γ1∂b2 ∂2l(θ) ∂b1∂γ1 ∂2l(θ) ∂b2 1 ∂2l(θ) ∂b1∂b2 ∂2l(θ) ∂b2∂γ1 ∂2l(θ) ∂b2∂b1 ∂2l(θ) ∂b2 2 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = − \uf8eb \uf8ec \uf8ec \uf8ed − n γ2 1 −T (1) 1 −T (1) 2 −T (1) 1 −n(0) 1 +n(1) 1 b12 0 −T (1) 2 0 −n(0) 2 +n(1) 2 b22 \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 9 Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' the Fisher information matrix is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='I(θ) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8ec ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' where N(j) k is the number of Y (j) i in [τ (j) k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' τ (j) k ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' j = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' An outline of calculations of the relevant expectations for the Fisher information matrix is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The inverse of the Fisher information matrix is obtained as � I−1(θ) � = 1 |I(θ)| \uf8eb \uf8ed A11(θ) −A12(θ) A13(θ) −A21(θ) A22(θ) −A23(θ) A31(θ) −A32(θ) A33(θ) \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' where the determinant of I(θ) is |I(θ)| = n � 2 − � e−2b1τ (0) 1 + e−γ1b1τ (1) 1 �� � e−2b1τ (0) 1 + e−γ1b1τ (1) 1 � γ2 1b2 1b2 2 − e−2γ1b1τ (1) 1 � 1 γ1b2 �2 � 2 − � e−2b1τ (0) 1 + e−γ1b1τ (1) 1 �� b2 1 − � 1 γ1b1 � 1 − (1 + γ1b1τ (1) 1 )e−γ1b1τ (1) 1 � + τ (1) 1 e−γ1b1τ (1) 1 �2 � e−2b1τ (0) 1 + e−γ1b1τ (1) 1 � b2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' A11(θ) = � 2 − � e−2b1τ (0) 1 + e−γ1b1τ (1) 1 �� � e−2b1τ (0) 1 + e−γ1b1τ (1) 1 � b2 1b2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' A22(θ) = n � e−2b1τ (0) 1 + e−γ1b1τ (1) 1 � γ2 1b2 2 − e−2γ1b1τ (1) 1 � 1 γ1b2 �2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' A33(θ) = n � 2 − � e−2b1τ (0) 1 + e−γ1b1τ (1) 1 �� γ2 1b2 1 − � 1 γ1b1 � 1 − (1 + γ1b1τ (1) 1 )e−γ1b1τ (1) 1 � + τ (1) 1 e−γ1b1τ (1) 1 �2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' A12(θ) = A21(θ) = � 1 γ1b1 � 1 − (1 + γ1b1τ (1) 1 )e−γ1b1τ (1) 1 � + τ (1) 1 e−γ1b1τ (1) 1 � � e−2b1τ (0) 1 + e−γ1b1τ (1) 1 � b2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 10 A13(θ) = A31(θ) = − e−γ1b1τ (1) 1 � 1 γ1b2 � � 2 − � e−2b1τ (0) 1 + e−γ1b1τ (1) 1 �� b2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' A23(θ) = A32(θ) = − �� 1 − (1 + γ1b1τ (1) 1 )e−γ1b1τ (1) 1 � + γ1b1τ (1) 1 e−γ1b1τ (1) 1 � e−γ1b1τ (1) 1 γ2 1b1b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Evaluating I−1(θ) at the MLE �θ, the asymptotic variance-covariance matrix of the MLEs is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Hence, 100(1 − α)% asymptotic confidence intervals for γ1, b1, and b2 are obtained as � �γ1 −zα/2 � A11(ˆθ) |I(ˆθ)| , �γ1 +zα/2 � A11(ˆθ) |I(ˆθ)| � , � �b1 −zα/2 � A22(ˆθ) |I(ˆθ)| , �b1 +zα/2 � A22(ˆθ) |I(ˆθ)| � , and � �b2 − zα/2 � A33(ˆθ) |I(ˆθ)| , �b2 + zα/2 � A33(ˆθ) |I(ˆθ)| � , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2 Bootstrap confidence intervals Using the MLE �θ, B bootstrap samples can be obtained in the same sampling framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' let �θ ∗ s = � �γ∗ 1s,�b∗ 1s,�b∗ 2s � denote the bootstrap estimates, s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Bootstrap bias and standard error are defined as biasb(�γ1) = �γ∗ 1 − �γ1, biasb(�b1) = �b∗ 1 −�b1, biasb(�b2) = �b∗ 2 −�b2 and SEb(�γ1) = � � � � 1 B − 1 B � s=1 � �γ∗ 1s − � γ∗ 1 �2 , SEb(�b1) = � � � � 1 B − 1 B � s=1 � �b∗ 1s − �b∗ 1 �2 , SEb(�b2) = � � � � 1 B − 1 B � s=1 � �b∗ 2s − �b∗ 2 �2 , where �γ∗ 1 = 1 B B � s=1 �γ∗ 1s, �b∗ 1 = 1 B B � s=1 �b∗ 1s, �b∗ 2 = 1 B B � s=1 �b∗ 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Finally, a 100(1 − α)% bootstrap confidence interval for γ1 can be calculated as � �γ1 − biasb(�γ1) − zα/2SEb(�γ1), �γ1 − biasb(�γ1) + zα/2SEb(�γ1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Bootstrap confidence intervals for b1 and b2 can be calculated similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For percentile bootstrap confidence intervals for, say γ1, the bootstrap estimates of �γ1 are first ordered in terms of magnitude: �γ∗ 1(1) < �γ∗ 1(2) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' < �γ∗ 1(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Then, a 100(1−α)% percentile bootstrap confidence interval for γ1 is � �γ∗ 1([ αB 2 ]), �γ∗ 1([(1− α 2 )B]) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Similarly, percentile bootstrap confidence intervals can be calculated for b1 and b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='3 Choice of Cut Points The number and position of the cut-points for constructing the PLA-based model need to be suitably chosen, so that the model can closely approximate the underlying CHF, but avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' A large number of cut points would provide a close local approximation to the underlying CHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' However, apart from being computationally expensive, a close local approximation may also lead to overfitting in which case it would be difficult to use the PLA-based model to predict future failures of components or systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' One of the possible ways to choose the number and position of the cut-points is by looking at the plot of the nonparametric estimator of CHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' From such a plot, observing the areas where the nonparametric estimate changes significantly, one can determine the positions and number of cut-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' More objectively, one can choose the positions of a given number of cut-points by max- imizing the log-likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For example, for three cut-points (N = 2), the natural choice for τ (j) 0 is min � y(j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , y(j) n � and τ (j) 2 is max � y(j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , y(j) n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Now to choose the position of τ (j) 1 , one may take τ (j) 1 equal to different sample quantiles of � y(j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , y(j) n � and choose one that provides the maximum value of log-likelihood function evaluated at MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' This process can be expressed as an algorithm as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Algorithm: Step 1: Fix 0 < p1 < p2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Step 2: Find the number of y(j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , y(j) n that are between p1-th and p2-th sample quantiles of � y(j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , y(j) n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Denote this number by l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Note that l does not depend on j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Step 3: Set aj1 = p1-th quantile of � y(j) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , y(j) n � , j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Step 4: Set LL1= the value of log-likelihood function evaluated at MLE taking τ (j) 1 = aj1, j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Step 5: Set aj2 = min � y(j) i > aj1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , n � , j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Step 6: Set LL2= the value of log-likelihood function evaluated at MLE taking τ (j) 1 = aj2, j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Step 7: Repeat the steps 5 and 6 to obtain LL1, LL2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , LLl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Step 8: Set k∗ = arg max 1≤k≤l LLk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Step 9: The final cut points are τ (j) 1 = ajk∗, j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , J − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 12 4 Estimation of various reliability characteristics The final goal of fitting a model to load-sharing data, naturally, is accurate estimation of reliability characteristics of load-sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' As the PLA-based model provides a good fit to load-sharing data due to the model’s flexible nature, it is natural that the important reliability characteristics of load-sharing systems can also be estimated quite accurately under this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In this section, we develop estimates of reliability characteristics such as the quantile function, MTTF, RMT, and MRT of load-sharing systems under the PLA-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Details of these derivations are given in Appendix B for interested readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Under the PLA-based model, the quantile function of Y (j) which is the system lifetime between the j-th and (j + 1)-st component failures, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=', J − 1, is given by η(p) = inf � y ∈ R : G(j)(y) ≥ p � , 0 < p < 1, where G(j)(y) = 1 − e−(J−j)Λ(j)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Using the expression of Λ(j)(y) given in Section 2, it is possible to work out an explicit formula for the quantile function η(p), as follows: η(p) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 τ (j) k−1 − log(1−p) (J−j)γjbk − 1 bk · k−1 � ℓ=1 bℓ(τ (j) ℓ − τ (j) ℓ−1), if p ∈ � G(j)(τ (j) k−1), G(j)(τ (j) k ) � , for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' τ (j) N−1 − log(1−p) (J−j)γjbN − 1 bN · N−1 � ℓ=1 bℓ(τ (j) ℓ − τ (j) ℓ−1), if p ∈ � G(j)(τ (j) N ), 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The mean time to failure or MTTF of a load-sharing system is the expected time the system operates till its failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Let T denote the system failure time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' then, T = �J−1 j=0 Y (j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The MTTF of a load-sharing system under the PLA-based model is given by E(T) = J−1 � j=0 N � s=1 �e−κj,s−1 − e−κj,s (J − j)γjbℓ � , where κj,s = (J − j)γj s � ℓ=1 bℓ � τ (j) ℓ − τ (j) ℓ−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Reliability at a mission time or RMT of a system is the probability that the system will operate till a desired time t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' it is calculated as the survival probability of the system at time t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=', S(t0) = P(T > t0) = P �J−1 � j=0 Y (j) > t0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' An explicit expression for RMT may be derived by using the distribution of the system lifetime T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' However, as Y (j)s, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=', J − 1 are independent but not identically distributed, it is difficult to obtain an explicit expression for the distribution of the system lifetime T, where T = �J−1 j=0 Y (j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' It is evident from the moment generating function φT(t) of T, which, under the PLA-based model, is given by φT(t) = J−1 � j=0 N � s=1 (J − j)bsγj (J − j)bsγj − t � etτ (j) s−1−κj,s−1 − etτ (j) s −κj,s � if t < γ1bN, 13 where κj,s = (J − j)γj s � ℓ=1 bℓ � τ (j) ℓ − τ (j) ℓ−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' From here, it is clear that it is difficult to find the RMT analytically under this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' However, for this model, RMT can be estimated using Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For a Monte Carlo estimate of the RMT at a pre-specified time t0, one needs to generate R data points ti, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , R, as realisations of the system lifetime T, and find R(t0) R , where R(t0) is the number of realisations of the system lifetime that exceed t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For a reasonably good estimate of RMT, a large value of R should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The mean residual time or MRT of a system is the expected additional time the system will survive if it has already survived a given time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' That is, MRT(t) = E(T − t|T > t) = � ∞ t sfT|T>t(s)ds − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Therefore, analytical derivation of MRT requires the truncated distribution of the system lifetime T, and it is difficult to obtain the truncated distribution of T in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Instead, an estimate of the MRT can be given using Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' We generate R data points t∗ i , i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , R, as realisations of the truncated lifetime T|T > t, and a Monte Carlo estimate of the MRT for load-sharing systems under the PLA-based model is then given by � MRT(t) = R � i=1 t∗ i R − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 5 Data Analysis In this section, we present an illustrative example using data from load-sharing systems comprising of two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Very recently, this data have been analysed by Sutar and Naik-Nimbalkar [34], Asha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [2] and Franco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The data consist of information on component lifetimes of 18 two-component load-sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Each system is a parallel combination of two motors - “A” and “B”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' When both motors A and B are in working condition, the total load on the system is shared between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' When one of the motors fails, the entire load goes to the operational motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Sutar and Naik-Nimbalkar [34] observed that the load-sharing phenomenon existed for the systems considered in this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Asha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [2] assumed Weibull lifetimes for the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' From the Weibull Q-Q plots for the lifetimes of motor A and B reported in Asha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [2], it was observed that although the Weibull model assumption for the lifetimes of motor B was reasonable, the lifetimes of motor A did not follow a Weibull distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' This motivated us to consider the PLA-based modelling approach for the lifetimes of the load-sharing systems in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The dataset is reproduced in Table 1 for ready reference of the readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The average and standard deviation of first component failure times are 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='61 and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='75, respectively, 14 0 100 200 300 0 100 200 300 Sample quantile Population quantile (a) Q-Q plot for Y (0) 0 25 50 75 100 125 0 25 50 75 100 125 Sample quantile Population quantile (b) Q-Q plot for Y (1) Figure 1: Q-Q plots 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='00 100 150 200 250 300 Time SF (a) Plot of SF for Y (0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='00 0 40 80 120 Time SF (b) Plot of SF for Y (1) Figure 2: Plots of SFs while those of the lifetime between first and second component failures are 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='72 and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='45, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' We consider three cut points for the PLA-based model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=', N = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The estimates of the model parameters are reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The Q-Q plots for Y (0) and Y (1) are given in Figures 1a and 1b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The plots of the estimated SF and CHF are given in Figures 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' These figures indicate that the PLA-based model fits the data quite adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='CHF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='(a) Plot of CHF for Y (0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='Time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='CHF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='(b) Plot of CHF for Y (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='Figure 3: Plots of CHFs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='Table 1: Time to failure (in days) data set for two motors in a load-sharing configuration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='System ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='Time to failure of motor A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='Time to failure of motor B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='Event description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='B failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='84 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='148 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='A failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='88 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='202 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='A failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='156 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='B failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='148 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='123 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='B failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='139 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='A failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='245 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='156 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='B failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='235 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='172 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='B failed first ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='A failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='212 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='B failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='212 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='220 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='A failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='213 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='265 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='A failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='220 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='275 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='A failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='243 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='A failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='248 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='B failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='257 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='330 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='A failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='263 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='A failed first ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='A Kolmogorov-Smirnov type test has been performed to test the following hypotheses: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='H0 : True model is specified by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='3) against H1 : True model is not specified by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='3) 16 Table 2: Point and interval estimates of parameters of the PLA-based model when applied to the two-motor load-sharing data Parameter MLE Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Error Asymptotic Percentile bootstrap Bootstrap γ1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2712 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1901 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='9386, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='6038) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0754, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0279) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='8456, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='8172) b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0008 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0019, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0048) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0021, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0062) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0008, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0052) b2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0039 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0056, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0212) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0061, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0209) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0083, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0232) Table 3: Mean residual time and reliability in mission time t0 MRTt0 RMTt0 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='00 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='963 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='50 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='706 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='50 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='466 272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='50 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='271 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='00 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='044 based on the test statistics Tn = max 1≤i≤n ���� �G(0) � Y (0) i:n � − i n ���� + max 1≤i≤n ���� �G(1) � Y (1) i:n � − i n ���� , where �G(j)(·) is the estimated cumulative distribution function corresponding to PLA-based model, and Y (j) i:n is the i-th order statistics corresponding to Y (j) i , j = 0, 1, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The observed value of the test statistics Tn is found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='414 based on this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The Monte Carlo estimate of the corresponding p-value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Therefore, the null hypothesis cannot be rejected at significance level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='05, and we conclude that it is quite reasonable to use the PLA-based model for this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' It may also be noted here that for this data, the value of the Akaike’s information criterion (AIC) for the model considered by Asha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [2] is 480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='50, and that for the best model considered by Franco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' [11] is 409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In contrast, the AIC value for the PLA- based model turns out to be 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='34, implying that the PLA-based model is more suitable for the two-motor load-sharing systems data considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For the PLA-based model, the estimated value of γ1 is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2712, which empirically implies that the load-sharing model is quite appropriate in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The same comment can also be made from the plots, by noting that the plot of the SF of the distribution of time between first and second failure component times diminishes to zero more quickly compared to that of first component failure times in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The reliability characteristics of the two-motor load-sharing systems are also estimated by using the expressions and techniques described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The MTTF is calculated to be 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='36 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Monte Carlo estimates of the MRT and RMT are calculated at different sample percentile points of the system failure times and are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 17 6 Simulation Study The accuracy of the proposed PLA-based model in fitting data from load-sharing systems is of utmost importance as the subsequent estimation of reliability characteristics depends on the PLA-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In this section, we present results of a Monte Carlo simulation study that examines the performance of the proposed PLA-based model in two directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' First, based on samples generated from a parent process with piecewise linear CHF, we assess the performance of the proposed estimation method that is presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Then, the efficacy of the PLA-based model in fitting data generated from a parent process represented by some parametric models is also assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The simulations are carried out by using R software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For the simulations, we consider two-component load-sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1 Assessing performance of the estimation method To assess the performance of the estimation methods, we consider an underlying cumulative hazard that is made up of two linear pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' To this effect, we generate samples from the model specified by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='3) with J = 2 and N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The true parameter values are taken to be b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' γ1 = 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' τ (0) 1 = ln 2 2b1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' τ (1) 1 = ln 2 γ1b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The estimation is performed based on samples of size n = 100 and 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The average estimates (AE), mean square errors (MSE), variance (VAR) of the MLEs based on 5000 Monte Carlo replications are reported in Tables 4, 5, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The coverage percentage (CP) and average lengths (AL) of 95% confidence intervals are also reported in the same tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' From the Tables 4, 5 and 6, we observe that the average estimates of γ1, b1 and b2 are very close to the true values, and the MSEs as well as VARs are quite small as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' It is also noticed that the performance of all the constructed confidence intervals is satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' These results demonstrate that the proposed inferential techniques can accurately estimate the parameters of the PLA-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Table 4: Performance measures for estimates of γ1 n b1 b2 AE MSE VAR Asymptotic Percentile bootstrap Bootstrap CP AL CP AL CP AL 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1725 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2 Assessing efficacy of the PLA-based model in fitting data from other models Now, we examine the robustness of the PLA-based model in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' We generate load-sharing data from parametric models, and then fit the PLA-based model to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The model fit is then assessed with respect to an integrated measure that is suitably defined to reflect the quality of approximation provided by the PLA-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The measure, which we call the Absolute Integrated Error (AIE), is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For j = 0, 1, let S(j) TGP(·) and H(j) TGP(·) denote the SF and CHF of the lifetimes between j-th and (j + 1)- st failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Also, assume that the estimated SF and CHF based on PLA-based model are denoted by �S(j) P LA(·) and �H(j) P LA(·), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Then the AIE, based on the SF and CHF, respectively, are defined as AIE(j) SF = 1 R R � k=1 1 y(j) max − y(j) min � y(j) max y(j) min ���S(j) TGP(t) − �S(j) P CA(t) ��� dt, 19 Table 7: AIE based on SF and CHF for Weibull distribution with k = 3, β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' n α AIE(0) SF AIE(1) SF AIE(0) CHF AIE(1) CHF 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0291 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1503 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1282 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2541 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0326 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2440 Table 8: AIE of the survival and cumulative hazard function of quadratic distribution for κ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='5, ˜κ1 = 2κ1 = 1, ˜κ2 > 2κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' n κ2 ˜κ2 AIE(0) SF AIE(1) SF AIE(0) CHF AIE(1) CHF 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='50 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2465 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0299 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2528 AIE(j) CHF = 1 R R � k=1 1 y(j) max − y(j) min � y(j) max y(j) min ���H(j) TGP(t) − �H(j) P CA(t) ��� dt, where y(j) min = min � y(j) 1 , y(j) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , y(j) n � , y(j) max = max � y(j) 1 , y(j) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , y(j) n � , j = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For generating load-sharing data from parametric models, two scenarios are considered: (a) Case - 1: It is assumed that the lifetimes of each components of a two-component load- sharing system are independent and identically distributed as Weibull distribution with shape parameter α and scale parameter β when both the components are working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' After the first failure, the lifetime of the surviving component is assumed to follow a Weibull distribution with same shape parameter α, but a different scale parameter kβ, where k > 2 is to ensure the increase of load on the surviving component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For β = 1, k = 3, we take α = 1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' (b) Case - 2: In the second scenario, the component lifetimes are assumed to be independent and identically distributed according to a distribution with quadratic CHF κ1t + κ2t2 when both components are working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' After the first failure, the lifetime of the surviving component is assumed to follow a quadratic CHF with different parameters ˜κ1 and ˜κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' We take several values of the parameters κ1, κ2, ˜κ1, and ˜κ2 ensuring the fact that the CHF increases after one component fails in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The numerical results are reported in Tables 7, 8, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For all cases, it is observed that the values of AIE based on SF and CHF are reasonably small, indicating that the 20 Table 9: AIE of the survival and cumulative hazard function of quadratic distribution for ˜κ1 > 2κ1, κ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='5, ˜κ2 = 2κ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' n κ1 ˜κ1 AIE(0) SF AIE(1) SF AIE(0) CHF AIE(1) CHF 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='0313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2570 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='2271 PLA-based model provides quite a satisfactory approximation to the data generated from different parent populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 7 Concluding Remarks In this article, a PLA-based model for the CHF is proposed for data from load-sharing sys- tems and then important reliability characteristics such as quantile function, RMT, MTTF, and MRT of load-sharing systems are estimated under the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The principal advantages of the model are that it is data-driven, and does not use strong parametric as- sumptions for the underlying lifetime variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Likelihood inference for the proposed model is discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' It is observed that for two-component load-sharing systems, it is possible to obtain explicit expressions for the MLEs of parameters of the PLA-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Construc- tion of confidence intervals using the Fisher information matrix and bootstrap approaches are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Derivations of the important reliability characteristics are provided in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' A Monte Carlo simulation study is performed to examine (a) the performance of the methods of inference, and (b) the efficacy of the PLA-based model to fit load-sharing data in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' It is shown that the PLA-based model performs quite satisfactorily in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Analysis of data pertaining to components lifetimes of a two-motor load-sharing system is provided as an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' It is illustrated that the PLA-based model is supe- rior to the models that have been considered for this data in the literature of load-sharing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' In summary, in this paper, an efficient PLA-based modelling framework using mini- mal assumptions for load-sharing systems is discussed, and estimates of important reliability characteristics for load-sharing systems in this setting are developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 21 Funding information The research of Ayon Ganguly is supported by the Mathematical Research Impact Cen- tric Support (File no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' MTR/2017/000700) from the Science and Engineering Research Board, Department of Science and Technology, Government of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' The research of Debanjan Mitra is supported by the Mathematical Research Impact Centric Support (File no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' MTR/2021/000533) from the Science and Engineering Re- search Board, Department of Science and Technology, Government of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Amari and R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Liu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Reliability modeling and analysis of load-sharing systems with continuously degrading components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' IEEE Transactions on Reliability, 67:1096– 1110, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Appendix A: Calculation of Fisher information ma- trix for two-component load sharing systems For calculating I(θ), the required expectations are E � N(0) 1 � , E � N(0) 2 � , E � N(1) 1 � , E � N(1) 2 � , E \uf8eb \uf8ec \uf8ed � i∈I(1) 1 Y (1) i \uf8f6 \uf8f7 \uf8f8 and E \uf8eb \uf8ec \uf8ed � i∈I(1) 2 Y (1) i \uf8f6 \uf8f7 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Note that N(0) k ∼ Bin(n, p(0) k ), N(1) k ∼ Bin(n, p(1) k ), with p(0) k = P � Y (0) i ∈ [τ (0) k−1, τ (0) k ) � , p(1) k = P � Y (1) i ∈ [τ (1) k−1, τ (1) k ) � , k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 24 In case of a two-component load-sharing system, PDF of Y (j) i , j = 1, 2, is given by gY (j) i (y) = (2 − j)λ(j)(y)e−(2−j) � y 0 λ(j)(u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Hence, p(0) 1 = � τ (0) 1 0 gY (0) i (y)dy = 1 − e−2b1τ (0) 1 , p(1) 1 = � τ (1) 1 0 gY (1) i (y)dy = 1 − e−γ1b1τ (1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Then, p(0) 2 = 1 − p(0) 1 = e−2b1τ (0) 1 and p(1) 2 = 1 − p(1) 1 = e−γ1b1τ (1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Therefore, E(N(0) 1 ) = 1−e−2b1τ (0) 1 , E(N(0) 2 ) = e−2b1τ (0) 1 , E(N(1) 1 ) = 1−e−γ1b1τ (1) 1 , E(N(1) 2 ) = e−γ1b1τ (1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Now, E \uf8eb \uf8ec \uf8ed � i∈I(1) 1 Y (1) i \uf8f6 \uf8f7 \uf8f8 = E \uf8eb \uf8ec \uf8edE \uf8eb \uf8ec \uf8ed � i∈I(1) 1 Y (1) i |N(1) 1 = n(1) 1 \uf8f6 \uf8f7 \uf8f8 \uf8f6 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For i ∈ I(1) 1 , Y (1) i follows a right truncated exponential distribution with PDF γ1b1e−γ1b1y 1−e−γ1b1τ(1) 1 for 0 < y < τ (1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Hence, for i ∈ I(1) 1 , E � Y (1) i � = � τ (1) 1 0 y γ1b1e−γ1b1y 1 − e−γ1b1τ (1) 1 dy = 1 γ1b1 � 1 − (1 + γ1b1τ (1) 1 )e−γ1b1τ (1) 1 1 − e−γ1b1τ (1) 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Therefore, E \uf8eb \uf8ec \uf8ed � i∈I(1) 1 Y (1) i \uf8f6 \uf8f7 \uf8f8 = 1 γ1b1 � 1 − (1 + γ1b1τ (1) 1 )e−γ1b1τ (1) 1 1 − e−γ1b1τ (1) 1 � E(N(1) 1 ) = 1 γ1b1 � 1 − (1 + γ1b1τ (1) 1 )e−γ1b1τ (1) 1 1 − e−γ1b1τ (1) 1 � � 1 − e−γ1b1τ (1) 1 � = 1 γ1b1 � 1 − (1 + γ1b1τ (1) 1 )e−γ1b1τ (1) 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Similarly, E \uf8eb \uf8ec \uf8ed � i∈I(1) 2 Y (1) i \uf8f6 \uf8f7 \uf8f8 = E \uf8eb \uf8ec \uf8edE \uf8eb \uf8ec \uf8ed � i∈I(1) 2 Y (1) i |N(1) 2 = n(1) 2 \uf8f6 \uf8f7 \uf8f8 \uf8f6 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For i ∈ I(1) 2 , Y (1) i follows a left truncated exponential distribution with PDF γ1b2e−γ1b2y e−γ1b2τ(1) 1 for y > τ (1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Hence, E � Y (1) i � = � ∞ τ (1) 1 yγ1b2e−γ1b2y e−γ1b2τ (1) 1 dy = 1 γ1b2 + τ (1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Therefore, E \uf8eb \uf8ec \uf8ed � i∈I(1) 2 Y (1) i \uf8f6 \uf8f7 \uf8f8 = � 1 γ1b2 + τ (1) 1 � E(N(1) 2 ) = � 1 γ1b2 + τ ′ 1 � e−γ1b1τ (1) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 25 Appendix B: Calculations of some important relia- bility characteristics Derivation of the quantile function: Denote p = G(j)(y) for y ∈ � τ (j) k−1, τ (j) k � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' then, y = η(p) for p ∈ � G(j)(τ (j) k−1), G(j)(τ (j) k ) � , k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Now, p = 1 − e −(J−j)γj ��k−1 ℓ=1 bℓ � τ (j) ℓ −τ (j) ℓ−1 � +bk � y−τ (j) k−1 �� =⇒ bk � y − τ (j) k−1 � = −log(1 − p) (J − j)γj − k−1 � ℓ=1 bℓ � τ (j) ℓ − τ (j) ℓ−1 � =⇒ y = τ (j) k−1 − log(1 − p) (J − j)γjbk − 1 bk k−1 � ℓ=1 bℓ � τ (j) ℓ − τ (j) ℓ−1 � , if p ∈ � G(j)(τ (j) k−1), G(j)(τ (j) k ) � , k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' If y ∈ � τ (j) N , ∞ � , then y = η(p) for p ∈ � G(j)(τ (j) N ), 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Therefore, p = 1 − e −(J−j)γj ��N−1 ℓ=1 bℓ � τ (j) ℓ −τ (j) ℓ−1 � +bN � y−τ (j) N−1 �� =⇒ y = τ (j) N−1 − log(1 − p) (J − j)γjbN − 1 bN N−1 � ℓ=1 bℓ � τ (j) ℓ − τ (j) ℓ−1 � , if p ∈ � G(j)(τ (j) N ), 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Derivation of MTTF: MTTF of the system lifetime T is given by E(T) = E �J−1 � j=0 Y (j) � = J−1 � j=0 E(Y (j)), where E(Y (j)) = � ∞ 0 P(Y (j) > y)dy = � τ (j) N−1 0 e−(J−j)Λ(j)(y)dy+ � ∞ τ (j) N−1 e−(J−j)Λ(j)(y)dy = I1+I2 (say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' I1 = � τ (j) N−1 0 e −(J−j)γj �N k=1 ��k−1 ℓ=1 bℓ � τ (j) ℓ −τ (j) ℓ−1 � +bk � y−τ (j) k−1 �� 1 [τ(0) k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' τ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=')(y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s−1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='+bs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='y−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s−1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−(J−j)γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�s−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ=1 bℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� � τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='e ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f3e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−(J−j)γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�s−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ=1 bℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� \uf8ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f01 − e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−(J−j)γjbs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='(J − j)γjbs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8fd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8fe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−(J−j)γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�s−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ=1 bℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='− e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−(J−j)γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ=1 bℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='(J − j)γjbs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8fd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8fe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='I2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−(J−j)γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='��k−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ=1 bℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='+bk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='y−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='k−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='[τ(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' τ(0) k )(y) dy = � ∞ τ (j) N−1 e −(J−j)γj ��N−1 ℓ=1 bℓ � τ (j) ℓ −τ (j) ℓ−1 � +bN � y−τ (j) N−1 �� dy = e −(J−j)γj �N−1 ℓ=1 bℓ � τ (j) ℓ −τ (j) ℓ−1 � � ∞ τ (j) N−1 e −(J−j)γjbN � y−τ (j) N−1 � dy = e −(J−j)γj �N−1 ℓ=1 bℓ � τ (j) ℓ −τ (j) ℓ−1 � � 1 (J − j)γjbN � = e −(J−j)γj �N−1 ℓ=1 bℓ � τ (j) ℓ −τ (j) ℓ−1 � (J − j)γjbN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Therefore, E(Y (j)) = N � s=1 \uf8f1 \uf8f2 \uf8f3 e −(J−j)γj �s−1 ℓ=1 bℓ � τ (j) ℓ −τ (j) ℓ−1 � − e −(J−j)γj �s ℓ=1 bℓ � τ (j) ℓ −τ (j) ℓ−1 � (J − j)γjbs \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' From here, the results follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Derivation of moment generating function of system lifetime: Note that the system lifetime MGF of T is T = J−1 � j=0 Y (j), where Y (j)’s are independent for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' , (J − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Therefore, the MGF of T is φT(t) = J−1 � j=0 φY (j)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Now, φY (j)(t) = E(etY (j)) = � ∞ 0 etygY (j)(y)dy = � τ (j) N−1 0 ety(J − j)λ(j)(y)e−(J−j)Λ(j)(y)dy + � ∞ τ (j) N−1 ety(J − j)λ(j)(y)e−(J−j)Λ(j)(y)dy = I1 + I2 (say) , where gY (j)(y) = (J − j)λ(j)(y)e−(J−j)Λ(j)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For t ∈ R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' I1 = � τ (j) N−1 0 ety(J − j)γj N � k=1 bk1[τ (j) k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=') (y) e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−(J−j)γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='��k−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ=1 bℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='+bk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='y−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='k−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='(J − j)bsγj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='(J−j)γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='��s−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ=1 bℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='+bs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='y−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−ty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='(J − j)bsγje ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−(J−j)γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�s−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ=1 bℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� � τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='− ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� \uf8ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='\uf8f0etτ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s−1 − e ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−tτ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' For t < (J − j)γjbN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' I2 = � ∞ τ (j) N−1 ety(J − j)γj N � k=1 bk1[τ (j) k−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=') (y) e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−(J−j)γj ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='+bk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='y−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='k−1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ty−(J−j)γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='��N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ=1 bℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='+bN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='y−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='= (J − j)bNγje ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−(J−j)γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ=1 bℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� � ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='e ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−ty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='dy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='= (J − j)bNγje ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−(J−j)γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ=1 bℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='etτ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='(J − j)γjbN − t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='= (J − j)bNγj · e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='tτ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='N−1−(J−j)γj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='�N−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ=1 bℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='−τ (j) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='ℓ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='(J − j)bNγj − t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' Therefore, for t < (J − j)γjbN, φY (j)(t) = N � s=1 (J − j)bsγj (J − j)bsγj − t � e − � (J−j)γj �s−1 ℓ=1 bℓ � τ (j) ℓ −τ (j) ℓ−1 � −tτ (j) s−1 � −e − � (J−j)γj �s ℓ=1 bℓ � τ (j) ℓ −τ (j) ℓ−1 � −tτ (j) s �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' From here the result follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AzT4oBgHgl3EQfg_xe/content/2301.01477v1.pdf'} +page_content=' 28' metadata={'source': 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a/CtE2T4oBgHgl3EQfSAdG/content/tmp_files/2301.03787v1.pdf.txt b/CtE2T4oBgHgl3EQfSAdG/content/tmp_files/2301.03787v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..07276be14cce2a461c6f735602a9f50c0d421514 --- /dev/null +++ b/CtE2T4oBgHgl3EQfSAdG/content/tmp_files/2301.03787v1.pdf.txt @@ -0,0 +1,2254 @@ +1 +Synchronization of Josephson junctions in series +array +Abhijit Bhattacharyya +Abstract—Multi-qubit quantum processors coupled to net- +working provides the state-of-the-art quantum computing plat- +form. However, each qubit has unique eigenfrequency even +though fabricated in the same process. To continue quantum +gate operations besides the detection and correction of errors it +is required that the qubits must be synchronized in the same +frequency. This study uses Kuramoto model which is a link +between statistical mean-field technique and non-linear dynamics +to synchronize the qubits applying small noise in the system. This +noise could be any externally applied noise function or just noise +from the difference of frequencies of qubits. The Kuramoto model +tunes the coupled oscillators adjusting the coupling strength +between the oscillators to evolve from the state of incoherence to +the synchronized state. +Index Terms—Josephson junction, Kuramoto Model, synchro- +nization, oscillators +I. INTRODUCTION +J +Osephson junction controls the flow of magnetic flux +quanta through frequency and voltage. Modern instruments +require measurement of voltage with a reproducible capability +exceeding the uncertainty of realization of the SI volt (cur- +rently 0.4 parts on 106). Before 1972, SI volt was represented +by using carefully stabilised Weston cell banks [1]. Drift and +transportability problems with these electrochemical artifact +standards limited the uniformity of voltage standards to about +1 part in 106. These uniformity was drastically improved by +the usage of Josephson junction [1]. +Josephson equation for supercurrent through a supercon- +ducting tunnel junction, called as DC Josephson Effect, is +defined as [2]–[4] +I = Ic sin +�4πe +h +� +V dt +� +, +(1) +where Ic is critical current, h is Planck’s constant and e is +electron charge. When a dc voltage is applied in equation +(1), the phase will vary linearly with time and current will +be sinusoidal with amplitude Ic and frequency fJ = 2eV/h. +The magnetic flux threading a superconducting loop or hole is +quantized [5]. The superconducting magnetic flux quantum Φ0 += h/(2e) is 2.0678×10−15 Wb. The inverse of flux quantum +1/Φ0 is called Josephson constant KJ defined as 2e/h has a +value of 483.597 GHz/mV . During each oscillation, a single +quantum of magnetic flux h/(2e) passes through the junction +which is very difficult to measure. However, if an alternating +current with frequency f is applied across the junction, there +Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400 +094, India +vega@barc.gov.in; abhihere@gmail.com +is a range of bias current for which flow of flux quanta +will phaselock to the applied frequency. Under this phase +locked condition, the average voltage across the junction is +precisely (h/2e)f. This effect is known as ac Josephson effect +observed as a constant voltage step at V =(h/2e)f in the I −V +characteristic curve. This means a Josephson junction can act +as a “Voltage to frequency converter”. It is also possible for +the junction to phaselock with the harmonics of fJ resulting +in a series of steps at voltages V =nf(h/2e), where n is an +integer denoting step number. This accuracy was limited to +the condition that a Josephson voltage higher than 10mV was +never used [6]. Therefore, if one obtain Josephson voltage +over 100 mV , the accuracy could be remarkably improved +besides the ability to vary the Jsephson voltage with the +frequency and step number could be utilized as potentiometer. +Series array of Josephson junction [6] has been effectively +used in development of a potentiometer system to produce (1- +10)V [1], [6] with uncertainty about 2.5 × 10−9 [6]. Larger +series arrays were initially considered as impractical due to +junction nonuniformity. The nonuniformity demanded each +junction to be biased separately. In 1977, Levinsen et al [7] +stated the important of the parameter βc=4πeIcR2C/h in +determining the characteristics of RF induced Josephson steps. +This βc is measure of the damping of Josephson oscillations +by the junction shunting resistance R. +The Josephson junction is also a natural choice for sub- +millimeter local oscillator [8], [9] as one may capitalize the +voltage controlled oscillator property. However, the disadvan- +tage, in this case lies in very low power output. The Josephson +constant clearly indicates that with dc voltage bias at 1 mV +at 483.6 GHz, the junction may accept 100 µA current +keeping under the limitation of Ic which limits the maximum +output RF power at about 100 nW. This requirement indicates +series array of junctions with a common current bias demands +keeping all the junctions in phase. +However, the issue with series array of junctions operated +with common current bias arises with nonuniformity of each +junction due to fabrication processes [1], [6]. When junctions +are connected in series, the system behaves as a coupled os- +cillator and understanding the periodic solutions is important. +Two special types of periodic solutions exist [10], namely, in- +phase state and splay state. +An in-phase state with period T is a state where all the +oscillators always possess the same phase at all times, i.e. +θi(t) = θj(t), and θi(t + T) = θi(t) + 2π. +The splay-phase or anti-phase or rotating wave state with +period T is a solution where the oscillators can be labeled +so that θi(t) = Θ(t + jT/N) for all j for some function +arXiv:2301.03787v1 [quant-ph] 10 Jan 2023 + +2 +Θ(t + T) = Θ(t) + 2π. Thus, this state indicates that all the +oscillators have the same waveform Θ(t) except for a shift in +time. As per [10], one may imagine that each oscillator “fires” +when it reaches a certain angle. For an in-phase solution, all +the oscillators fire simultaneously at every instant T, while +splay-phase state has a single oscillator firing every T/N +instant. Therefore, for splay-phase state, oscillators nearly +coincide or coincide when ˙θ is small where as for large +values, oscillators are not coherent. The definition of splay- +phase does not imply that the phases of the oscillators are +equi-spaced around he circle. The oscillators bunch up for +smaller ˙θ while spread out for large ˙θ. Therefore, splay-state +shows non-uniformity in the distribution of oscillators as they +are coherent for smaller ˙θ. It has been shown that [10], [11], +the non-uniformity can be removed by determining a set of +“natural” angles ϕj, so that the splay-phase solution satisfies +ϕj(t) = 2πj/N + 2πt/T + const. The “natural” angle based +dynamical system gets locked. This provides an idea of phase- +locking N oscillators, like N Josephson junctions, having +eigenfrequencies with smaller spread which may get locked +to some resonating frequency. +Kuramoto model provides an exactly solvable mean-field +model of coupled nonlinear oscillators connecting a large of +them having distributed natural frequencies. This model links +mean-field techniques and nonlinear dynamics together and +also provides precise technique to tune the synchronization. +Section II discusses the theory of the Kuramoto model, +Section III discusses on the reduction of the equations for +the Josephson junctions connected in series to the Kuramoto +Model framework and section IV discusses on the numerical +analysis of the results for the generalised Kuramoto Model +theory and Kuramoto model for Josephson junctions. +II. KURAMOTO MODEL +Let us consider a system of N globally coupled differential +equations with the stable limits cycles. Yoshiki Kuramoto de- +veloped a mathematical model for coupled oscillators (n ⩾ 2) +to synchronize which is known as “Kuramoto model” [12]. +In this model, each jth oscillator is represented by a phase +variable θj(t), possessing its own natural frequency ωj ∈ R. +The dynamics of the system of coupled N oscillators becomes +˙θj(t) = ωj + +N +� +i=1,j̸=i +Kji sin (θj(t) − θi(t)) , j ∈ {1, . . . , N} , +(2) +where Kji is coupling coefficient of the jth oscillator with all +other oscillators in the system. Kuramoto assumed mean field +coupling among phase oscillators such that Kji ≈ K/N ⩾ 0 +where K is mean coupling strength which changes (2) as +˙θj(t) = ωj + K +N +N +� +i=1,j̸=i +sin (θj(t) − θi(t)) , j ∈ {1, . . . , N} , +(3) +where, K ⩾ 0 is the coupling strength among the oscillators +whose frequencies are distributed with a probability density +g(ω). One may find a suitable rotating frame like θj → θj − +Ωt transforming the system so that natural frequencies of the +oscillators may have zero mean, where Ω is the first moment of +the distribution function of natural frequencies g(ω). Therfore, +one may consider the normal form calculation for the system +such that one may define the system of equations as +˙θj = fj(θj) + K +N +N +� +i=1,i̸=j +g (θi, θj) , θj ∈ Rd, j = 1, . . . , N, +(4) +where, function fj(θj) are eigenfrequencies defining the nat- +ural dynamics in the system. Here coupling parameter K +has been added with coupling strength K/N, g is the phase +response curve defining the interaction of the system. In the +following section, we are not discussing with the stability of +the dynamical system, bifurcation etc while one may consult +other references like [13]. +In the original paper [12], Kuramoto considered the proba- +bility density g(ω) to be uni-modal and symmetric centered at +mean frequency ω so that, without loss of generality, one can +assume that the mean frequency ω = 0 after a shift leading to +g(ω) = g(−ω) for the even and symmetric distribution g(ω). +To diagnose the feasibility of synchronization, Kuramoto +introduced the order parameter R(t) projecting the oscillation +on unit circle where R(t) : 0 ⩽ R(t) ⩽ 1 is a measure of the +coherence of oscillators as +R(t)eȷψ(t) = 1 +N +N +� +i=1 +eȷθi(t), +(5) +where R(t) = 0 for asynchronised oscillators, +and R(t) > 0 for synchronization. +The quantity ψ(t) refers to average phase of all the oscillators +at an instant t. Physically, this order parameter R(t) is the +centroid of a set of N points eȷθi distributed in the unit circle in +the complex plane at the instant t. If the phases are uniformly +spread in the range [−π, π], then R → 0 indicates that the +oscillators are not synchronized. All the oscillators become +synchronized with the same average phase ψ(t) for R(t) ≈ 1. +If the dynamics show stability of R(t) at 1, then the oscillators +are synchronized and phaselocked. Eq. (3) may be re-written +by multiplying Ke−ȷθj on both sides of (5) and equating the +imaginary parts of the both sides to reduce (3) to +˙θj(t) = ωj + KR(t) sin (ψ(t) − θj(t)) = vj(θ, ω, t) (say). +(6) +Here, vj(θ, ω, t) is the angular velocity of a given oscillator +with phase θ and natural frequency ω at the instant t. The +equation (6) reveals that the interaction is set through R(t) +and ψ(t) while the phases θj seem to evolve independently +from each other. Also the effective coupling is proportional to +the order parameter R(t) creating a feedback relation between +coupling and synchronization. In the limit K → 0, (6) reduces +to +θj(t) ≈ ωjt + θ(0), +(7) +where, θj(0) denotes initial phase of the jth oscillator and +(7) suggests that each oscillator oscillates with own natural +frequencies in the absence of coupling. + +3 +In the limit of infinite number of oscillators having a +distribution of frequency, phase over time, Kuramoto de- +scribed the system by the probability density ρ (θ, ω, t) so +that ρ (θ, ω, t) dθ gives the fraction of oscillators with phase +between θ(t) and θ(t) + dθ(t) at the instant t for a given +natural frequency ω. Since ρ is non-negative and 2π-periodic +in θ satisfying the normalization condition +� π +−π +ρ (θ, ω, t) dθ = 1. +(8) +The probability density function g must also obey the equation +of continuity using the angular velocity v(θ, ω, t) as +∂ρ(θ, ω, t) +∂t ++ ∂ +∂θ {ρ(θ, ω, t).v} = 0, +∂ρ(θ, ω, t) +∂t ++ +∂ +∂θ [ρ(θ, ω, t) {ω + KR(t) sin (ψ(t) − θ(t))}] = 0.(9) +In the limit R(t) → 0, the dynamics provides stationary +solution for ρ(θ, ω, t) = 1/(2π). +In the continuum limit, (5) gets re-defined by the order +parameter R(t) and the average phase ψ(t) incorporating +previously described frequency distribution as +R(t)eȷψ(t) = +� π +−π +� ∞ +−∞ +eȷθρ (θ, ω, t) g(ω)dωdθ. +(10) +In the strong coupling limit where K → ∞ indicate K ≫ +Kc where Kc is critical coupling strength and (6) reduces to +system having phases reduced to the average phase as θ(t) = +ωt + θ(0) = ψ(t). +From (6), if oscillators get into phaselocked condition, +vi(t) → 0 which provides +ωj = KR(t) sin (θj(t) − ψ(t)) , −π +2 ⩽ (θj(t) − ψ(t)) ⩽ π +2 . +(11) +From (9), partially synchronized state leading to a locked +system can be described as +∂ +∂t(ρ(θ, ω, t)) = 0 which also +means +∂ +∂θ (ρ(θ, ω, t).v(t)) = 0. Eq. (11), in this partial +synchronized state for vj(t) → 0 and +∂ +∂t (ρ(θ, ω, t)) = 0, +reduces to +ω +KR(t) → sin(θj(t) − ψ(t)), +which means +ρ(θ, ω, t) = δ +� +θj(t) − ψ(t) − sin−1 +� +ω +KR(t) +�� +H(cos θ), +(12) +such that |ω| ⩽ KR(t) and +H(x) = +1, +x > 0, +0, +elsewhere.. +(13) +Now, for the other condition +∂ +∂θ (ρ(θ, ω, t)v(t)) = 0 using +(6), +ρ(θ, ω, t)v(t) = C(say) = constant, +or, +ρ(θ, ω, t) = +C +|ω + KR(t) sin(θj(t) − ψ(t))|, +|ω| � KR(t). +(14) +The constant C can be determined from (8) such that (14) +reduces to +ρ(θ, ω, t) = +� +ω2 − K2R2(t) +2π|ω − KR(t) sin(θj(t) − ψ(t))|, +|ω| � KR(t). +(15) +Therefore, the constraint on the probablity density of the +oscillators may be +ρ(θ, ω, t) = δ +� +θj(t) − ψ(t) − sin−1 +� +ω +KR(t) +�� +H(cos θ), +for |ω| ⩽ KR(t) +(16) +and +ρ(θ, ω, t) = +� +ω2 − K2R2(t) +2π|ω − KR(t) sin(θj(t) − ψ(t))|, elsewhere . +(17) +Here δ is the Dirac delta function. Eqs. (16) and (17) indicate +that partial synchronized states are divided into two groups +depending on the natural frequencies. Oscillators having con- +straint |ω| ⩽ KR(t) operate in mean-field resulting in locking +in a common average phase ψ(t) = Ωt where Ω is the average +frequency of the ensemble of the oscillators in this regime. On +the other side, the second group of oscillators having constraint +|ω| > KR(t) rotate incoherently which are called as drifting +oscillators. +Inserting (16) and (17) in (10) we get +R(t) += +� π +−π +� ∞ +−∞ +eȷ(φ(t)−ψ(t)) +δ +� +θ(t) − ψ(t) − sin−1 +� +ω +KR(t) +�� +g(ω)dθdω ++ +� π +−π +� +|ω|⩽KR(t) +� +ω2 − K2R2(t)g(ω)dθdω +2π|ω − KR(t) sin(θ(t) − ψ(t))|. +(18) +Since g(ω) is even and symmetric, g(ω) = g(−ω) and +ρ(θ + π, −ω) = ρ(θ, ω). The even function condition makes +the second term of (18) vanish which physically means all +the incoherent oscillator solutions vanish resulting in order +parameter R(t) only for coherent synchronized oscillators that +reform as +R(t) += +� +|ω|⩽KR(t) +cos +� +sin−1 +� +ω +KR(t) +�� +g(ω)dωdθ, += +� +π +2 +− π +2 +cos θg (KR(t) sin θ) KR(t) cos θdθ, += +KR(t) +� +π +2 +− π +2 +cos2 θg (KR(t) sin θ) dθ. +(19) +Here, (19) shows a trivial solution for which order parameter +R(t) = 0 which actually shows incoherence as discussed +earlier for ρ (θ, ω, t) = 1/(2π). However, (19) also suggests +1 = K +� +π +2 +− π +2 +cos2 θ g (KR(t) sin θ) dθ. + +4 +Setting R(t) = 0, considering K = Kc - the critical coupling +strength we get, +Kc = +2 +πg(0), +(20) +that triggers the synchronization. In general, expanding the +right hand side of (19) in terms of powers of KR(t) and +considering g′′(0) < 0 the order parameter can be written as +R(t) ∼ +� +−8 (K − Kc) +K3c g′′(0) , +(21) +which shows that near the transition point, the order parameter +[12], [14] yields the form R(t) ∼ (K − Kc)β with β = 1/2 +like second order phase transition. +The Kuramoto model can be generalized for a complex +network including the connectivity parameter in the coupling +term as +˙θj = ωj + +N +� +i=1 +KjiAji sin(θj − θi), +(22) +where, Kji is the coupling strength between nodes j and i. +Aji is the element of the adjacency matrix A (Aji = 1 if there +is a connection between j and i else Aji = 0 otherwise). +Any real system may have noise. Let us discuss on the +effect of the noise for the Kuramoto model. The noise may +arise from the variation of frequency of incoherent oscillators +as they may not be identical or there may either be an external +white noise or white noise inherent to the system. Therefore +the model (3) could be reframed as +˙θj += +σωj + K +N +N +� +i=1 +sin (θj(t) − θi(t)) + +√ +Γηj(t), +: +j ∈ {1, . . . , N} , +(23) +where, both ωj and ηj(t) are Gaussian distributions having +zero mean and unit variance while σ and Γ behave as am- +plitudes of the noise. Here last term refers to white noise in +the system. Therefore (23) physically indicates locally coupled +oscillators having natural frequencies of oscillators derived +from Gaussian distribution in presence of stochastic effects +like white noise due to fluctuations in the system. The reason +for stochastic behavior may vary for different systems while +any natural process exhibit stochastic behavior. . +The situation of lim σ → 0 refers to the Kuramoto model +having identical oscillators in presence of gaussian white +noise. The system behaves as if the system is in contact with +a heat source and the dynamics is evolving in the statistical +equilibrium. +The situation for lim Γ → 0 indicates that the Kuramoto +model has been constructed with oscillators having distributed +natural frequencies in absence of gaussian white noise. The +system behaves as nonlinear dynamical system relaxing to the +non-equilibrium stationary state. +Beside this brief summary, one may also consult articles +like [15]. +Next, let us transform the Josephson equations for series +array of junctions to Kuramoto model. +III. KURAMOTO MODEL FOR JOSEPHSON JUNCTION +SERIES +The Josephson junction array can be constructed using +Kirchhoff’s laws considering each Josephson junction as a +parallel circuit of two elements: an ideal resistance ρ carrying +ideal current Iρ and a junction carrying critical current Ic. +Actual Josephson junction also contains a capacitor in parallel +to the nonlinear inductor which we have neglected due to +its very small value. Let each of N junctions be connected +serially and then coupled to external load having inductance +L, resistance R and capacitance C. +C +R +L +Ib +ρ1 +I1 +ρ2 +I2 +ρN +IN +Fig. 1. Schematic circuit of qubits connected in series parallel to a Load. +Let us consider Josephson junction in the series array, say +jth junction and following Josephson equation; we express the +circuit shown in the Fig.1 as +V (t) +ρj ++ Ij sin φj + dQ +dt = Ib, +which can be written as, +dφj +dt = 2πρj +Φ0 +� +Ib − Ij sin φj − dQ +dt +� +. +(24) +Further, +L ¨Q + R ˙Q + Q +C = +N +� +k=1 +Vk, +or, +L ¨Q + +� +R + +N +� +k=1 +ρk +� +dQ +dt + Q +C = − +N +� +k=1 +Ikρk sin φk, +(25) +where Q is the charge on load capacitor, Φ0 = h/(2e) is +magnetic flux quantum, h is Planck’s constant, e being the +charge of an electron. Here, junction resistance ρk for any +junction k is very small compared to the load variable Q/C +such that one may consider, Q/C − � +k ρkIb ≈ Q/C. To +understand the effect of external parameters like L, C and R +on each junction, one may consider a scaled version of those +parameters by choosing +l = L +N , r = R +N , c = NC. +(26) + +5 +Here, it is to be noted that +Φ0 +Ij += +1 +2πf 2 +j Cj +where fj is the frequency and Cj is the capacitance of jth +junction. +Let us now consider transformation of time t and charge Q +so that (24) and (25) become dimensionless. From (24) +Φ0 +2πρjIj +dφj +dt + sin φj + 1 +Ij +dQ +dt = Ib +Ij += αj. +Let us consider the following transformation relation to +transform time t to dimensionless form τ as +Φ0 +2πρjIj +d +dt ≡ d +dτ . +(27) +such that we may write +dφj +dτ + sin φj + 2πρj +Φ0 +dQ +dτ = αj. +(28) +Substituting dimensionless time τ and scaled parameters as +in (26) in (25) we get, +L +N +�2πρjIj +Φ0 +�2 d2Q +dτ 2 ++ +(R + �N +k=1 ρk) +N +�2πρjIj +Φ0 +� dQ +dτ ++ +Q +NC = 1 +N +N +� +k=1 +−Ikρk sin φk, +or, l +�2πρjIj +Φ0 +�2 d2Q +dτ 2 ++ +� +r + +�N +k=1 ρk +N +� �2πρjIj +Φ0 +� dQ +dτ ++ +Q +c = − 1 +N +N +� +k=1 +Ikρk sin φk. +(29) +Let us also consider the following transformation to transform +charge Q to dimensionless form q as +2πρjIj +Φ0 +Q ≡ qj. +(30) +Therefore, through (29), (25) transforms as +d2qj +dτ 2 + γj +dqj +dτ + ω2 +0jqj = −δj +N +N +� +k=1 +Ikρk sin φk. +(31) +Eq. (30) can be used to rewrite (28) as +dφj +dτ + sin φj + ϵj +dqj +dτ = αj, +(32) +where coefficients may be written as +γj += +� +Φ0 +2πρjIj +� �1 +l +� � +r + +�N +k=1 ρk +N +� +, +(33) +ω2 +0j += +� +Φ0 +2πρjIj +�2 1 +lc, +(34) +δj += +� +Φ0 +2πρjIj +� 1 +l , +(35) +and ϵj += +1 +Ij +. +(36) +Let us write the equation (32) in the uncoupled form for +ϵj → 0 or ˙Q → 0 such that we get, +dφj +dτ = αj − sin φj. +(37) +As discussed in the Section I, the splay-state shows that +transforming the dynamical system equations make a rigid sys- +tem with coherent frequencies in weak coupling or uncoupled +limit. Hence, let us transform φj in (24) into ‘natural’ angle +ψj such that dψj +dt = constant. Eq. (37) can be transformed in +terms of the ‘natural’ angle ψj such that dψj/dt − c, where c +is constant to be determined, i.e. transformation as φj → ψj as +uniform rotation with first derivative remaining constant. The +constant ‘c’ may be determined with the fact that the time to +complete one cycle by these two sets of coordinates must be +same. Thus, +T += +� T +0 +dτ += +� 2π +0 +dψj +c += +� 2π +0 +dψj +ωj += +� 2π +0 +dφj +(αj − sin φj). +or, 2π +ωj += +2π +�� +α2 +j − 1 +�, for αj ⩾ 0 i.e. Ib ⩾ Ij, +which shows +ωj = +� +α2 +j − 1. +(38) +Then the transformation to the natural angles satisfies +dψj = +� +α2 +j − 1 +αj − sin φj +dφj, +(39) +which on integration yields +ψj = 2 tan−1 +�� +αj − 1 +αj + 1 tan +�φj +2 + π +4 +�� +. +(40) +At this point, one may construct a transformation function +ψ(φj) to translate any angle φj to its natural angle ψj while +another transformation function φ(ψj) may be used to invert +as +ψ (φ) = 2 tan−1 +�� +α − 1 +α + 1 tan +�φ +2 + π +4 +�� +, (41) +φ (ψ) = 2 tan−1 +�� +α + 1 +α − 1 tan +�ψ +2 +�� +− π +2 . (42) +Here, we use the shorthand: ψj ≡ ψ(φj) and φj ≡ φ(ψj). +From (40), +sin φj = 1 − αj cos ψj +αj − cos ψj += αj − +α2 +j − 1 +αj − cos ψj +. +(43) +Detailed derivation of (43) from (40) is shown in appendix A. +Therefore, one may rewrite (32) using (39) and (43) as +dψj +dτ += +dψj +dφj +dφj +dτ = +� +α2 +j − 1 +αj − sin φj +. +� +αj − sin φj − ϵj +dqj +dτ +� +, += +� +α2 +j − 1 − +ϵj +� +α2 +j − 1 +αj − sin φj +dqj +dτ . +(44) + +6 +Let us rescale non-dimensional quantity τ as ˜τ such that +τ = +˜τ +� +α2 +j − 1 +=⇒ +d +d˜τ ≡ +1 +� +α2 +j − 1 +d +dτ +=⇒ +d2 +dτ 2 ≡ +� +α2 +j − 1 +� d2 +d˜τ 2 . +(45) +Eq. (44), using (45), transforms as +dψj +d˜τ = 1 − +ϵj +� +α2 +j − 1 +αj − sin φj +dqj +d˜τ , +(46) +The weak-coupling solution of (44) may be written as +ψj(τ) ≡ +�� +α2 +j − 1 +� +τ + cj = ˜τ + ψj0, +(47) +where cj is the integration constant. Initially, at τ += 0, +one may assume initial phase as ψj0 such that cj=ψj0. The +reference [11] discusses about the importance of the weak +coupling condition for the Josephson junction arrays and drift +in ψj may be obtained by averaging (46) over one cycle as +�dψj +d˜τ +� += 1 − 1 +2π +� 2π +0 +ϵj +� +α2 +j − 1 +αj − sin φj +�dqj +d˜τ +� +d˜τ. +(48) +Similarly, one may rewrite non-dimensional charge equation +(31) in terms of ˜τ as +� +α2 +j − 1 +� d2qj +d˜τ 2 ++ +γj +� +α2 +j − 1dqj +d˜τ + ω2 +0jqj += +−δj +N +N +� +k=1 +Ikρk sin φk. +(49) +It is usually convenient to write sin(φj)=sin(φ(ψj)) in terms +of its Fourier series as +sin φ(ψk) += +∞ +� +n=0 +Akn cos (nψkn) += +∞ +� +n=0 +Akn cos {n (˜τ + ck)} . +(50) +Then (49) reduces to +� +α2 +j − 1 +� d2qj +d˜τ 2 ++ +γj +� +α2 +j − 1dqj +d˜τ + ω2 +0jqj += +−δj +N +N +� +k=1 +∞ +� +n=0 +IkρkAkn cos {n (˜τ + ck)} . +(51) +One may obtain the steady-state solution of (51) as +qj(˜τ) += +−δj +N IkρkBkn cos {n (˜τ + ck) + βkn} ,(52) +dqj(˜τ) +d˜τ += +δj +N nIkρkBkn sin {n (˜τ + ck) + βkn} , (53) +d2qj(˜τ) +d˜τ 2 += +δj +N n2IkρkBkn cos {n (˜τ + ck) + βkn} ,(54) +where +B2 +kn += +A2 +kn +n2γ2 +j +� +α2 +j − 1 +� ++ +� +n2 � +α2 +j − 1 +� +− ω2 +0j +�2 , (55) +βkn += +tan−1 +� +� +nγj +� +α2 +j − 1 +n2 � +α2 +j − 1 +� +− ω2 +0j +� +� = βn. +(56) +Using the expression (43), one may derive Akn and obtain +Ak0 += +1 +π +� π +−π +1 − αk cos ψk +αk − cos ψk +dψk, +(57) +Akn += +1 +π +� π +−π +1 − αk cos ψk +αk − cos ψk +cos +�nπψk +π +� +dψk (58) +where n ̸= 0. +Bkn denotes the amplitude of the linear damped oscillator +while βkn denotes its phase. Therefore, Bkn must be chosen +to be positive. +Now, (48) may be re-written as +�dψj +d˜τ +� += +1 − +ϵjδj +� +α2 +j − 1 +2πN +� 2π +0 +� +1 +αj − sin φj +× +N +� +k=1 +∞ +� +n=0 +nIkρkBkn sin {n (˜τ + ck) + βkn} +� +d˜τ. +(59) +Using (43), +sin φj = αj − +α2 +j − 1 +αj − cos ψj +, +or, αj − sin φj = +α2 +j − 1 +αj − cos ψj +. +(60) +With this (59) may be modified using (60) to +or, +�dψj +d˜τ +� += 1 + Kj +N +N +� +k=1 +Ak sin (cj − ck − ζj) , +(61) +where, +Kj += +ϵjδj +� +α2 +j − 1 +� +γ2 +j +� +α2 +j − 1 +�2 + +� +ω2 +0j − +� +α2 +j − 1 +�2�2 , +(62) +AK += +Ikρk +� +1 − α2 +k + αk +� +α2 +k − 1 +� +, +(63) +ζj += +tan−1 +� +� +γj +� +α2 +j − 1 +α2 +j − 1 − ω2 +0j +� +� = β1j. +(64) +Reader may check detailed description of the derivation in the +appendix B. +In the final step, one may replace the ‘initial values’ of +phases by their slowly evolving components like ⟨ψj(˜τ)⟩ and +⟨ψk(˜τ)⟩. Also one may get firstorder averaged equation by +dropping the angular brackets so that (61) transforms to +dψj +d˜τ = 1 + Kj +N +N +� +k=1 +Ak sin (ψj(˜τ) − ψk(˜τ) − δ) . +(65) + +7 +Eq. (65) resembles the Kuramoto model in a generalized +form. For the sake of mathematical formalities, it is important +to note that except terms corresponding to n = 1 terms for +other values of n becomes zero. +To arrive at (65), it was assumed that the fabrication process +may not guarantee exactly same values of parameters for each +junction and hence one may consider that each junction has +different internal resistance and different critical current. The +difference may be very small for junctions prepared in the +same batch. If the fabrication process is done in very skilled +sequence (65) may turn into special form for assuming ρ1 = +ρ2 = . . . = ρN = ρ (say) and I1 = I2 = . . . = IN = Ic(say) +so that each junction has nearly same frequency f (say). This +case of identical junctions has been studied extensively in may +literatures. +The transformation (27) for time leads to +Φ0 +2πρIc +d +dt ≡ d +dτ . +(66) +while (30) entails +2πρIc +Φ0 +Q ≡ q. +(67) +Consequently, (31) reduces to +d2q +dτ 2 + γ dq +dτ + ω2 +0q = − β +N +N +� +k=1 +sin φk, +(68) +where +γ += +� Φ0 +2πρIc +� � 1 +lρ +� +(r + ρ) , +(69) +ω2 +0 += +� Φ0 +2πρIc +�2 1 +lc, +(70) +β += +� Φ0 +2πρIc +� 1 +l , +(71) +(72) +Eqs. (55) and (56) become +B2 +n = +A2 +n +n2γ2 (α2 − 1) + {n2 (α2 − 1) − ω2 +0}2 , (73) +βn = tan−1 +� +nγ +√ +α2 − 1 +ω2 +0 − n2 (α2 − 1) +� +. +(74) +Repeating the earlier exercise, one may obtain the final phase +equation (59) as +�dψj +d˜τ +� += +1 − +β +2πN +√ +α2 − 1 +� 2π +0 +(α − cos (τ + cj)) +× +N +� +k=1 +∞ +� +n=0 +nBn sin {n (˜τ + ck) + βn} d˜τ. +(75) +In the case of identical junctions, computation shows that +only B1 exists while others are evaluated to zero. Thus, (75) +becomes +�dψj +d˜τ +� += +1 − +B1β +2πN +√ +α2 − 1 +� 2π +0 +(α − cos (τ + cj)) +× +N +� +k=1 +∞ +� +n=0 +n sin {n (˜τ + ck) + βn} d˜τ. +Integrating, we get +dψj +d˜τ = 1 + K +N +N +� +k=1 +sin (ψj(˜τ) − ψk(˜τ) − β1) , +(76) +where +K = +πB1β +2π +√ +α2 − 1 +. +(77) +Eq. (76) exactly resembles as the Kuramoto model. +In the following section, let us try to understand general +characteristics of the Kuramoto model in general and in the +context of Josephson junction array. +IV. ANALYSIS +A C + + code has been developed alongwith DISLIN +code to analyse the equations. DISLIN [16] is a freely +available graph plotting routine that plots during runtime and +can be stored. In this section, let us first investigate basic +Fig. 2. +Kuramoto model in arbitrary unit for 100 oscillators with K=4 +showing synchronization after a certain settling time within a band of +frequency range. +Kuramoto model as discussed in (3) including the effect of +coupling strength (K). If K is properly tuned, one may expect +synchronization as shown in Fig.2. +Here we consider that the oscillators are oscillating possess- +ing a frequency distribution g(ω). One may control width of +the distribution while keeping the zero mean. +We consider Logistic and Lorentzian fuctions having +width β. Oscillators tend get to be synchronized if K is equal +to or more than some threshold value Kc as discussed in (20). +g(ω) = +exp (−ω/β) +β [1 + exp (−ω/β)]2 +(78) +The Logistic function is described as (78) which shows +g(0)=1/(4β) where, β is the width. Likewise, one may define +the Lorentzian function as (79) +g(ω) = +b +(ω2 + b2), +(79) + +Kuramoto model for 1oo oscillators having K=4 without any noise +0.5 +a +sin +0 +-0.5 +0 +2 +4 +6 +8 +10 +time8 +Fig. 3. +100 oscillators with K=0.1 having Logistic distribution of width +0.001. +Fig. 4. 100 oscillators with K=0.1 having Lorentzian distribution of width +0.001. +so that one get g(0)=2/(πb). This g(0) estimates thresh- +old value of the coupling strength as Kc=2/πg(0). +One +may compare Fig.3 with Fig.5 where the latter is operating +with threshold coupling. The synchronization for the latter +shows phase space of order parameter as a dot denoting +synchronization. Figs.4 and 6 also show similar observation +of synchronization. +This theoretical study clearly heps us to understand the sig- +nificance of coupling strength and the treatment of frequency +range of oscillators to start with. +Next one may apply this understanding in the case of +Josephson junction. The situation is very much different hereas +the definition of K is complex for both non-identical and +Fig. 5. 100 oscillators oscillating with k=Kc=0.509 with Logistic function +of width 0.2. +Fig. 6. 100 oscillators oscillating with k=Kc=0.4 with Lorentzian function +of width 0.2. +identical junction arrays as evident from either (65) or (76) +respectively. Let us consider mean frequency may be around +5 GHz. Figs.7 and 8 show simulated results of systems +of 100 Josephson junctions in non-identical and identical +configurations respectively operated for ˜τ = 25. The interesting +part is that synchronization is not pulling the oscillators to a +certain unique frequency. Rather, oscillators tend to cool down +to a narrow band of frequencies resulting in an arc in phase +space diagram which resembles as if oscillators have a certain +‘viscosity’ in the combined system. For the non identical case, +the spread of Ic is considered very small like 0.1% while +variation in ρj is about 0.05 % as fabrication is much better +and junctions fabricated in the same substrate will not vary + +Phase graph of 10o oscillators forK=0.100 dt=0.001simulation time=10 +(a) Phase graph of 100 oscillators +(b) Order Paraneter of acillators +1.0 +10 +sin +P +-1.心 +0.0 +2.0 +4.0 +B.0 +08 +10.0 +0.0 +2.0 +4.0 +08 +10.0 +time (T) +time (r) +(o)orderParameterof oscillatorsatT=0 +(d) Order Parameter of oncillatore at T =io +1.1 +1.1 +40 +40 +90 +90 +0.3 +UTS +0.1 +sin +0.1 +0.1 +0.1 +0.8 +0.3 +0.5 +0.5 +0.7 +0.7 +0.9 +8'0- +-L1 +11 +0.9 +20 +0.6 +0.3 +心1 +2 +6'0- +0 +0.5 +0.3 +01 +0.5 +COSPhase graph of 10o oscillators forK=0.100 dt=0.001simulation time=10 +(a) Phase graph of 100 oscillators +(b) Order Paraneter of acillators +1.0 +10 +sin +P +1.心 +0.0 +2.0 +4.0 +B.0 +08 +10.0 +0.0 +2.0 +4.0 +08 +10.0 +time (r) +time (r) +(o)orderParameterof oscillatorsatT=0 +(d) Order Parameter of oncillatore at T =io +1.1 +1.1 +40 +40 +90 +90 +0 +UTS +0.1 +++ +sin +0.1 +0.1 +0.1 +0.8 +0.3 +0.5 +0.5 +0.7 +0.7 +8'0- +-L1 +11 +6'0 +0.3 +心1 +0.8 +0 +0.5 +0.3 +01 +0.5 +F'T +COSPhasegraph of 100oscillatorsforK=0.509dt=0.100simulation time=30 +(a) Phase graph of 100 oscillators +[b) Order Paraneter of acillators +1.0 +20 +P 0.5 +一 +1.心 +0.0 +6.0 +12.0 +18.0 +24.0 +30.0 +0.0 +6.0 +12.0 +18.0 +24. +30.0 +time (-) +time (-) +(o)Order Parameter of oscillators atT=0 +(d) Order Parameter of oBcillatorB at T =90 +1.1 +1.1 +B0 +2'0 +90 +90 +sin +UTS +0.1 +++++ +0.1 +-0.1 +0.1 +0.8 +0.3 +0.5 +-0.5 +0.7 +0.7 +0.9 +-0.8 +-L1 +-L1 +0.E +0.3 +心1 +0.9 +0.7 +0.5 +0.3 +01 +0.5 +COSPhasegraphof100oscillatorsforK=0.400dt=0.100simulationtime=30 +(a) Phase graph of 1o0 oscillators +(b) Order Paraneter of oacillators +1.0 +二 +0.5 +> +1.心 +0.0 +6.0 +12.0 +18.0 +24.0 +30.0 +0.0 +6.0 +12.0 +18.0 +24. +30.0 +time (-) +time (-) +(o)orderParameterof oscillators atT=0 +(d) Order Parameter of oBcillatore at T=go +1.1 +1.1 +10 +2'0 +90 +9'0 +sin +sin +0.1 ++++ +. +0.1 +-0.1 +0.1 +0.8 +0.3 +0.5 +-0.5 +0.7 +0.7 +0.9 +-0.8 +-L1 +11 +0 +0.E +0.3 +心1 +0.9 +0.7 +0.5 +0.3 +01 +0.5 +COS9 +Fig. 7. 100 non-identical Josephson junctions operating with mean frequency +of 5 GHz having mean Ic = 10 µA, mean internal resistance ρj = 4.2 kΩ +connected in series array to external load with parameters L=1 nH, C = 1 +µF and R = 2 Ω treated with bias current Ib = 12 µA synchronizes within +a narrow band of distribution. The final phase space is not a dot! +Fig. 8. 100 identical Josephson junctions operating at mean frequency of 5 +GHz having mean Ic = 10 µA, internal resistance ρ = 4.2 kΩ connected in +series array to external load with parameters L=1 nH, C = 1 µF and R = 2 +Ω treated with bias current Ib = 12 µA synchronizes within a narrow band +of distribution. The final phase space is not a dot! +too much. Another point to note is that the oscillators in the +non-identical case tend to syncronize faster and better than the +other case, possibly due to the noisy environment. +It has already been discussed that Kuramoto model stands +on the assumption that a large number of oscillators have +been considered. In our experimental regime, one may need +to use smaller number of oscillators say 5 or 10 oscillators as +shown in Fig.9 as asynchronized. The order parameter R is +Fig. 9. 5 identical oscillators having Ic = 10 µA and ρ = 4.2 kΩ operating +with 5 GHz frequency. +also shown to be oscillating at a lower value. The observation +was made for ˜τ = 25. The circuit parameters were kept same +as those for 100 oscillators. Evidently oscillators were not +syhronized. The case for the 5 non-identical oscillators is same +as 9. Now, to tune the circuit, let us select Ic as 10 µA and ρj +Fig. 10. +5 non identical Josephson junctions are partially syncronized +changing Ib to 10.8785 µA. += 4.2 kΩ as before as we wish to experiment with the same +junctions while we change Ib - the bias current. In the Fig.10, +the synchronization is observed where one oscillator is out of +sync while the rest 4 oscillators come closer to lie in a band +very fast ˜τ ≈ 1. + +Phase graph of 100 oseillators for K=12205.95dt=0.001 simulation time=25 +(a) Phase graph of 100 oscillators +[b) Order Paraneter of acillators +1.0 +B'0 +0.5 +一 +-1.心 +0.1 +0.0 +6.0 +10.0 +15.0 +20.心 +25.0 +0.0 +5.0 +10.0 +15.0 +20.心 +25.0 +time (r) +time (r) +(o) Order Parameter of oscillators at T=0 +(d) Order Parameter of oBcillatorB at T=z5 +1.1 +1.1 ++ + +2'0 +2'0 +90 +90 +us +0 +sin +0.1 +0.1 +0.1 +0.1 +0.8 +0.3 +0.5 +0.5 +0.7 +0.7 +0.9 +-0.9 +-L1 +11 +0.9 +20 +0.E +0.3 +心1 +0.9 +0.7 +0.5 +0.3 +心1 +cosu +cosyPhase graph of 100 oseillators for K=12205.95dt=0.001 simulation time=25 +(a) Phase graph of 1oo oscillators +(b) Order Paraneter of oacillators +1.0 +10 +TTTTT +0.8 +sin +一 +0.4 +0.2 +-1.心 +0.0 +0.0 +6.0 +10.0 +15.0 +20.心 +25.0 +0.0 +5.0 +10.0 +15.0 +20.心 +25.0 +time (r) +time (r) +(o)orderParameterof oscillators atT=0 +(d) Order Parameter of oBcillatorB at T=z5 +1.1 +1.1 +2'0 +2'0 +90 +90 +sin +us +0 +0.1 +0.1 +-0.1 +0.1 +0.8 +0.3 +0.5 +0.5 +0.7 +0.7 +0.9 +-0.9 +-L1 +11 +0.9 +20 +0.E +0.3 +心1 +0.9 +0.7 +0.5 +0.3 +心1 +cosu +cosyPhasegraph of 5oseillators forK=4307.83 dt=0.001 simulationtime=25 +(a) Phase graph of 5 oscillators +(b) Order Paraneter of acillators +1.0 +0.9 +0.7 +sin +9'0 +1.心 +0.1 +0.0 +6.0 +10.0 +15.0 +20.0 +25.0 +0.0 +5.0 +10.0 +15.0 +20.0 +25.0 +time (r) +time (r) +(o)OrderParameter of oscillators atT=0 +(d) Order Parameter of oBcillator at T =25 +1.1 +1.1 +2'0 +2'0 +90 +90 +us +0 +0.1 +0.1 +-0.1 +0.1 +0.8 +0.3 +0.5 +-0.5 +0.7 +0.7 +0.9 +-0.9 +-L1 +11 +0.E +0.3 +TO- +0.9 +0.7 +0.5 +0.3 +心1 +0.5 +cosyPhase graph of 5 oscillators for K=30333.88dt=0.00lsimulation time=15 +(a) Phase graph of 5oscillators +(b) Order Paraneter of oacillators +1.0 +80 +P 0.5 +0.1 - +-1.心 +0.0 +12.心 +15.0 +0.0 +6.0 +12.心 +15.0 +time (r) +time (r) +(o)orderParameterof oscillators atT=0 +(d) Order Parameter of oBcillatore at T =15 +1.1 +1.1 +2'0 +++ +20 +90 +90 +uIs +0.1 +0.1 +0.1 +-0.1 +0.8 +0.3 +0.5 +0.5 +0.7 +0.7 +-0.9 +-L1 +1L1 +0.E +0.3 +心1 +0.7 +0.5 +E'O +心1 +cosW +cosu10 +Fig. 11. 5 identical Josephson junctions are partially syncronized changing +Ib to 10.877 µA. +V. CONCLUSION +The exercises demonstrated in Figs.10 and 11 show the +possibility of synchronization for few oscillators following +Kuramoto model. However, order parameter show in-course +instability which later settles down. +This study helps to understand applicability of junctions in +series array and steps to control the level of synchronization. +The process is easier and synchronization is performed well +for larger number of junctions while partial synchronization is +also possible following the Kuramoto model. However, this +study does not state any conclusive equation for threshold +coupling for Josephson junction as it discussed in case of +general oscillators. This aspect will be discussed in future. +APPENDIX A +From (40), +tan +�φj +2 + π +4 +� += +� +αj + 1 +αj − 1 tan ψj +2 , +or, +� +tan +�φj +2 + π +4 +� ++ 1 +� � +tan +�φj +2 + π +4 +� +− 1 +� += +�� +αj + 1 +αj − 1 tan ψj +2 + 1 +� �� +αj + 1 +αj − 1 tan ψj +2 − 1 +� +, +or, +� +1 + tan φj +2 +1 − tan φj +2 ++ 1 +� � +1 + tan φj +2 +1 − tan φj +2 +− 1 +� += αj + 1 +αj − 1 tan2 ψj +2 − 1, +or, +� +cos φj +2 + sin φj +2 +cos φj +2 − sin φj +2 ++ 1 +� � +cos φj +2 + sin φj +2 +cos φj +2 − sin φj +2 +− 1 +� += αj + 1 +αj − 1 tan2 ψj +2 − 1, +or, +2 cos φj +2 × 2 sin φj +2 +� +cos φj +2 − sin φj +2 +�2 = +2 sin φj +� +cos φj +2 − sin φj +2 +�2 += αj + 1 +αj − 1 tan2 ψj +2 − 1, +or, +2 sin φj +1 − sin φj += αj tan2 ψj +2 + tan2 ψj +2 − αj + 1 +αj − 1 +, +or, +1 − sin φj +2 sin φj += +1 − αj cos ψj +(αj − 1) cos2 ψj +2 +, +or, +1 +2 sin φj += 1 +2 + +1 − αj cos ψj +(αj − 1) cos2 ψj +2 +, +or, +sin φj = 1 − αj cos ψj +αj − cos ψj +, +or, +sin φj ≡ sin φ(ψj) = αj − +� +α2 +j − 1 +� +αj − cos ψj +. + +Phasegraph of 5oscillators forK=30453.72dt=0.00l simulationtime=15 +(a) Phase graph of 5 oscillators +(b) Order Paraneter of oacillators +1.0 +E0 +sin + 0.5 +1.心 +0.0 +12.心 +15.0 +0.0 +6.0 +12.心 +15.0 +time (T) +time (r) +(o)orderParameterof oscillators at T=Q +(d) Order Parameter of oBcillatore at T =15 +1.1 +1.1 +2'0 +20 +90 +90 +sin +uIs +0.1 +0.1 +-0.1 +0.1 +0.8 +0.3 +-0.8 +0.5 +0.7 +0.7 +0.9 +-L1 +-11 +0.3 +心1 +0.7 +0.5 +心1 +cos +cosu11 +APPENDIX B +�dψj +d˜τ +� += 1 − +ϵjδj +� +α2 +j − 1 +2πN +� 2π +0 +� +αj − cos ψj +α2 +j − 1 +× +N +� +k=1 +∞ +� +n=0 +nIkρkBkn sin {n (˜τ + ck) + βkn} +� +d˜τ. +or, +�dψj +d˜τ +� += 1 − +ϵjδj +2πN +� +α2 +j − 1 +� 2π +0 +(αj − cos (˜τ + cj)) +× +N +� +k=1 +∞ +� +n=0 +nIkρkBkn sin {n (˜τ + ck) + βkn} d˜τ. +or, +�dψj +d˜τ +� += 1 ++ +ϵjδj +N +� +α2 +j − 1 +� +γ2 +j +� +α2 +j − 1 +�2 + +� +ω2 +0j − +� +α2 +j − 1 +�2�2 +× +N +� +k=1 +Ikρk +� +1 − α2 +k + αk +� +α2 +k − 1 +� +sin (cj − ck − ζj) . +or, +�dψj +d˜τ +� += 1 + Kj +N +N +� +k=1 +Ikρk +� +1 − α2 +k + αk +� +α2 +k − 1 +� +× sin (cj − ck − ζj) , +or, +�dψj +d˜τ +� += 1 + Kj +N +N +� +k=1 +Ak sin (cj − ck − ζj) . +ACKNOWLEDGMENT +The author would like to Sudhir R Jain, for his ideas, inspi- +ration and continuous support to conceptualize, understand and +formulate the problem. 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Available: http://www.dislin.de/ + diff --git a/CtE2T4oBgHgl3EQfSAdG/content/tmp_files/load_file.txt b/CtE2T4oBgHgl3EQfSAdG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab2674547c254d26b6dbc8ef5f22f10fd1a42e48 --- /dev/null +++ b/CtE2T4oBgHgl3EQfSAdG/content/tmp_files/load_file.txt @@ -0,0 +1,769 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf,len=768 +page_content='1 Synchronization of Josephson junctions in series array Abhijit Bhattacharyya Abstract—Multi-qubit quantum processors coupled to net- working provides the state-of-the-art quantum computing plat- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, each qubit has unique eigenfrequency even though fabricated in the same process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' To continue quantum gate operations besides the detection and correction of errors it is required that the qubits must be synchronized in the same frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This study uses Kuramoto model which is a link between statistical mean-field technique and non-linear dynamics to synchronize the qubits applying small noise in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This noise could be any externally applied noise function or just noise from the difference of frequencies of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The Kuramoto model tunes the coupled oscillators adjusting the coupling strength between the oscillators to evolve from the state of incoherence to the synchronized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Index Terms—Josephson junction, Kuramoto Model, synchro- nization, oscillators I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' INTRODUCTION J Osephson junction controls the flow of magnetic flux quanta through frequency and voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Modern instruments require measurement of voltage with a reproducible capability exceeding the uncertainty of realization of the SI volt (cur- rently 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='4 parts on 106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Before 1972, SI volt was represented by using carefully stabilised Weston cell banks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Drift and transportability problems with these electrochemical artifact standards limited the uniformity of voltage standards to about 1 part in 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' These uniformity was drastically improved by the usage of Josephson junction [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Josephson equation for supercurrent through a supercon- ducting tunnel junction, called as DC Josephson Effect, is defined as [2]–[4] I = Ic sin �4πe h � V dt � , (1) where Ic is critical current, h is Planck’s constant and e is electron charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' When a dc voltage is applied in equation (1), the phase will vary linearly with time and current will be sinusoidal with amplitude Ic and frequency fJ = 2eV/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The magnetic flux threading a superconducting loop or hole is quantized [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The superconducting magnetic flux quantum Φ0 = h/(2e) is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0678×10−15 Wb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The inverse of flux quantum 1/Φ0 is called Josephson constant KJ defined as 2e/h has a value of 483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='597 GHz/mV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' During each oscillation, a single quantum of magnetic flux h/(2e) passes through the junction which is very difficult to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, if an alternating current with frequency f is applied across the junction, there Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400 094, India vega@barc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' abhihere@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='com is a range of bias current for which flow of flux quanta will phaselock to the applied frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Under this phase locked condition, the average voltage across the junction is precisely (h/2e)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This effect is known as ac Josephson effect observed as a constant voltage step at V =(h/2e)f in the I −V characteristic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This means a Josephson junction can act as a “Voltage to frequency converter”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' It is also possible for the junction to phaselock with the harmonics of fJ resulting in a series of steps at voltages V =nf(h/2e), where n is an integer denoting step number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This accuracy was limited to the condition that a Josephson voltage higher than 10mV was never used [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore, if one obtain Josephson voltage over 100 mV , the accuracy could be remarkably improved besides the ability to vary the Jsephson voltage with the frequency and step number could be utilized as potentiometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Series array of Josephson junction [6] has been effectively used in development of a potentiometer system to produce (1- 10)V [1], [6] with uncertainty about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 × 10−9 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Larger series arrays were initially considered as impractical due to junction nonuniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The nonuniformity demanded each junction to be biased separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In 1977, Levinsen et al [7] stated the important of the parameter βc=4πeIcR2C/h in determining the characteristics of RF induced Josephson steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This βc is measure of the damping of Josephson oscillations by the junction shunting resistance R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The Josephson junction is also a natural choice for sub- millimeter local oscillator [8], [9] as one may capitalize the voltage controlled oscillator property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, the disadvan- tage, in this case lies in very low power output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The Josephson constant clearly indicates that with dc voltage bias at 1 mV at 483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='6 GHz, the junction may accept 100 µA current keeping under the limitation of Ic which limits the maximum output RF power at about 100 nW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This requirement indicates series array of junctions with a common current bias demands keeping all the junctions in phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, the issue with series array of junctions operated with common current bias arises with nonuniformity of each junction due to fabrication processes [1], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' When junctions are connected in series, the system behaves as a coupled os- cillator and understanding the periodic solutions is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Two special types of periodic solutions exist [10], namely, in- phase state and splay state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' An in-phase state with period T is a state where all the oscillators always possess the same phase at all times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' θi(t) = θj(t), and θi(t + T) = θi(t) + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The splay-phase or anti-phase or rotating wave state with period T is a solution where the oscillators can be labeled so that θi(t) = Θ(t + jT/N) for all j for some function arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='03787v1 [quant-ph] 10 Jan 2023 2 Θ(t + T) = Θ(t) + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Thus, this state indicates that all the oscillators have the same waveform Θ(t) except for a shift in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' As per [10], one may imagine that each oscillator “fires” when it reaches a certain angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' For an in-phase solution, all the oscillators fire simultaneously at every instant T, while splay-phase state has a single oscillator firing every T/N instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore, for splay-phase state, oscillators nearly coincide or coincide when ˙θ is small where as for large values, oscillators are not coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The definition of splay- phase does not imply that the phases of the oscillators are equi-spaced around he circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The oscillators bunch up for smaller ˙θ while spread out for large ˙θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore, splay-state shows non-uniformity in the distribution of oscillators as they are coherent for smaller ˙θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' It has been shown that [10], [11], the non-uniformity can be removed by determining a set of “natural” angles ϕj, so that the splay-phase solution satisfies ϕj(t) = 2πj/N + 2πt/T + const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The “natural” angle based dynamical system gets locked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This provides an idea of phase- locking N oscillators, like N Josephson junctions, having eigenfrequencies with smaller spread which may get locked to some resonating frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Kuramoto model provides an exactly solvable mean-field model of coupled nonlinear oscillators connecting a large of them having distributed natural frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This model links mean-field techniques and nonlinear dynamics together and also provides precise technique to tune the synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Section II discusses the theory of the Kuramoto model, Section III discusses on the reduction of the equations for the Josephson junctions connected in series to the Kuramoto Model framework and section IV discusses on the numerical analysis of the results for the generalised Kuramoto Model theory and Kuramoto model for Josephson junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' KURAMOTO MODEL Let us consider a system of N globally coupled differential equations with the stable limits cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Yoshiki Kuramoto de- veloped a mathematical model for coupled oscillators (n ⩾ 2) to synchronize which is known as “Kuramoto model” [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In this model, each jth oscillator is represented by a phase variable θj(t), possessing its own natural frequency ωj ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The dynamics of the system of coupled N oscillators becomes ˙θj(t) = ωj + N � i=1,j̸=i Kji sin (θj(t) − θi(t)) , j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' , N} , (2) where Kji is coupling coefficient of the jth oscillator with all other oscillators in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Kuramoto assumed mean field coupling among phase oscillators such that Kji ≈ K/N ⩾ 0 where K is mean coupling strength which changes (2) as ˙θj(t) = ωj + K N N � i=1,j̸=i sin (θj(t) − θi(t)) , j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' , N} , (3) where, K ⩾ 0 is the coupling strength among the oscillators whose frequencies are distributed with a probability density g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' One may find a suitable rotating frame like θj → θj − Ωt transforming the system so that natural frequencies of the oscillators may have zero mean, where Ω is the first moment of the distribution function of natural frequencies g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therfore, one may consider the normal form calculation for the system such that one may define the system of equations as ˙θj = fj(θj) + K N N � i=1,i̸=j g (θi, θj) , θj ∈ Rd, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' , N, (4) where, function fj(θj) are eigenfrequencies defining the nat- ural dynamics in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Here coupling parameter K has been added with coupling strength K/N, g is the phase response curve defining the interaction of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the following section, we are not discussing with the stability of the dynamical system, bifurcation etc while one may consult other references like [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the original paper [12], Kuramoto considered the proba- bility density g(ω) to be uni-modal and symmetric centered at mean frequency ω so that, without loss of generality, one can assume that the mean frequency ω = 0 after a shift leading to g(ω) = g(−ω) for the even and symmetric distribution g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' To diagnose the feasibility of synchronization, Kuramoto introduced the order parameter R(t) projecting the oscillation on unit circle where R(t) : 0 ⩽ R(t) ⩽ 1 is a measure of the coherence of oscillators as R(t)eȷψ(t) = 1 N N � i=1 eȷθi(t), (5) where R(t) = 0 for asynchronised oscillators, and R(t) > 0 for synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The quantity ψ(t) refers to average phase of all the oscillators at an instant t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Physically, this order parameter R(t) is the centroid of a set of N points eȷθi distributed in the unit circle in the complex plane at the instant t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' If the phases are uniformly spread in the range [−π, π], then R → 0 indicates that the oscillators are not synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' All the oscillators become synchronized with the same average phase ψ(t) for R(t) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' If the dynamics show stability of R(t) at 1, then the oscillators are synchronized and phaselocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (3) may be re-written by multiplying Ke−ȷθj on both sides of (5) and equating the imaginary parts of the both sides to reduce (3) to ˙θj(t) = ωj + KR(t) sin (ψ(t) − θj(t)) = vj(θ, ω, t) (say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (6) Here, vj(θ, ω, t) is the angular velocity of a given oscillator with phase θ and natural frequency ω at the instant t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The equation (6) reveals that the interaction is set through R(t) and ψ(t) while the phases θj seem to evolve independently from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Also the effective coupling is proportional to the order parameter R(t) creating a feedback relation between coupling and synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the limit K → 0, (6) reduces to θj(t) ≈ ωjt + θ(0), (7) where, θj(0) denotes initial phase of the jth oscillator and (7) suggests that each oscillator oscillates with own natural frequencies in the absence of coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 3 In the limit of infinite number of oscillators having a distribution of frequency, phase over time, Kuramoto de- scribed the system by the probability density ρ (θ, ω, t) so that ρ (θ, ω, t) dθ gives the fraction of oscillators with phase between θ(t) and θ(t) + dθ(t) at the instant t for a given natural frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Since ρ is non-negative and 2π-periodic in θ satisfying the normalization condition � π −π ρ (θ, ω, t) dθ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (8) The probability density function g must also obey the equation of continuity using the angular velocity v(θ, ω, t) as ∂ρ(θ, ω, t) ∂t + ∂ ∂θ {ρ(θ, ω, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='v} = 0, ∂ρ(θ, ω, t) ∂t + ∂ ∂θ [ρ(θ, ω, t) {ω + KR(t) sin (ψ(t) − θ(t))}] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (9) In the limit R(t) → 0, the dynamics provides stationary solution for ρ(θ, ω, t) = 1/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the continuum limit, (5) gets re-defined by the order parameter R(t) and the average phase ψ(t) incorporating previously described frequency distribution as R(t)eȷψ(t) = � π −π � ∞ −∞ eȷθρ (θ, ω, t) g(ω)dωdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (10) In the strong coupling limit where K → ∞ indicate K ≫ Kc where Kc is critical coupling strength and (6) reduces to system having phases reduced to the average phase as θ(t) = ωt + θ(0) = ψ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' From (6), if oscillators get into phaselocked condition, vi(t) → 0 which provides ωj = KR(t) sin (θj(t) − ψ(t)) , −π 2 ⩽ (θj(t) − ψ(t)) ⩽ π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (11) From (9), partially synchronized state leading to a locked system can be described as ∂ ∂t(ρ(θ, ω, t)) = 0 which also means ∂ ∂θ (ρ(θ, ω, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='v(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (11), in this partial synchronized state for vj(t) → 0 and ∂ ∂t (ρ(θ, ω, t)) = 0, reduces to ω KR(t) → sin(θj(t) − ψ(t)), which means ρ(θ, ω, t) = δ � θj(t) − ψ(t) − sin−1 � ω KR(t) �� H(cos θ), (12) such that |ω| ⩽ KR(t) and H(x) = 1, x > 0, 0, elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='. (13) Now, for the other condition ∂ ∂θ (ρ(θ, ω, t)v(t)) = 0 using (6), ρ(θ, ω, t)v(t) = C(say) = constant, or, ρ(θ, ω, t) = C |ω + KR(t) sin(θj(t) − ψ(t))|, |ω| � KR(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (14) The constant C can be determined from (8) such that (14) reduces to ρ(θ, ω, t) = � ω2 − K2R2(t) 2π|ω − KR(t) sin(θj(t) − ψ(t))|, |ω| � KR(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (15) Therefore, the constraint on the probablity density of the oscillators may be ρ(θ, ω, t) = δ � θj(t) − ψ(t) − sin−1 � ω KR(t) �� H(cos θ), for |ω| ⩽ KR(t) (16) and ρ(θ, ω, t) = � ω2 − K2R2(t) 2π|ω − KR(t) sin(θj(t) − ψ(t))|, elsewhere .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (17) Here δ is the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (16) and (17) indicate that partial synchronized states are divided into two groups depending on the natural frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Oscillators having con- straint |ω| ⩽ KR(t) operate in mean-field resulting in locking in a common average phase ψ(t) = Ωt where Ω is the average frequency of the ensemble of the oscillators in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' On the other side, the second group of oscillators having constraint |ω| > KR(t) rotate incoherently which are called as drifting oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Inserting (16) and (17) in (10) we get R(t) = � π −π � ∞ −∞ eȷ(φ(t)−ψ(t)) δ � θ(t) − ψ(t) − sin−1 � ω KR(t) �� g(ω)dθdω + � π −π � |ω|⩽KR(t) � ω2 − K2R2(t)g(ω)dθdω 2π|ω − KR(t) sin(θ(t) − ψ(t))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (18) Since g(ω) is even and symmetric, g(ω) = g(−ω) and ρ(θ + π, −ω) = ρ(θ, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The even function condition makes the second term of (18) vanish which physically means all the incoherent oscillator solutions vanish resulting in order parameter R(t) only for coherent synchronized oscillators that reform as R(t) = � |ω|⩽KR(t) cos � sin−1 � ω KR(t) �� g(ω)dωdθ, = � π 2 − π 2 cos θg (KR(t) sin θ) KR(t) cos θdθ, = KR(t) � π 2 − π 2 cos2 θg (KR(t) sin θ) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (19) Here, (19) shows a trivial solution for which order parameter R(t) = 0 which actually shows incoherence as discussed earlier for ρ (θ, ω, t) = 1/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, (19) also suggests 1 = K � π 2 − π 2 cos2 θ g (KR(t) sin θ) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 4 Setting R(t) = 0, considering K = Kc - the critical coupling strength we get, Kc = 2 πg(0), (20) that triggers the synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In general, expanding the right hand side of (19) in terms of powers of KR(t) and considering g′′(0) < 0 the order parameter can be written as R(t) ∼ � −8 (K − Kc) K3c g′′(0) , (21) which shows that near the transition point, the order parameter [12], [14] yields the form R(t) ∼ (K − Kc)β with β = 1/2 like second order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The Kuramoto model can be generalized for a complex network including the connectivity parameter in the coupling term as ˙θj = ωj + N � i=1 KjiAji sin(θj − θi), (22) where, Kji is the coupling strength between nodes j and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Aji is the element of the adjacency matrix A (Aji = 1 if there is a connection between j and i else Aji = 0 otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Any real system may have noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Let us discuss on the effect of the noise for the Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The noise may arise from the variation of frequency of incoherent oscillators as they may not be identical or there may either be an external white noise or white noise inherent to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore the model (3) could be reframed as ˙θj = σωj + K N N � i=1 sin (θj(t) − θi(t)) + √ Γηj(t), : j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' , N} , (23) where, both ωj and ηj(t) are Gaussian distributions having zero mean and unit variance while σ and Γ behave as am- plitudes of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Here last term refers to white noise in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore (23) physically indicates locally coupled oscillators having natural frequencies of oscillators derived from Gaussian distribution in presence of stochastic effects like white noise due to fluctuations in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The reason for stochastic behavior may vary for different systems while any natural process exhibit stochastic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The situation of lim σ → 0 refers to the Kuramoto model having identical oscillators in presence of gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The system behaves as if the system is in contact with a heat source and the dynamics is evolving in the statistical equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The situation for lim Γ → 0 indicates that the Kuramoto model has been constructed with oscillators having distributed natural frequencies in absence of gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The system behaves as nonlinear dynamical system relaxing to the non-equilibrium stationary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Beside this brief summary, one may also consult articles like [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Next, let us transform the Josephson equations for series array of junctions to Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' KURAMOTO MODEL FOR JOSEPHSON JUNCTION SERIES The Josephson junction array can be constructed using Kirchhoff’s laws considering each Josephson junction as a parallel circuit of two elements: an ideal resistance ρ carrying ideal current Iρ and a junction carrying critical current Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Actual Josephson junction also contains a capacitor in parallel to the nonlinear inductor which we have neglected due to its very small value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Let each of N junctions be connected serially and then coupled to external load having inductance L, resistance R and capacitance C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' C R L Ib ρ1 I1 ρ2 I2 ρN IN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Schematic circuit of qubits connected in series parallel to a Load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Let us consider Josephson junction in the series array, say jth junction and following Josephson equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' we express the circuit shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 as V (t) ρj + Ij sin φj + dQ dt = Ib, which can be written as, dφj dt = 2πρj Φ0 � Ib − Ij sin φj − dQ dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (24) Further, L ¨Q + R ˙Q + Q C = N � k=1 Vk, or, L ¨Q + � R + N � k=1 ρk � dQ dt + Q C = − N � k=1 Ikρk sin φk, (25) where Q is the charge on load capacitor, Φ0 = h/(2e) is magnetic flux quantum, h is Planck’s constant, e being the charge of an electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Here, junction resistance ρk for any junction k is very small compared to the load variable Q/C such that one may consider, Q/C − � k ρkIb ≈ Q/C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' To understand the effect of external parameters like L, C and R on each junction, one may consider a scaled version of those parameters by choosing l = L N , r = R N , c = NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (26) 5 Here, it is to be noted that Φ0 Ij = 1 2πf 2 j Cj where fj is the frequency and Cj is the capacitance of jth junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Let us now consider transformation of time t and charge Q so that (24) and (25) become dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' From (24) Φ0 2πρjIj dφj dt + sin φj + 1 Ij dQ dt = Ib Ij = αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Let us consider the following transformation relation to transform time t to dimensionless form τ as Φ0 2πρjIj d dt ≡ d dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (27) such that we may write dφj dτ + sin φj + 2πρj Φ0 dQ dτ = αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (28) Substituting dimensionless time τ and scaled parameters as in (26) in (25) we get, L N �2πρjIj Φ0 �2 d2Q dτ 2 + (R + �N k=1 ρk) N �2πρjIj Φ0 � dQ dτ + Q NC = 1 N N � k=1 −Ikρk sin φk, or, l �2πρjIj Φ0 �2 d2Q dτ 2 + � r + �N k=1 ρk N � �2πρjIj Φ0 � dQ dτ + Q c = − 1 N N � k=1 Ikρk sin φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (29) Let us also consider the following transformation to transform charge Q to dimensionless form q as 2πρjIj Φ0 Q ≡ qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (30) Therefore, through (29), (25) transforms as d2qj dτ 2 + γj dqj dτ + ω2 0jqj = −δj N N � k=1 Ikρk sin φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (31) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (30) can be used to rewrite (28) as dφj dτ + sin φj + ϵj dqj dτ = αj, (32) where coefficients may be written as γj = � Φ0 2πρjIj � �1 l � � r + �N k=1 ρk N � , (33) ω2 0j = � Φ0 2πρjIj �2 1 lc, (34) δj = � Φ0 2πρjIj � 1 l , (35) and ϵj = 1 Ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (36) Let us write the equation (32) in the uncoupled form for ϵj → 0 or ˙Q → 0 such that we get, dφj dτ = αj − sin φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (37) As discussed in the Section I, the splay-state shows that transforming the dynamical system equations make a rigid sys- tem with coherent frequencies in weak coupling or uncoupled limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Hence, let us transform φj in (24) into ‘natural’ angle ψj such that dψj dt = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (37) can be transformed in terms of the ‘natural’ angle ψj such that dψj/dt − c, where c is constant to be determined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' transformation as φj → ψj as uniform rotation with first derivative remaining constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The constant ‘c’ may be determined with the fact that the time to complete one cycle by these two sets of coordinates must be same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Thus, T = � T 0 dτ = � 2π 0 dψj c = � 2π 0 dψj ωj = � 2π 0 dφj (αj − sin φj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or, 2π ωj = 2π �� α2 j − 1 �, for αj ⩾ 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Ib ⩾ Ij, which shows ωj = � α2 j − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (38) Then the transformation to the natural angles satisfies dψj = � α2 j − 1 αj − sin φj dφj, (39) which on integration yields ψj = 2 tan−1 �� αj − 1 αj + 1 tan �φj 2 + π 4 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (40) At this point, one may construct a transformation function ψ(φj) to translate any angle φj to its natural angle ψj while another transformation function φ(ψj) may be used to invert as ψ (φ) = 2 tan−1 �� α − 1 α + 1 tan �φ 2 + π 4 �� , (41) φ (ψ) = 2 tan−1 �� α + 1 α − 1 tan �ψ 2 �� − π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (42) Here, we use the shorthand: ψj ≡ ψ(φj) and φj ≡ φ(ψj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' From (40), sin φj = 1 − αj cos ψj αj − cos ψj = αj − α2 j − 1 αj − cos ψj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (43) Detailed derivation of (43) from (40) is shown in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore, one may rewrite (32) using (39) and (43) as dψj dτ = dψj dφj dφj dτ = � α2 j − 1 αj − sin φj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' � αj − sin φj − ϵj dqj dτ � , = � α2 j − 1 − ϵj � α2 j − 1 αj − sin φj dqj dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (44) 6 Let us rescale non-dimensional quantity τ as ˜τ such that τ = ˜τ � α2 j − 1 =⇒ d d˜τ ≡ 1 � α2 j − 1 d dτ =⇒ d2 dτ 2 ≡ � α2 j − 1 � d2 d˜τ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (45) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (44), using (45), transforms as dψj d˜τ = 1 − ϵj � α2 j − 1 αj − sin φj dqj d˜τ , (46) The weak-coupling solution of (44) may be written as ψj(τ) ≡ �� α2 j − 1 � τ + cj = ˜τ + ψj0, (47) where cj is the integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Initially, at τ = 0, one may assume initial phase as ψj0 such that cj=ψj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The reference [11] discusses about the importance of the weak coupling condition for the Josephson junction arrays and drift in ψj may be obtained by averaging (46) over one cycle as �dψj d˜τ � = 1 − 1 2π � 2π 0 ϵj � α2 j − 1 αj − sin φj �dqj d˜τ � d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (48) Similarly, one may rewrite non-dimensional charge equation (31) in terms of ˜τ as � α2 j − 1 � d2qj d˜τ 2 + γj � α2 j − 1dqj d˜τ + ω2 0jqj = −δj N N � k=1 Ikρk sin φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (49) It is usually convenient to write sin(φj)=sin(φ(ψj)) in terms of its Fourier series as sin φ(ψk) = ∞ � n=0 Akn cos (nψkn) = ∞ � n=0 Akn cos {n (˜τ + ck)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (50) Then (49) reduces to � α2 j − 1 � d2qj d˜τ 2 + γj � α2 j − 1dqj d˜τ + ω2 0jqj = −δj N N � k=1 ∞ � n=0 IkρkAkn cos {n (˜τ + ck)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (51) One may obtain the steady-state solution of (51) as qj(˜τ) = −δj N IkρkBkn cos {n (˜τ + ck) + βkn} ,(52) dqj(˜τ) d˜τ = δj N nIkρkBkn sin {n (˜τ + ck) + βkn} , (53) d2qj(˜τ) d˜τ 2 = δj N n2IkρkBkn cos {n (˜τ + ck) + βkn} ,(54) where B2 kn = A2 kn n2γ2 j � α2 j − 1 � + � n2 � α2 j − 1 � − ω2 0j �2 , (55) βkn = tan−1 � � nγj � α2 j − 1 n2 � α2 j − 1 � − ω2 0j � � = βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (56) Using the expression (43), one may derive Akn and obtain Ak0 = 1 π � π −π 1 − αk cos ψk αk − cos ψk dψk, (57) Akn = 1 π � π −π 1 − αk cos ψk αk − cos ψk cos �nπψk π � dψk (58) where n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Bkn denotes the amplitude of the linear damped oscillator while βkn denotes its phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Therefore, Bkn must be chosen to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Now, (48) may be re-written as �dψj d˜τ � = 1 − ϵjδj � α2 j − 1 2πN � 2π 0 � 1 αj − sin φj × N � k=1 ∞ � n=0 nIkρkBkn sin {n (˜τ + ck) + βkn} � d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (59) Using (43), sin φj = αj − α2 j − 1 αj − cos ψj , or, αj − sin φj = α2 j − 1 αj − cos ψj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (60) With this (59) may be modified using (60) to or, �dψj d˜τ � = 1 + Kj N N � k=1 Ak sin (cj − ck − ζj) , (61) where, Kj = ϵjδj � α2 j − 1 � γ2 j � α2 j − 1 �2 + � ω2 0j − � α2 j − 1 �2�2 , (62) AK = Ikρk � 1 − α2 k + αk � α2 k − 1 � , (63) ζj = tan−1 � � γj � α2 j − 1 α2 j − 1 − ω2 0j � � = β1j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (64) Reader may check detailed description of the derivation in the appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the final step, one may replace the ‘initial values’ of phases by their slowly evolving components like ⟨ψj(˜τ)⟩ and ⟨ψk(˜τ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Also one may get firstorder averaged equation by dropping the angular brackets so that (61) transforms to dψj d˜τ = 1 + Kj N N � k=1 Ak sin (ψj(˜τ) − ψk(˜τ) − δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (65) 7 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (65) resembles the Kuramoto model in a generalized form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' For the sake of mathematical formalities, it is important to note that except terms corresponding to n = 1 terms for other values of n becomes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' To arrive at (65), it was assumed that the fabrication process may not guarantee exactly same values of parameters for each junction and hence one may consider that each junction has different internal resistance and different critical current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The difference may be very small for junctions prepared in the same batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' If the fabrication process is done in very skilled sequence (65) may turn into special form for assuming ρ1 = ρ2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' = ρN = ρ (say) and I1 = I2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' = IN = Ic(say) so that each junction has nearly same frequency f (say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This case of identical junctions has been studied extensively in may literatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The transformation (27) for time leads to Φ0 2πρIc d dt ≡ d dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (66) while (30) entails 2πρIc Φ0 Q ≡ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (67) Consequently, (31) reduces to d2q dτ 2 + γ dq dτ + ω2 0q = − β N N � k=1 sin φk, (68) where γ = � Φ0 2πρIc � � 1 lρ � (r + ρ) , (69) ω2 0 = � Φ0 2πρIc �2 1 lc, (70) β = � Φ0 2πρIc � 1 l , (71) (72) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (55) and (56) become B2 n = A2 n n2γ2 (α2 − 1) + {n2 (α2 − 1) − ω2 0}2 , (73) βn = tan−1 � nγ √ α2 − 1 ω2 0 − n2 (α2 − 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (74) Repeating the earlier exercise, one may obtain the final phase equation (59) as �dψj d˜τ � = 1 − β 2πN √ α2 − 1 � 2π 0 (α − cos (τ + cj)) × N � k=1 ∞ � n=0 nBn sin {n (˜τ + ck) + βn} d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (75) In the case of identical junctions, computation shows that only B1 exists while others are evaluated to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Thus, (75) becomes �dψj d˜τ � = 1 − B1β 2πN √ α2 − 1 � 2π 0 (α − cos (τ + cj)) × N � k=1 ∞ � n=0 n sin {n (˜τ + ck) + βn} d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Integrating, we get dψj d˜τ = 1 + K N N � k=1 sin (ψj(˜τ) − ψk(˜τ) − β1) , (76) where K = πB1β 2π √ α2 − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (77) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' (76) exactly resembles as the Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the following section, let us try to understand general characteristics of the Kuramoto model in general and in the context of Josephson junction array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' ANALYSIS A C + + code has been developed alongwith DISLIN code to analyse the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' DISLIN [16] is a freely available graph plotting routine that plots during runtime and can be stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In this section, let us first investigate basic Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Kuramoto model in arbitrary unit for 100 oscillators with K=4 showing synchronization after a certain settling time within a band of frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Kuramoto model as discussed in (3) including the effect of coupling strength (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' If K is properly tuned, one may expect synchronization as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Here we consider that the oscillators are oscillating possess- ing a frequency distribution g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' One may control width of the distribution while keeping the zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' We consider Logistic and Lorentzian fuctions having width β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Oscillators tend get to be synchronized if K is equal to or more than some threshold value Kc as discussed in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' g(ω) = exp (−ω/β) β [1 + exp (−ω/β)]2 (78) The Logistic function is described as (78) which shows g(0)=1/(4β) where, β is the width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Likewise, one may define the Lorentzian function as (79) g(ω) = b (ω2 + b2), (79) Kuramoto model for 1oo oscillators having K=4 without any noise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 a sin 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0 2 4 6 8 10 time8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 100 oscillators with K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 having Logistic distribution of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 100 oscillators with K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 having Lorentzian distribution of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' so that one get g(0)=2/(πb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This g(0) estimates thresh- old value of the coupling strength as Kc=2/πg(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' One may compare Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 where the latter is operating with threshold coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The synchronization for the latter shows phase space of order parameter as a dot denoting synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='4 and 6 also show similar observation of synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This theoretical study clearly heps us to understand the sig- nificance of coupling strength and the treatment of frequency range of oscillators to start with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Next one may apply this understanding in the case of Josephson junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The situation is very much different hereas the definition of K is complex for both non-identical and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 100 oscillators oscillating with k=Kc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='509 with Logistic function of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 100 oscillators oscillating with k=Kc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='4 with Lorentzian function of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' identical junction arrays as evident from either (65) or (76) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Let us consider mean frequency may be around 5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 and 8 show simulated results of systems of 100 Josephson junctions in non-identical and identical configurations respectively operated for ˜τ = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The interesting part is that synchronization is not pulling the oscillators to a certain unique frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Rather, oscillators tend to cool down to a narrow band of frequencies resulting in an arc in phase space diagram which resembles as if oscillators have a certain ‘viscosity’ in the combined system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' For the non identical case, the spread of Ic is considered very small like 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1% while variation in ρj is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='05 % as fabrication is much better and junctions fabricated in the same substrate will not vary Phase graph of 10o oscillators forK=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='100 dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='001simulation time=10 (a) Phase graph of 100 oscillators (b) Order Paraneter of acillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 10 sin P 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 08 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 08 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (T) time (r) (o)orderParameterof oscillatorsatT=0 (d) Order Parameter of oncillatore at T =io 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 40 40 90 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 UTS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 sin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="9 8'0- L1 11 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="3 心1 2 6'0- 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 COSPhase graph of 10o oscillators forK=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='100 dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='001simulation time=10 (a) Phase graph of 100 oscillators (b) Order Paraneter of acillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 10 sin P 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 08 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 08 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (r) time (r) (o)orderParameterof oscillatorsatT=0 (d) Order Parameter of oncillatore at T =io 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 40 40 90 90 0 UTS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 ++ sin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 一 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (-) time (-) (o)Order Parameter of oscillators 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (-) time (-) (o)orderParameterof oscillators atT=0 (d) Order Parameter of oBcillatore at T=go 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 100 non-identical Josephson junctions operating with mean frequency of 5 GHz having mean Ic = 10 µA, mean internal resistance ρj = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2 kΩ connected in series array to external load with parameters L=1 nH, C = 1 µF and R = 2 Ω treated with bias current Ib = 12 µA synchronizes within a narrow band of distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The final phase space is not a dot!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 100 identical Josephson junctions operating at mean frequency of 5 GHz having mean Ic = 10 µA, internal resistance ρ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2 kΩ connected in series array to external load with parameters L=1 nH, C = 1 µF and R = 2 Ω treated with bias current Ib = 12 µA synchronizes within a narrow band of distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The final phase space is not a dot!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Another point to note is that the oscillators in the non-identical case tend to syncronize faster and better than the other case, possibly due to the noisy environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' It has already been discussed that Kuramoto model stands on the assumption that a large number of oscillators have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In our experimental regime, one may need to use smaller number of oscillators say 5 or 10 oscillators as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 as asynchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The order parameter R is Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 5 identical oscillators having Ic = 10 µA and ρ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2 kΩ operating with 5 GHz frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' also shown to be oscillating at a lower value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The observation was made for ˜τ = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The circuit parameters were kept same as those for 100 oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Evidently oscillators were not syhronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The case for the 5 non-identical oscillators is same as 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Now, to tune the circuit, let us select Ic as 10 µA and ρj Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 5 non identical Josephson junctions are partially syncronized changing Ib to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='8785 µA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='2 kΩ as before as we wish to experiment with the same junctions while we change Ib - the bias current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' In the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='10, the synchronization is observed where one oscillator is out of sync while the rest 4 oscillators come closer to lie in a band very fast ˜τ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Phase graph of 100 oseillators for K=12205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='95dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='001 simulation time=25 (a) Phase graph of 100 oscillators [b) Order Paraneter of acillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="0 B'0 0." metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (r) time (r) (o) Order Parameter of oscillators at T=0 (d) Order Parameter of oBcillatorB at T=z5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="1 + + 2'0 2'0 90 90 us 0 sin 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (r) time (r) (o)OrderParameter of oscillators atT=0 (d) Order Parameter of oBcillator at T =25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="1 2'0 2'0 90 90 us 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 L1 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 TO- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 心1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 cosyPhase graph of 5 oscillators for K=30333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='88dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='00lsimulation time=15 (a) Phase graph of 5oscillators (b) Order Paraneter of oacillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 80 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (r) time (r) (o)orderParameterof oscillators atT=0 (d) Order Parameter of oBcillatore at T =15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="1 2'0 ++ 20 90 90 uIs 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 L1 1L1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 心1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="5 E'O 心1 cosW cosu10 Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 5 identical Josephson junctions are partially syncronized changing Ib to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='877 µA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' CONCLUSION The exercises demonstrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='10 and 11 show the possibility of synchronization for few oscillators following Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, order parameter show in-course instability which later settles down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This study helps to understand applicability of junctions in series array and steps to control the level of synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The process is easier and synchronization is performed well for larger number of junctions while partial synchronization is also possible following the Kuramoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' However, this study does not state any conclusive equation for threshold coupling for Josephson junction as it discussed in case of general oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' This aspect will be discussed in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' APPENDIX A From (40),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' tan �φj 2 + π 4 � = � αj + 1 αj − 1 tan ψj 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' � tan �φj 2 + π 4 � + 1 � � tan �φj 2 + π 4 � − 1 � = �� αj + 1 αj − 1 tan ψj 2 + 1 � �� αj + 1 αj − 1 tan ψj 2 − 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' � 1 + tan φj 2 1 − tan φj 2 + 1 � � 1 + tan φj 2 1 − tan φj 2 − 1 � = αj + 1 αj − 1 tan2 ψj 2 − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' � cos φj 2 + sin φj 2 cos φj 2 − sin φj 2 + 1 � � cos φj 2 + sin φj 2 cos φj 2 − sin φj 2 − 1 � = αj + 1 αj − 1 tan2 ψj 2 − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 2 cos φj 2 × 2 sin φj 2 � cos φj 2 − sin φj 2 �2 = 2 sin φj � cos φj 2 − sin φj 2 �2 = αj + 1 αj − 1 tan2 ψj 2 − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 2 sin φj 1 − sin φj = αj tan2 ψj 2 + tan2 ψj 2 − αj + 1 αj − 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 1 − sin φj 2 sin φj = 1 − αj cos ψj (αj − 1) cos2 ψj 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' 1 2 sin φj = 1 2 + 1 − αj cos ψj (αj − 1) cos2 ψj 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' sin φj = 1 − αj cos ψj αj − cos ψj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' sin φj ≡ sin φ(ψj) = αj − � α2 j − 1 � αj − cos ψj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' Phasegraph of 5oscillators forK=30453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='72dt=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='00l simulationtime=15 (a) Phase graph of 5 oscillators (b) Order Paraneter of oacillators 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 E0 sin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='心 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='0 time (T) time (r) (o)orderParameterof oscillators at T=Q (d) Order Parameter of oBcillatore at T =15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content="1 2'0 20 90 90 sin uIs 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='9 L1 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='3 心1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='5 心1 cos cosu11 APPENDIX B �dψj d˜τ � = 1 − ϵjδj � α2 j − 1 2πN � 2π 0 � αj − cos ψj α2 j − 1 × N � k=1 ∞ � n=0 nIkρkBkn sin {n (˜τ + ck) + βkn} � d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or, �dψj d˜τ � = 1 − ϵjδj 2πN � α2 j − 1 � 2π 0 (αj − cos (˜τ + cj)) × N � k=1 ∞ � n=0 nIkρkBkn sin {n (˜τ + ck) + βkn} d˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or, �dψj d˜τ � = 1 + ϵjδj N � α2 j − 1 � γ2 j � α2 j − 1 �2 + � ω2 0j − � α2 j − 1 �2�2 × N � k=1 Ikρk � 1 − α2 k + αk � α2 k − 1 � sin (cj − ck − ζj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' or, �dψj d˜τ � = 1 + Kj N N � k=1 Ikρk � 1 − α2 k + αk � α2 k − 1 � × sin (cj − ck − ζj) , or, �dψj d˜τ � = 1 + Kj N N � k=1 Ak sin (cj − ck − ζj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' ACKNOWLEDGMENT The author would like to Sudhir R Jain, for his ideas, inspi- ration and continuous support to conceptualize, understand and formulate the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content=' The author also expresses gratitude to Susmita Bhattacharyya and Tilottoma Bhattacharyya for their guidance.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} +page_content='de/' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfSAdG/content/2301.03787v1.pdf'} diff --git a/DNE3T4oBgHgl3EQfUwpD/content/tmp_files/2301.04453v1.pdf.txt b/DNE3T4oBgHgl3EQfUwpD/content/tmp_files/2301.04453v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d4696d81460821df8f840021e04badf5b164b089 --- /dev/null +++ b/DNE3T4oBgHgl3EQfUwpD/content/tmp_files/2301.04453v1.pdf.txt @@ -0,0 +1,927 @@ +Nakayama et al.: Preparation of Papers for IEEE Access +. +. +VOLUME 4, 2016 +1 +arXiv:2301.04453v1 [eess.SY] 11 Jan 2023 + +TEEEAccesSDate of publication xxxx 00, 0000, date of current version xxxx 00, 0000. +Digital Object Identifier 10.1109/ACCESS.2017.DOI +Trajectory Tracking Control of +The Second-order Chained Form System +by Using State Transitions +MAYU NAKAYAMA1, MASAHIDE ITO1, (Member, IEEE) +1School of Information Science and Technology, Aichi Prefectural University, Nagakute, Aichi, Japan +Corresponding author: Masahide Ito (e-mail: masa-ito@ist.aichi-pu.ac.jp). +ABSTRACT This paper proposes a novel control approach composed of sinusoidal reference trajectories +and trajectory tracking controller for the second-order chained form system. The system is well-known as +a canonical form for a class of second-order nonholonomic systems obtained by appropriate transformation +of the generalized coordinates and control inputs. The system is decomposed into three subsystems, two of +them are the so-called double integrators and the other subsystem is a nonlinear system depending on one of +the double integrators. The double integrators are linearly controllable, which enables to transit the value of +the position state in order to modify the nature of the nonlinear system that depends on them. Transiting the +value to “one” corresponds to modifying the nonlinear subsystem into the double integrator; transiting the +value to “zero” corresponds to modifying the nonlinear subsystem into an uncontrollable linear autonomous +system. Focusing on this nature, this paper proposes a feedforward control strategy. Furthermore, from the +perspective of practical usefulness, the control strategy is extended into trajectory tracking control by using +proportional-derivative feedback. The effectiveness of the proposed method is demonstrated through several +numerical experiments including an application to an underactuated manipulator. +INDEX TERMS +nonholonomic systems; state transitions; the second-order chained form; trajectory +tracking control +I. INTRODUCTION +N +ONHOLONOMIC systems are nonlinear dynamical +systems with non-integrable differential constraints, +whose control problems have been attracting many re- +searchers and engineers for the last three decades. The main +reason is that the nonholonomic systems do not satisfy +Brockett’s theorem [1]. The challenging and negative fact +means that there is not any smooth time-invariant feedback +control law to be able to stabilize them. The applications +include various types of robotic vehicles and manipulation. +Some of them have been often used as a kind of bench- +mark platform to demonstrate the performance of a proposed +controller for not only a control problem of a single robotic +system and also a distributed control problem of multiagent +robotic systems. +The class subject to acceleration constraints—called +second-order nonholonomic systems—includes real exam- +ples such as a V/STOL aircraft [2], an underactuated manip- +ulator [3], an underactuated hovercraft [4], and a crane [5]. +These systems can be represented in a canonical system +called the second-order chained form by coordinate and in- +put transformations. The second-order chained form system +is also affected by Brockett’s theorem [1]. To avoid this +difficulty, there are several ingenious control approaches. +The stabilizing controllers proposed in [4], [6]–[8] exploit +discontinuity or time-variance; [3], [9] and [10] reduce the +control problem into a trajectory tracking problem. Other +than those, [11] and [12] consider a motion planning problem +(in other words, a feedforward control problem). +For the second-order chained form system, this paper +presents a novel control approach composed of sinusoidal +reference trajectories and a simple trajectory tracking con- +troller. The second-order chained form system is decomposed +into three subsystems. Two of them are the so-called dou- +ble integrators; the other subsystem is a nonlinear system +depending on one of the double integrators. The double +integrator is linearly controllable, which enables to transit the +value of the position state in order to modify the nature of the +nonlinear subsystem. Transiting the value into “one” corre- +sponds to modifying the nonlinear subsystem into the double +2 +VOLUME 4, 2016 + +IEEEAccesS +Multidisciplinary Rapid Review Open Access JournalNakayama et al.: Preparation of Papers for IEEE Access +integrator; transiting the value into “zero” corresponds to +modifying the nonlinear subsystem into a linear autonomous +system. Focusing on this nature, this paper proposes a feed- +forward control strategy. Furthermore, from the perspective +of practical usefulness, the control strategy is extended into +trajectory tracking control by using proportional-derivative +(PD) feedback. +The remainder of this paper is organized as follows: Sec- +tion II presents that the second-order chained form system +can be decomposed to linear subsystems by using state +transitions. On the basis of such system nature, Section III +proposes a feedforward control strategy and also a trajectory +tracking controller of PD feedback. Section IV applies the +proposed control approach to an underactuated manipulator +and evaluates it through numerical experiments. The last +section concludes the paper with a summary and future work. +II. SUBSYSTEM DECOMPOSITION OF THE +SECOND-ORDER CHAINED FORM SYSTEM BY USING +STATE TRANSITIONS +Consider the following second-order chained form system: +d2 +dt2 ξ = +� +� +1 +0 +0 +1 +ξ2 +0 +� +� u, +(1) +where ξ = [ξ1, ξ2, ξ3]⊤ and u = [u1, u2]⊤ are the gen- +eralized coordinate vector and the generalized input vector, +respectively. This system is well-known as a canonical form +for a class of second-order nonholonomic systems, which +can be resulted from the original dynamical model via an +appropriate transformation of the generalized coordinates +and control inputs. Representing the system (1) as an affine +nonlinear system: +d +dt +� +������� +ξ1 +ξ2 +ξ3 +˙ξ1 +˙ξ2 +˙ξ3 +� +������� += +� +������� +˙ξ1 +˙ξ2 +˙ξ3 +0 +0 +0 +� +������� ++ +� +������� +0 +0 +0 +1 +0 +ξ2 +� +������� +u1 + +� +������� +0 +0 +0 +0 +1 +0 +� +������� +u2, +(2) +we +can +easily +confirm +that +the +equilibrium +points +(ξ⋆ +1, ξ⋆ +2, ξ⋆ +3, 0, 0, 0), ξ⋆ +1, ξ⋆ +2, ξ⋆ +3 +∈ +R are small-time local +controllable (STLC) via Sussmann’s theorem [13]. +By focusing on the control inputs, the system (1) can be +decomposed into the following two subsystems: +d +dt +� +��� +ξ1 +ξ3 +˙ξ1 +˙ξ3 +� +��� = +� +��� +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +� +��� +� +��� +ξ1 +ξ3 +˙ξ1 +˙ξ3 +� +��� + +� +��� +0 +0 +1 +ξ2 +� +��� u1, +(3a) +d +dt +� ξ2 +˙ξ2 +� += +� 0 +1 +0 +0 +� � ξ2 +˙ξ2 +� ++ +� 0 +1 +� +u2. +(3b) +The subsystem (3b) with respect to the control input u2 is +a linear and controllable system represented by the double +integrator. On the other hand, the subsystem (3a) with respect +to the input u1 is a four-dimensional nonlinear system whose +input matrix depends on the state variable ξ2. The subsys- +tem (3a) can be further decomposed as follows: +d +dt +� ξ1 +˙ξ1 +� += +� 0 +1 +0 +0 +� � ξ1 +˙ξ1 +� ++ +� 0 +1 +� +u1, +(4a) +d +dt +� ξ3 +˙ξ3 +� += +� 0 +1 +0 +0 +� � ξ3 +˙ξ3 +� ++ +� 0 +ξ2 +� +u1. +(4b) +The subsystem (4a) of the double integrator is linear and +controllable; the subsystem (4b) inherits the nonlinearity of +the system (3a). +Fig. 1 shows a block diagram describing the above- +mentioned subsystem decomposition explicitly. The state of +the subsystem (3b) can be transited to be a constant value +because of the linear controllability. For example, by setting +time intervals where ξ2 is “zero” and also ξ2 is “one”, the +nonlinear subsystem (4b) can be treated as a linear system. +During the time interval of ξ2 = 1, the subsystems (4a) +and (4b) are linear which have the same double integrator +structure and control input u1. On the other hand, during +the time interval of ξ2 = 0, the subsystem (3a) becomes +a linear autonomous (i.e., uncontrollable) system and the +subsystem (4a) can be controlled independently from sub- +system (4b) by the control input u1. +Remark 1. Some conventional approaches such as in [14], +[10] and [15] exploit a different subsystem decomposition +that can decompose the system (1) as follows: +d +dt +� ξ1 +˙ξ1 +� += +� 0 +1 +0 +0 +� � ξ1 +˙ξ1 +� ++ +� 0 +1 +� +u1, +(5a) +d +dt +� +��� +ξ2 +ξ3 +˙ξ2 +˙ξ3 +� +��� = +� +��� +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +u1 +0 +0 +0 +� +��� +� +��� +ξ2 +ξ3 +˙ξ2 +˙ξ3 +� +��� + +� +��� +0 +0 +1 +0 +� +��� u2. +(5b) +The subsystem (5a) is the same with (4a); the subsystem (5b) +has a variable structure depending on u1. The subsystem (5b) +is linear when u1 is a non-zero constant, which reduces a +control problem of the second-order chained form system into +a simultaneous stabilizing problem of the two subsystems (5a) +and (5b). When u1 becomes zero before the end of control, +however, the subsystem (5b) will be uncontrollable with a +pole at the origin and then the whole of the subsystem loses +the controllability. This subsystem decomposition, therefore, +needs control in consideration with u1. +III. PROPOSED CONTROL APPROACH +In this paper, a control task of a rest-to-rest motion is ad- +dressed. For this task, the authors propose a control approach +composed of sinusoidal reference trajectories and a trajec- +tory tracking controller. In particular, a feedforward control +strategy that generates the reference trajectories exploits the +system decomposition based on state transition described in +the previous section. +The feedforward control strategy using system switching +based on state transitions in ξ2 is as follows: +VOLUME 4, 2016 +3 + +TEEEAccesSNakayama et al.: Preparation of Papers for IEEE Access +� +� +u1 +˙ξ1 +ξ1 +× +� +� +˙ξ3 +ξ3 +� +� +u2 +˙ξ2 +ξ2 +� +� +u1 +˙ξ1 +ξ1 +� +� +˙ξ3 +ξ3 +� +� +u1 +˙ξ1 +ξ1 +� +� +˙ξ3 +ξ3 +⇐⇒ +when ξ2 = 1 +when ξ2 = 0 +FIGURE 1. Subsystem decomposition of the second-order chained form by using ξ2’s state transitions between 0 and 1. +Step 1 +Transit ξ2 from any initial value to 1 by using +u1(t) = 0, u2(t) = q2(t); +Step 2 +Transit ξ3 from any initial value to any desired +value (in conjunction with it, ξ1 is also driven) +by using u1(t) = q3(t), u2(t) = 0; +Step 3 +Transit ξ2 from 1 to 0 by using u1(t) += +0, u2(t) = q2(t); +Step 4 +Transit ξ1 from any value in Step 2 to any de- +sired value by using u1(t) = q1(t), u2(t) = 0; +Step 5 +Transit ξ2 from 0 to any desired value by using +u1(t) = 0, u2(t) = q2(t). +A control input in Step k (k = 1, 2, . . . , 5) is designed +by an appropriate sinusoidal function qi(t) (i = 1, 2, 3) +without any feedback. This control strategy is namely mo- +tion planning, which naturally cannot deal with disturbance. +Therefore, we provide a trajectory tracking controller that +follow the reference trajectory. +Consider to drive the state variables ξi(t), ˙ξi(t) of the +system (1) by the following sinusoidal functions with pe- +riod T = 2π/ω and amplitude ak: +qi(t) = akω2 sin ωt. +(6) +Then, at time t (≤ kT), trajectories of a subsystem with non- +zero input are derived as +˙ξi(t) = ˙ξi((k − 1)T) − akω cos ωt + akω, +(7) +ξi(t) = ξi((k − 1)T) + ˙ξi((k − 1)T)t +− ˙ξi((k − 1)T)(k − 1)T +− ak sin ωt + akωt − ak(k − 1)ωT, +(8) +respectively, where ξi((k − 1)T) and ˙ξi((k − 1)T) are initial +values of the state variables in Step k. Thus, at the end of +k-th period (t = kT), the state transitions are represented as +˙ξi(kT) = ˙ξi((k − 1)T), +(9) +ξi(kT) = ξi((k − 1)T) + ˙ξi((k − 1)T)T + 2πak, +(10) +which means that a displacement of 2πak on ξi is obtained. +This can be seen that the desired displacement is extracted by +using the amplitude ak as a tuning parameter. +By setting the trajectories (6), (7), (8) as reference trajec- +tories qref +i (t), ξref +i (t), ˙ξref +i (t), a PD feedback control system +can be designed for trajectory tracking. A linear system of a +double integrator can be represented in the following state- +space form with the state zi = [ξi, ˙ξi]⊤ and control input qi: +˙zi = +� 0 +1 +0 +0 +� +� �� � +A +zi + +� 0 +1 +� +���� +b +qi(t, zi). +(11) +In Step k, a feedback controller for trajectory tracking to zref +i +is given as follows: +qi(t, zi) = qref +i (t) + k ei, +(12) +where ei := zref +i +− zi and k = [kp, kd] is a feedback gain +matrix. The system (11) yields the closed-loop system ˙ei = +(A − bk)ei. By choosing the feedback gain k so that (A − +bk) is Hurwitz-stable, the closed-loop system is stabilized, +that is, zi tracks zref +i . +IV. NUMERICAL EXPERIMENTS +In this section, we evaluate the effectiveness of the proposed +control approach through numerical experiments. +Firstly, we validate the proposed controller for the second- +order chained form system. A numerical experiment was per- +formed with T = 1 s, ξ(0) = [3, 0.5, 1]⊤, ˙ξ(0) = 03, ξ⋆ = +[1, 1, 0]⊤, and ˙ξ⋆ = 03. Fig. 2 shows the simulation results +when choosing a1 = 1/(4π), a2 = a3 = a4 = −1/(2π), +and a5 = 1/(2π). The ordinary differential equations was +numerically solved by ODE45 of MATLAB [16] with a rela- +tive tolerance of 1×10−3. The results indicate that each state +reached to the target value ξ⋆ with the remaining errors at t = +5T: ξ(5T)−ξ⋆ = [−2.7×10−8, 1.0×10−10, −4.7×10−8]⊤ +and ˙ξ(5T)− ˙ξ⋆ = [1.3×10−8, −8.9×10−9, −2.1×10−8]⊤, +which means that the desired control is achieved. +Secondly, the proposed controller is applied to an underac- +tuated manipulator—a typical example of second-order non- +holonomic systems—as shown in Fig. 3. This manipulator +has first two joints being actuated and the last joint being +unactuated. The system representation can be converted to +the second-order chained form system. Even if the third joint +cannot be driven due to no actuator, the acceleration (α1, α2) +acting on the center of percussion of the third link can be +treated equivalently as a control input owing to dynamic cou- +pling effect—the rotational actuation of the first and second +4 +VOLUME 4, 2016 + +TEEEAccesSNakayama et al.: Preparation of Papers for IEEE Access +TABLE 1. Definition of variables and parameters +(x, y) : position of the center of percussion of the third link in the frame O-XY ; +θ +: angle of the third link relative to X-axis; +d3 +: distance between the third joint and the center of mass of the third link; +m3 +: mass of the third link; +I3 +: moment of inertia mass of the third link; +LCoP : distance between the third joint and the center of percussion of the third link +� +LCoP := (I3 + m3d2 +3)/(m3d3) +� +; +α1 +: translational acceleration along the third link; +α2 +: angular acceleration around the center of percussion of the third link. +FIGURE 2. Simulation results of trajectory tracking control +joints propagates through the links. For simplicity, assume +that there is no disturbance such as load, friction, linear and +nonlinear damping, etc. The main variables are defined as in +Table 1. +Let χ := [x, y, θ]⊤ and α = [α1, α2]⊤. Yoshikawa, et +al. [11] provided a set of coordinate and input transforma- +tions to convert the manipulator dynamics derived from the +Lagrange’s equation of motion into the following system +𝜃 +1st revolute joint +(actuated) +𝑥 +𝑑! +2nd revolute joint +(actuated) +3rd revolute joint +(unactuated) +𝑦 +𝑋 +𝑌 +𝑂 +center of percussion +of 3rd link +: +𝛼" +𝛼# +FIGURE 3. A three-joint manipulator with passive third joint +representation: +¨χ = +� +� +cos θ +0 +sin θ +0 +0 +1 +� +� α. +(13) +Using the coordinate transformation +� +� +ξ1 +ξ2 +ξ3 +� +� = +� +� +x − LCoP +tan θ +y +� +� , +� +� +˙ξ1 +˙ξ2 +˙ξ3 +� +� = +� +� +˙x +˙θ sec2 θ +˙y +� +� +(14) +and the input transformation +� α1 +α2 +� += +� +u1 sec θ +u2 cos2 θ − 2 ˙θ2 tan θ +� +, +(15) +the system (13) can be transformed into the second-order +chained form system (1). Note that both transformation are +singular point at θ = ±π/2. +For the third joint of the underactuated manipulator with +m3 = 0.6 kg, d3 = 0.3 m, and I3 = 4.5 × 10−3 kg · m2, +steer from initial values χ(0) += +[3.33 m, 1 m, 4.6 × +10−1 rad]⊤, ˙χ(0) += +03 to the desired ones χ⋆ += +[1 m, 0 m, 0 rad]⊤, ˙χ⋆ = 03. +Fig. 4 shows a simulation result with the period T = 1 s +and the feedback gain kp = kd = 1. In this case, from (14), +we have ξ(0) = [3, 0.5, 1]⊤ and ξ⋆ = [0.67, 0, 0]⊤. It can +be confirmed that each state converges to the desired value in +the both system representation. +VOLUME 4, 2016 +5 + +TEEEAccesSNakayama et al.: Preparation of Papers for IEEE Access +(a) States and inputs of the second-order chained form system +(b) Status and inputs of a three-joint underactuated manipulator +FIGURE 4. Numerical results +Furthermore, to verify the effect of feedback control, an- +other case with an initial value error was simulated. For +a rest-to-rest motion from χ(0) += +[3.33 m, 1 m, 4.6 × +10−1 rad]⊤ to χ⋆ = [1.33 m, 0 m, 7.8 × 10−1 rad]⊤ with +the zero velocities, the initial value error of +10% is given to +θ, i.e., χ(0) = [3.33 m, 1 m, 5.1 × 10−1 rad]⊤. The result +is shown in Fig. 5. The dashed lines indicate the target +trajectories. It can be observed that tracking error due to the +initial value error is alleviated over time. +Similarly, when initial value errors of ±1%, ±10%, and +±30% on θ are given the tracking errors at the end of control +at t = 5T are summarized in Table 2. The terminal values +of the tracking errors do not increase greatly even if the +magnitude of the initial value error increases. Consequently, +it is confirmed that the feedback of trajectory tracking has a +sufficient effect on initial value errors. Note that the terminal +error on x is relatively larger than the one on θ. The proposed +control method attempts to settle the system by focusing on a +single state every step. In addition, the state in which the con- +trol step ends has no chance to be controlled directly. For such +a state, there can be a secondary state transition that yields +in control steps that focus on the other states. Therefore, if +a state fails to converge into its reference trajectory within +the control step due to initial value error or disturbance, it +behaves unexpectedly until the end of the control strategy. +In particular, ξ2—the state used for switching the systems— +has a negative effect on the other states because the reference +trajectory is not computed correctly. Furthermore, the error +remaining in the velocity state (ξ4, ξ5, ξ6) causes a drift in +the position state (ξ1, ξ2, ξ3) even if the input is zero in +the following control steps. This is explained by numerical +experiments shown in Fig. 5. Note that θ is related to ξ2 +as specified in (14). This means that θ affects the other +states (x, y) when not converging completely. On the other +hand, since ξ2 is settled in the final step (i.e., Step 5), the +propagation from the error in the velocity state is small. +Therefore, the error remaining in θ is considered to be smaller +than in x. +V. CONCLUSION +In this paper, a novel control approach composed of sinu- +soidal reference trajectories and a simple trajectory tracking +6 +VOLUME 4, 2016 + +TEEEAccesSNakayama et al.: Preparation of Papers for IEEE Access +FIGURE 5. Given an initial value error(+10%) +controller for the second-order chained form system was +proposed. The key idea is a subsystem decomposition of the +second-order chained form system by using state transitions. +The effectiveness of the proposed algorithm was demon- +strated by numerical results including an application to a +three-joint underactuated manipulator. In particular, it can be +confirmed that the feedback control works well against the +initial value error. +The future work of this research is to verify the proposed +approach via experiments on an actual robot. +REFERENCES +[1] R. W. Brockett: “Asymptotic stability and feedback stabilization,” in +Differential Geometric Control Theory (Eds. by R. W. Brockett, R. S. +Millmann and H. J. Sussmann), Birkhauser, Boston, pp. 181–191, 1983. +[2] J. Hauser, S. Sastry, and G. Meyer: “Nonlinear control design for slightly +TABLE 2. Error from target value by the initial value error +Case +χ(0) − χ⋆ +χ(5T) − χ⋆ +w/o init. err. +� +� +0 m +0 m +0 rad +� +� +� +� +4.5 × 10−8 m +4.4 × 10−7 m +4.9 × 10−9 rad +� +� +w/ +1 % init. err. +� +� +0 m +0 m +4.6 × 10−3 rad +� +� +� +� +2.9 × 10−3 m +−6.9 × 10−4 m +−8.5 × 10−4 rad +� +� +w/ −1 % init. err. +� +� +0 m +0 m +−4.6 × 10−3 rad +� +� +� +� +−2.9 × 10−3 m +7.4 × 10−4 m +8.5 × 10−4 rad +� +� +w/ +10 % init. err. +� +� +0 m +0 m +4.6 × 10−2 rad +� +� +� +� +3.0 × 10−2 m +−4.8 × 10−3 m +−8.8 × 10−3 rad +� +� +w/ −10 % init. err. +� +� +0 m +0 m +−4.6 × 10−2 rad +� +� +� +� +−2.9 × 10−2 m +9.9 × 10−3 m +8.3 × 10−3 rad +� +� +w/ +30 % init. err. +� +� +0 m +0 m +1.4 × 10−1 rad +� +� +� +� +9.1 × 10−2 m +3.3 × 10−3 m +−2.8 × 10−2 rad +� +� +w/ −30 % init. err. +� +� +0 m +0 m +−1.4 × 10−1 rad +� +� +� +� +−8.5 × 10−2 m +4.3 × 10−2 m +2.3 × 10−2 rad +� +� +non-minimum phase systems: application to V/STOL aircraft,” Automat- +ica, Vol. 28, No. 4, pp. 665–679, 1992. +[3] H. Arai, K. Tanie, and N. Shiroma: “Nonholonomic control of a three- +DOF planar underactuated manipulator,” IEEE Transactions on Robotics +Automation, Vol. 14, No. 5, pp. 681–695, 1998. +[4] G. He, C. Zhang, W. Sun, and Z. Geng: “Stabilizing the second-order +nonholonomic systems with chained form by finite-time stabilizing con- +trollers,” Robotica, Vol. 34, pp. 2344–2367, 2016. +[5] M. Nowicki, W. Respondek, J. Piasek, and K. Kozłowski: “Geometry and +flatness of m-crane systems,” Bulletin of The Polish Academy of Sciences, +Technical Sciences, Vol. 67, No. 5, pp. 893–903, 2019. +[6] S.S. Ge, Z. Sun, T.H. Lee, and M.W. Spong: “Feedback linearization and +stabilization of second-order nonholonomic chained systems,” Interna- +tional Journal of Control, Vol. 74, pp. 1383–1392, 2001. +[7] K. Pettersen and O. Egeland: “Exponential stabilization of an underac- +tuated surface vessel,” in Proceedings of the 35th IEEE International +Conference on Decision and Control (CDC’96), Vol. 1, pp. 967–972, 1996. +[8] K. Pettersen and O. Egeland: “Position and attitude control of an au- +tonomous underwater vehicle,” in Proceedings of the 35th IEEE Interna- +tional Conference on Decision and Control (CDC’96), pp. 987–991, 1996. +[9] A. De Luca and G. Oriolo: “Trajectory planning and control for planar +robots with passive last joint,” International Journal of Robotics Research, +Vol. 21, No. 5–6, pp. 575–590, 2002. +[10] N.P.I. Aneke, H. Nijmeijer, and A.G. de Jager: “Tracking control of +second-order chained form systems by cascaded backstepping,” Interna- +tional Journal of Robust and Nonlinear Control, Vol. 13, No. 2, pp. 95– +115. +[11] T. Yoshikawa, K. Kobayashi, and T. Watanabe, “Design of a desirable +trajectory and convergent control for 3-D.O.F manipulator with a nonholo- +nomic constraint,” in Proceedings of the IEEE International Conference +on Robotics and Automation (ICRA’00), San Francisco, CA, USA, Vol. 2, +pp. 1805–1810, 2000. +[12] M. Ito: “Motion planning of a second-order nonholonomic chained form +system based on holonomy extraction,” Electronics, Vol. 8, No. 11, +pp. 1337, 2019. +[13] H. Sussmann: “A general theorem on local controllability,” SIAM Journal +on Control and Optimization, Vol. 25, No. 1, pp. 158–194, 1987. +[14] T. Nam, T. Tamura, T. Mita, and Y. Kim: “Control of the high-order +VOLUME 4, 2016 +7 + +TEEEAccesSNakayama et al.: Preparation of Papers for IEEE Access +chained form system,” in Proceedings of the 41st SICE Annual Confer- +ence, Vol. 4, pp. 2196–2201, 2002. +[15] A. Hably and N. Marchand: “Bounded control of a general extended +chained form systems,” in Proceedings of the 53rd IEEE Conference on +Decision and Control (CDC’14), pp. 6342–6347, Los Angeles, CA, USA, +2014. +[16] MathWorks, “ode45: Solve nonstiff differential equations—medium order +method,” Documentation for MATLAB R2022b, 2022. [Online]. Avail- +able: https://www.mathworks.com/help/matlab/ref/ode45.html. Accessed +on: Jan 11, 2023. +MAYU NAKAYAMA was born in Kiyosu, Aichi, +Japan in 1997. She received the B.S. and M.S. de- +grees in information science and technology from +Aichi Prefectural University (APU), Nagakute, +Aichi, Japan, in 2020 and 2022. +She is currently with DENSO Corporation. Her +research interests include nonlinear control for +underactuated systems. +MASAHIDE ITO (M’10) was born in Nagoya, +Aichi, Japan in 1979. He received the B.S., M.S., +and Ph.D. degrees in information science and tech- +nology from Aichi Prefectural University (APU), +Nagakute, Aichi, Japan, in 2002, 2004, and 2008. +He is currently an Associate Professor with +the School of Information Science and Technol- +ogy, APU. His research interests include visual +feedback control of robotic systems and nonlinear +control for underactuated systems. +8 +VOLUME 4, 2016 + +TEEEAccesS \ No newline at end of file diff --git a/DNE3T4oBgHgl3EQfUwpD/content/tmp_files/load_file.txt b/DNE3T4oBgHgl3EQfUwpD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c874ee77241c3cde9ff31c6d068d40930db28d0 --- /dev/null +++ b/DNE3T4oBgHgl3EQfUwpD/content/tmp_files/load_file.txt @@ -0,0 +1,372 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf,len=371 +page_content='Nakayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' : Preparation of Papers for IEEE Access .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' VOLUME 4, 2016 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='04453v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='SY] 11 Jan 2023 TEEEAccesSDate of publication xxxx 00, 0000, date of current version xxxx 00, 0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Digital Object Identifier 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='DOI Trajectory Tracking Control of The Second-order Chained Form System by Using State Transitions MAYU NAKAYAMA1, MASAHIDE ITO1, (Member, IEEE) 1School of Information Science and Technology, Aichi Prefectural University, Nagakute, Aichi, Japan Corresponding author: Masahide Ito (e-mail: masa-ito@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='aichi-pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='jp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' ABSTRACT This paper proposes a novel control approach composed of sinusoidal reference trajectories and trajectory tracking controller for the second-order chained form system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The system is well-known as a canonical form for a class of second-order nonholonomic systems obtained by appropriate transformation of the generalized coordinates and control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The system is decomposed into three subsystems, two of them are the so-called double integrators and the other subsystem is a nonlinear system depending on one of the double integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The double integrators are linearly controllable, which enables to transit the value of the position state in order to modify the nature of the nonlinear system that depends on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Transiting the value to “one” corresponds to modifying the nonlinear subsystem into the double integrator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' transiting the value to “zero” corresponds to modifying the nonlinear subsystem into an uncontrollable linear autonomous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Focusing on this nature, this paper proposes a feedforward control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Furthermore, from the perspective of practical usefulness, the control strategy is extended into trajectory tracking control by using proportional-derivative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The effectiveness of the proposed method is demonstrated through several numerical experiments including an application to an underactuated manipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' INDEX TERMS nonholonomic systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' state transitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' the second-order chained form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' trajectory tracking control I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' INTRODUCTION N ONHOLONOMIC systems are nonlinear dynamical systems with non-integrable differential constraints, whose control problems have been attracting many re- searchers and engineers for the last three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The main reason is that the nonholonomic systems do not satisfy Brockett’s theorem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The challenging and negative fact means that there is not any smooth time-invariant feedback control law to be able to stabilize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The applications include various types of robotic vehicles and manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Some of them have been often used as a kind of bench- mark platform to demonstrate the performance of a proposed controller for not only a control problem of a single robotic system and also a distributed control problem of multiagent robotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The class subject to acceleration constraints—called second-order nonholonomic systems—includes real exam- ples such as a V/STOL aircraft [2], an underactuated manip- ulator [3], an underactuated hovercraft [4], and a crane [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' These systems can be represented in a canonical system called the second-order chained form by coordinate and in- put transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The second-order chained form system is also affected by Brockett’s theorem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' To avoid this difficulty, there are several ingenious control approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The stabilizing controllers proposed in [4], [6]–[8] exploit discontinuity or time-variance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' [3], [9] and [10] reduce the control problem into a trajectory tracking problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Other than those, [11] and [12] consider a motion planning problem (in other words, a feedforward control problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' For the second-order chained form system, this paper presents a novel control approach composed of sinusoidal reference trajectories and a simple trajectory tracking con- troller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The second-order chained form system is decomposed into three subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Two of them are the so-called dou- ble integrators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' the other subsystem is a nonlinear system depending on one of the double integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The double integrator is linearly controllable, which enables to transit the value of the position state in order to modify the nature of the nonlinear subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Transiting the value into “one” corre- sponds to modifying the nonlinear subsystem into the double 2 VOLUME 4, 2016 IEEEAccesS Multidisciplinary Rapid Review Open Access JournalNakayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' : Preparation of Papers for IEEE Access integrator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' transiting the value into “zero” corresponds to modifying the nonlinear subsystem into a linear autonomous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Focusing on this nature, this paper proposes a feed- forward control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Furthermore, from the perspective of practical usefulness, the control strategy is extended into trajectory tracking control by using proportional-derivative (PD) feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The remainder of this paper is organized as follows: Sec- tion II presents that the second-order chained form system can be decomposed to linear subsystems by using state transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' On the basis of such system nature, Section III proposes a feedforward control strategy and also a trajectory tracking controller of PD feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Section IV applies the proposed control approach to an underactuated manipulator and evaluates it through numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The last section concludes the paper with a summary and future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' SUBSYSTEM DECOMPOSITION OF THE SECOND-ORDER CHAINED FORM SYSTEM BY USING STATE TRANSITIONS Consider the following second-order chained form system: d2 dt2 ξ = � � 1 0 0 1 ξ2 0 � � u, (1) where ξ = [ξ1, ξ2, ξ3]⊤ and u = [u1, u2]⊤ are the gen- eralized coordinate vector and the generalized input vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' This system is well-known as a canonical form for a class of second-order nonholonomic systems, which can be resulted from the original dynamical model via an appropriate transformation of the generalized coordinates and control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Representing the system (1) as an affine nonlinear system: d dt � ������� ξ1 ξ2 ξ3 ˙ξ1 ˙ξ2 ˙ξ3 � ������� = � ������� ˙ξ1 ˙ξ2 ˙ξ3 0 0 0 � ������� + � ������� 0 0 0 1 0 ξ2 � ������� u1 + � ������� 0 0 0 0 1 0 � ������� u2, (2) we can easily confirm that the equilibrium points (ξ⋆ 1, ξ⋆ 2, ξ⋆ 3, 0, 0, 0), ξ⋆ 1, ξ⋆ 2, ξ⋆ 3 ∈ R are small-time local controllable (STLC) via Sussmann’s theorem [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' By focusing on the control inputs, the system (1) can be decomposed into the following two subsystems: d dt � ��� ξ1 ξ3 ˙ξ1 ˙ξ3 � ��� = � ��� 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 � ��� � ��� ξ1 ξ3 ˙ξ1 ˙ξ3 � ��� + � ��� 0 0 1 ξ2 � ��� u1, (3a) d dt � ξ2 ˙ξ2 � = � 0 1 0 0 � � ξ2 ˙ξ2 � + � 0 1 � u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' (3b) The subsystem (3b) with respect to the control input u2 is a linear and controllable system represented by the double integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' On the other hand, the subsystem (3a) with respect to the input u1 is a four-dimensional nonlinear system whose input matrix depends on the state variable ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The subsys- tem (3a) can be further decomposed as follows: d dt � ξ1 ˙ξ1 � = � 0 1 0 0 � � ξ1 ˙ξ1 � + � 0 1 � u1, (4a) d dt � ξ3 ˙ξ3 � = � 0 1 0 0 � � ξ3 ˙ξ3 � + � 0 ξ2 � u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' (4b) The subsystem (4a) of the double integrator is linear and controllable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' the subsystem (4b) inherits the nonlinearity of the system (3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' 1 shows a block diagram describing the above- mentioned subsystem decomposition explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The state of the subsystem (3b) can be transited to be a constant value because of the linear controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' For example, by setting time intervals where ξ2 is “zero” and also ξ2 is “one”, the nonlinear subsystem (4b) can be treated as a linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' During the time interval of ξ2 = 1, the subsystems (4a) and (4b) are linear which have the same double integrator structure and control input u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' On the other hand, during the time interval of ξ2 = 0, the subsystem (3a) becomes a linear autonomous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=', uncontrollable) system and the subsystem (4a) can be controlled independently from sub- system (4b) by the control input u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Some conventional approaches such as in [14], [10] and [15] exploit a different subsystem decomposition that can decompose the system (1) as follows: d dt � ξ1 ˙ξ1 � = � 0 1 0 0 � � ξ1 ˙ξ1 � + � 0 1 � u1, (5a) d dt � ��� ξ2 ξ3 ˙ξ2 ˙ξ3 � ��� = � ��� 0 0 1 0 0 0 0 1 0 0 0 0 u1 0 0 0 � ��� � ��� ξ2 ξ3 ˙ξ2 ˙ξ3 � ��� + � ��� 0 0 1 0 � ��� u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' (5b) The subsystem (5a) is the same with (4a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' the subsystem (5b) has a variable structure depending on u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The subsystem (5b) is linear when u1 is a non-zero constant, which reduces a control problem of the second-order chained form system into a simultaneous stabilizing problem of the two subsystems (5a) and (5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' When u1 becomes zero before the end of control, however, the subsystem (5b) will be uncontrollable with a pole at the origin and then the whole of the subsystem loses the controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' This subsystem decomposition, therefore, needs control in consideration with u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' PROPOSED CONTROL APPROACH In this paper, a control task of a rest-to-rest motion is ad- dressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' For this task, the authors propose a control approach composed of sinusoidal reference trajectories and a trajec- tory tracking controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' In particular, a feedforward control strategy that generates the reference trajectories exploits the system decomposition based on state transition described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The feedforward control strategy using system switching based on state transitions in ξ2 is as follows: VOLUME 4, 2016 3 TEEEAccesSNakayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' : Preparation of Papers for IEEE Access � � u1 ˙ξ1 ξ1 × � � ˙ξ3 ξ3 � � u2 ˙ξ2 ξ2 � � u1 ˙ξ1 ξ1 � � ˙ξ3 ξ3 � � u1 ˙ξ1 ξ1 � � ˙ξ3 ξ3 ⇐⇒ when ξ2 = 1 when ξ2 = 0 FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Subsystem decomposition of the second-order chained form by using ξ2’s state transitions between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Step 1 Transit ξ2 from any initial value to 1 by using u1(t) = 0, u2(t) = q2(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Step 2 Transit ξ3 from any initial value to any desired value (in conjunction with it, ξ1 is also driven) by using u1(t) = q3(t), u2(t) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Step 3 Transit ξ2 from 1 to 0 by using u1(t) = 0, u2(t) = q2(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Step 4 Transit ξ1 from any value in Step 2 to any de- sired value by using u1(t) = q1(t), u2(t) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Step 5 Transit ξ2 from 0 to any desired value by using u1(t) = 0, u2(t) = q2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' A control input in Step k (k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' , 5) is designed by an appropriate sinusoidal function qi(t) (i = 1, 2, 3) without any feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' This control strategy is namely mo- tion planning, which naturally cannot deal with disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Therefore, we provide a trajectory tracking controller that follow the reference trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Consider to drive the state variables ξi(t), ˙ξi(t) of the system (1) by the following sinusoidal functions with pe- riod T = 2π/ω and amplitude ak: qi(t) = akω2 sin ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' (6) Then, at time t (≤ kT), trajectories of a subsystem with non- zero input are derived as ˙ξi(t) = ˙ξi((k − 1)T) − akω cos ωt + akω, (7) ξi(t) = ξi((k − 1)T) + ˙ξi((k − 1)T)t − ˙ξi((k − 1)T)(k − 1)T − ak sin ωt + akωt − ak(k − 1)ωT, (8) respectively, where ξi((k − 1)T) and ˙ξi((k − 1)T) are initial values of the state variables in Step k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Thus, at the end of k-th period (t = kT), the state transitions are represented as ˙ξi(kT) = ˙ξi((k − 1)T), (9) ξi(kT) = ξi((k − 1)T) + ˙ξi((k − 1)T)T + 2πak, (10) which means that a displacement of 2πak on ξi is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' This can be seen that the desired displacement is extracted by using the amplitude ak as a tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' By setting the trajectories (6), (7), (8) as reference trajec- tories qref i (t), ξref i (t), ˙ξref i (t), a PD feedback control system can be designed for trajectory tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' A linear system of a double integrator can be represented in the following state- space form with the state zi = [ξi, ˙ξi]⊤ and control input qi: ˙zi = � 0 1 0 0 � � �� � A zi + � 0 1 � ���� b qi(t, zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' (11) In Step k, a feedback controller for trajectory tracking to zref i is given as follows: qi(t, zi) = qref i (t) + k ei, (12) where ei := zref i − zi and k = [kp, kd] is a feedback gain matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The system (11) yields the closed-loop system ˙ei = (A − bk)ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' By choosing the feedback gain k so that (A − bk) is Hurwitz-stable, the closed-loop system is stabilized, that is, zi tracks zref i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' NUMERICAL EXPERIMENTS In this section, we evaluate the effectiveness of the proposed control approach through numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Firstly, we validate the proposed controller for the second- order chained form system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' A numerical experiment was per- formed with T = 1 s, ξ(0) = [3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='5, 1]⊤, ˙ξ(0) = 03, ξ⋆ = [1, 1, 0]⊤, and ˙ξ⋆ = 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' 2 shows the simulation results when choosing a1 = 1/(4π), a2 = a3 = a4 = −1/(2π), and a5 = 1/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The ordinary differential equations was numerically solved by ODE45 of MATLAB [16] with a rela- tive tolerance of 1×10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The results indicate that each state reached to the target value ξ⋆ with the remaining errors at t = 5T: ξ(5T)−ξ⋆ = [−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='7×10−8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='0×10−10, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='7×10−8]⊤ and ˙ξ(5T)− ˙ξ⋆ = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='3×10−8, −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='9×10−9, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='1×10−8]⊤, which means that the desired control is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Secondly, the proposed controller is applied to an underac- tuated manipulator—a typical example of second-order non- holonomic systems—as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' This manipulator has first two joints being actuated and the last joint being unactuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The system representation can be converted to the second-order chained form system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Even if the third joint cannot be driven due to no actuator, the acceleration (α1, α2) acting on the center of percussion of the third link can be treated equivalently as a control input owing to dynamic cou- pling effect—the rotational actuation of the first and second 4 VOLUME 4, 2016 TEEEAccesSNakayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' : Preparation of Papers for IEEE Access TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Definition of variables and parameters (x, y) : position of the center of percussion of the third link in the frame O-XY ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' θ : angle of the third link relative to X-axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' d3 : distance between the third joint and the center of mass of the third link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' m3 : mass of the third link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' I3 : moment of inertia mass of the third link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' LCoP : distance between the third joint and the center of percussion of the third link � LCoP := (I3 + m3d2 3)/(m3d3) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' α1 : translational acceleration along the third link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' α2 : angular acceleration around the center of percussion of the third link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Simulation results of trajectory tracking control joints propagates through the links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' For simplicity, assume that there is no disturbance such as load, friction, linear and nonlinear damping, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The main variables are defined as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Let χ := [x, y, θ]⊤ and α = [α1, α2]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Yoshikawa, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' [11] provided a set of coordinate and input transforma- tions to convert the manipulator dynamics derived from the Lagrange’s equation of motion into the following system 𝜃 1st revolute joint (actuated) 𝑥 𝑑!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' 2nd revolute joint (actuated) 3rd revolute joint (unactuated) 𝑦 𝑋 𝑌 𝑂 center of percussion of 3rd link : 𝛼" 𝛼# FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' A three-joint manipulator with passive third joint representation: ¨χ = � � cos θ 0 sin θ 0 0 1 � � α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' (13) Using the coordinate transformation � � ξ1 ξ2 ξ3 � � = � � x − LCoP tan θ y � � , � � ˙ξ1 ˙ξ2 ˙ξ3 � � = � � ˙x ˙θ sec2 θ ˙y � � (14) and the input transformation � α1 α2 � = � u1 sec θ u2 cos2 θ − 2 ˙θ2 tan θ � , (15) the system (13) can be transformed into the second-order chained form system (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Note that both transformation are singular point at θ = ±π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' For the third joint of the underactuated manipulator with m3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='6 kg, d3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='3 m, and I3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='5 × 10−3 kg · m2, steer from initial values χ(0) = [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='33 m, 1 m, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='6 × 10−1 rad]⊤, ˙χ(0) = 03 to the desired ones χ⋆ = [1 m, 0 m, 0 rad]⊤, ˙χ⋆ = 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' 4 shows a simulation result with the period T = 1 s and the feedback gain kp = kd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' In this case, from (14), we have ξ(0) = [3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='5, 1]⊤ and ξ⋆ = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='67, 0, 0]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' It can be confirmed that each state converges to the desired value in the both system representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' VOLUME 4, 2016 5 TEEEAccesSNakayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' : Preparation of Papers for IEEE Access (a) States and inputs of the second-order chained form system (b) Status and inputs of a three-joint underactuated manipulator FIGURE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Numerical results Furthermore, to verify the effect of feedback control, an- other case with an initial value error was simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' For a rest-to-rest motion from χ(0) = [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='33 m, 1 m, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='6 × 10−1 rad]⊤ to χ⋆ = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='33 m, 0 m, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='8 × 10−1 rad]⊤ with the zero velocities, the initial value error of +10% is given to θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=', χ(0) = [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='33 m, 1 m, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='1 × 10−1 rad]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The dashed lines indicate the target trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' It can be observed that tracking error due to the initial value error is alleviated over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Similarly, when initial value errors of ±1%, ±10%, and ±30% on θ are given the tracking errors at the end of control at t = 5T are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The terminal values of the tracking errors do not increase greatly even if the magnitude of the initial value error increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Consequently, it is confirmed that the feedback of trajectory tracking has a sufficient effect on initial value errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Note that the terminal error on x is relatively larger than the one on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The proposed control method attempts to settle the system by focusing on a single state every step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' In addition, the state in which the con- trol step ends has no chance to be controlled directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' For such a state, there can be a secondary state transition that yields in control steps that focus on the other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Therefore, if a state fails to converge into its reference trajectory within the control step due to initial value error or disturbance, it behaves unexpectedly until the end of the control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' In particular, ξ2—the state used for switching the systems— has a negative effect on the other states because the reference trajectory is not computed correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Furthermore, the error remaining in the velocity state (ξ4, ξ5, ξ6) causes a drift in the position state (ξ1, ξ2, ξ3) even if the input is zero in the following control steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' This is explained by numerical experiments shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Note that θ is related to ξ2 as specified in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' This means that θ affects the other states (x, y) when not converging completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' On the other hand, since ξ2 is settled in the final step (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=', Step 5), the propagation from the error in the velocity state is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Therefore, the error remaining in θ is considered to be smaller than in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' CONCLUSION In this paper, a novel control approach composed of sinu- soidal reference trajectories and a simple trajectory tracking 6 VOLUME 4, 2016 TEEEAccesSNakayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' : Preparation of Papers for IEEE Access FIGURE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Given an initial value error(+10%) controller for the second-order chained form system was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The key idea is a subsystem decomposition of the second-order chained form system by using state transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The effectiveness of the proposed algorithm was demon- strated by numerical results including an application to a three-joint underactuated manipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' In particular, it can be confirmed that the feedback control works well against the initial value error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' The future work of this research is to verify the proposed approach via experiments on an actual robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' REFERENCES [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Brockett: “Asymptotic stability and feedback stabilization,” in Differential Geometric Control Theory (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Brockett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Millmann and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Sussmann), Birkhauser, Boston, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' 181–191, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Hauser, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Sastry, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Meyer: “Nonlinear control design for slightly TABLE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Error from target value by the initial value error Case χ(0) − χ⋆ χ(5T) − χ⋆ w/o init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' � � 0 m 0 m 0 rad � � � � 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='5 × 10−8 m 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='4 × 10−7 m 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='9 × 10−9 rad � � w/ +1 % init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' � � 0 m 0 m 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='6 × 10−3 rad � � � � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='9 × 10−3 m −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='9 × 10−4 m −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='5 × 10−4 rad � � w/ −1 % init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' � � 0 m 0 m −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='6 × 10−3 rad � � � � −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='9 × 10−3 m 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='4 × 10−4 m 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='5 × 10−4 rad � � w/ +10 % init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' � � 0 m 0 m 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='6 × 10−2 rad � � � � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='0 × 10−2 m −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='8 × 10−3 m −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='8 × 10−3 rad � � w/ −10 % init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' � � 0 m 0 m −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='6 × 10−2 rad � � � � −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='9 × 10−2 m 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='9 × 10−3 m 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='3 × 10−3 rad � � w/ +30 % init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' � � 0 m 0 m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='4 × 10−1 rad � � � � 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='1 × 10−2 m 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='3 × 10−3 m −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='8 × 10−2 rad � � w/ −30 % init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' � � 0 m 0 m −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='4 × 10−1 rad � � � � −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='5 × 10−2 m 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='3 × 10−2 m 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Marchand: “Bounded control of a general extended chained form systems,” in Proceedings of the 53rd IEEE Conference on Decision and Control (CDC’14), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' 6342–6347, Los Angeles, CA, USA, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' [16] MathWorks, “ode45: Solve nonstiff differential equations—medium order method,” Documentation for MATLAB R2022b, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Avail- able: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='mathworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='com/help/matlab/ref/ode45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Accessed on: Jan 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' MAYU NAKAYAMA was born in Kiyosu, Aichi, Japan in 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' She received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' de- grees in information science and technology from Aichi Prefectural University (APU), Nagakute, Aichi, Japan, in 2020 and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' She is currently with DENSO Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' Her research interests include nonlinear control for underactuated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' MASAHIDE ITO (M’10) was born in Nagoya, Aichi, Japan in 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' He received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=', and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' degrees in information science and tech- nology from Aichi Prefectural University (APU), Nagakute, Aichi, Japan, in 2002, 2004, and 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' He is currently an Associate Professor with the School of Information Science and Technol- ogy, APU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} +page_content=' His research interests include visual feedback control of robotic systems and nonlinear control for underactuated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'} 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sha256:46ac581fd420982fa9283086c636d3f1868a32f0a0d5470f835d07c5ce3817f5 +size 5701677 diff --git a/LtFLT4oBgHgl3EQfMS8S/content/tmp_files/2301.12015v1.pdf.txt b/LtFLT4oBgHgl3EQfMS8S/content/tmp_files/2301.12015v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf4f219f61034a83c5934a43b2c81b116670d2a3 --- /dev/null +++ b/LtFLT4oBgHgl3EQfMS8S/content/tmp_files/2301.12015v1.pdf.txt @@ -0,0 +1,3109 @@ +A Variant Prescribed Curvature Flow on Closed Surfaces with +Negative Euler Characteristic +Franziska Borer∗ +Peter Elbau† +Tobias Weth‡ +Abstract +On a closed Riemannian surface (M, ¯g) with negative Euler characteristic, we study the problem of +finding conformal metrics with prescribed volume A > 0 and the property that their Gauss curvatures +fλ = f + λ are given as the sum of a prescribed function f ∈ C∞(M) and an additive constant λ. Our +main tool in this study is a new variant of the prescribed Gauss curvature flow, for which we establish local +well-posedness and global compactness results. In contrast to previous work, our approach does not require +any sign conditions on f. Moreover, we exhibit conditions under which the function fλ is sign changing and +the standard prescribed Gauss curvature flow is not applicable. +Acknowledgment +This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project +408275461 (Smoothing and Non-Smoothing via Ricci Flow). +We would like to thank Esther Cabezas–Rivas for helpful discussions. +1. Introduction +Let (M, ¯g) be a two-dimensional, smooth, closed, connected, oriented Riemann manifold endowed with a smooth +background metric ¯g. A classical problem raised by Kazdan and Warner in [11] and [10] is the question which +smooth functions f : M → R arise as the Gauss curvature Kg of a conformal metric g(x) = e2u(x)¯g(x) on M +and to characterise the set of all such metrics. +For a constant function f, this prescribed Gauss curvature problem is exactly the statement of the Uni- +formisation Theorem (see e.g. [16], [12]): +There exists a metric g which is pointwise conformal to ¯g and has constant Gauss curvature Kg ≡ ¯K ∈ R. +We now use this statement to assume in the following without loss of generality that the background metric +¯g itself has constant Gauss curvature K¯g ≡ ¯K ∈ R. Furthermore we can normalise the volume of (M, ¯g) to +one. We recall that the Gauss curvature of a conformal metric g(x) = e2u(x)¯g(x) on M is given by the Gauss +equation +Kg(x) = e−2u(x)(−∆¯gu(x) + ¯K). +(1.1) +Therefore the problem reduces to the question for which functions f there exists a conformal factor u solving +the equation +− ∆¯gu(x) + ¯K = f(x)e2u(x) +in M. +(1.2) +Given a solution u, we may integrate (1.2) with respect to the measure µ¯g on M induced by the Riemannian +volume form. Using the Gauss–Bonnet Theorem, we then obtain the identity +� +M +f(x)dµg(x) = +� +M +¯Kdµ¯g(x) = ¯K vol¯g = ¯K = 2πχ(M), +(1.3) +where dµg(x) = e2u(x)dµ¯g(x) is the element of area in the metric g(x) = e2u(x)¯g(x). +We note that (1.3) +immediately yields necessary conditions on f for the solvability of the prescribed Gauss curvature problem. In +particular, if ±χ(M) > 0, then ±f must be positive somewhere. Moreover, if χ(M) = 0, then f must change +sign or must be identically zero. +∗Technical University of Berlin, Faculty II—Mathematics and Natural Sciences, Straße des 17. Juni 136, 10623 Berlin, Germany +email: borer@tu-berlin.de +†Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria +email: peter.elbau@univie.ac.at +‡Goethe University Frankfurt, Institut f¨ur Mathematik, Robert-Mayer-Straße 10, 60629 Frankfurt, Germany +email: weth@math.uni-frankfurt.de +1 +arXiv:2301.12015v1 [math.AP] 27 Jan 2023 + +2 +Franziska Borer, Peter Elbau, Tobias Weth +In the present paper we focus on the case χ(M) < 0, so M is a surface of genus greater than one and +¯K < 0. The complementary cases χ(M) ≥ 0—i.e., the cases where M = S2 or M = T, the 2-torus—will be +discussed briefly at the end of this introduction, and we also refer the reader to [18, 19, 2, 8] and the references +therein. Multiplying equation (1.2) with the factor e−2u and integrating over M with respect to the measure µ¯g, +we get the following necessary condition—already mentioned by Kazdan and Warner in [11]—for the average +¯f := +1 +vol¯g +� +M f(x)dµ¯g(x), with vol¯g := +� +M dµ¯g(x): +¯f = +1 +vol¯g +� +M +f(x)dµ¯g(x) = +� +M +(−∆¯gu(x) + ¯K)e−2u(x)dµ¯g(x) += +� +M +(−2|∇¯gu(x)|2 +¯g + ¯K)e−2u(x)dµ¯g(x) < 0. +(1.4) +This condition is not sufficient. Indeed, it has already been pointed out in [11, Theorem 10.5] that in the case +χ(M) < 0 there always exist functions f ∈ C∞(M) with ¯f < 0 and the property that (1.2) has no solution. +We recall that solutions of (1.2) can be characterised as critical points of the functional +Ef : H1(M, ¯g) → R; +Ef(u) := 1 +2 +� +M +� +|∇¯gu(x)|2 +¯g + 2 ¯Ku(x) − f(x)e2u(x)� +dµ¯g(x). +(1.5) +Under the assumption χ(M) < 0, i.e., ¯K < 0, the functional Ef is strictly convex and coercive on H1(M, ¯g) +if f ≤ 0 and f does not vanish identically. Hence, as noted in [7], the functional Ef admits a unique critical +point uf ∈ H1(M, ¯g) in this case, which is a strict absolute minimiser of Ef and a (weak) solution of (1.2). +The situation is more delicate in the case where fλ = f0 + λ, where f0 ≤ 0 is a smooth, nonconstant function +on M with maxx∈M f0(x) = 0, and λ > 0. In the case where λ > 0 sufficiently small (depending on f0), it was +shown in [7] and [1] that the corresponding functional Efλ admits a local minimiser uλ and a further critical +point uλ ̸= uλ of mountain pass type. +These results motivate our present work, where we suggest a new flow approach to the prescribed Gausss +curvature problem in the case χ(M) < 0. It is important to note here that there is an intrinsic motivation to +formulate the static problem in a flow context. Typically, elliptic theories are regarded as the static case of the +corresponding parabolic problem; in that sense, many times the better-understood elliptic theory has been a +source of intuition to generalise the corresponding results in the parabolic case. Examples of this feedback are +minimal surfaces/mean curvature flow, harmonic maps/solutions of the heat equation, and the uniformisation +theorem/the two-dimensional normalised Ricci flow. +In this spirit, a flow approach to (1.2), the so-called prescribed Gauss curvature flow, was first introduced +by Struwe in [18] (and [2]) for the case M = S2 with the standard background metric and a positive function +f ∈ C2(M). More precisely, he considers a family of metrics (g(t, ·))t≥0 which fulfils the initial value problem +∂tg(t, x) = 2(α(t)f(x) − Kg(t,·)(x))g(t, x) +in (0, T) × M; +(1.6) +g(0, x) = g0(x) +on {0} × M, +(1.7) +with +α(t) = +� +M Kg(t,·)(x)dµg(t,·)(x) +� +M f(x)dµg(t,·)(x) += +2πχ(M) +� +M f(x)dµg(t,·)(x). +(1.8) +This choice of α(t) ensures that the volume of (M, g(t, ·)) remains constant throughout the deformation, i.e., +� +M +dµg(t,·)(x) = +� +M +e2u(t,x)dµ¯g(x) ≡ volg0 +for all t ≥ 0, +where g0 denotes the initial metric on M. +Equivalently one may consider the evolution equation for the +associated conformal factor u given by g(t, x) = e2u(t,x)¯g(x): +∂tu(t, x) = α(t)f(x) − Kg(t,·)(x) +in (0, T) × M; +(1.9) +u(0, x) = u0(x) +on {0} × M. +(1.10) +Here the initial value u0 is given by g0(x) = e2u0(x)¯g(x). The flow associated to this parabolic equation is +usually called the prescribed Gauss curvature flow. With the help of this flow, Struwe [18] provided a new proof +of a result by Chang and Yang [6] on sufficient criteria for a function f to be the Gauss curvature of a metric +g(x) = e2u(x)gS2(x) on S2. He also proved the sharpness of these criteria. +In the case of surfaces with genus greater than one, i.e., with negative Euler characteristic, the prescribed +Gauss curvature flow was used by Ho in [9] to prove that any smooth, strictly negative function on a surface +with negative Euler characteristic can be realised as the Gaussian curvature of some metric. More precisely, + +Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic +3 +assuming that χ(M) < 0 and that f ∈ C∞(M) is a strictly negative function, he proves that equation (1.9) has +a solution which is defined for all times and converges to a metric g∞ with Gaussian curvature Kg∞ satisfying +Kg∞(x) = α∞f(x) +for some constant α∞. +While the prescribed Gauss curvature flow is a higly useful tool in the cases where f is of fixed sign, it +cannot be used in the case where f is sign-changing. Indeed, in this case we may have +� +M f(x)dµg(t,·)(x) = 0 +along the flow and then the normalising factor α(t) is not well-defined by (1.8). As a consequence, a long-time +solution of (1.9) might not exist. In particular, the static existence results of [7] and [1] can not be recovered +and reinterpreted with the standard prescribed Gauss curvature flow. +In this paper we develop a new flow approach to (1.2) in the case χ(M) < 0 for general f ∈ C∞(M), which +sheds new light on the results in [7], [1] and [9]. The main idea is to replace the multiplicative normalisation in +(1.9) by an additive normalisation, as will be described in details in the next chapter. +At this point, it should be noted that the normalisation factor α(t) in the prescribed Gauss curvature flow +given by (1.8) is also not the appropriate choice in the case of the torus, where, as noted before, f has to +change sign or be identically zero in order to arise as the Gauss curvature of a conformal metric. The case of +the torus was considered by Struwe in [19], where, in particular, he used to a flow approach to reprove and +partially improve a result by Galimberti [8] on the static problem. In this approach, the normalisation in (1.8) +is replaced by +α(t) = +� +M f(x)Kg(t,·)(x)dµg(t,·)(x) +� +M f 2(x)dµg(t,·)(x) +. +(1.11) +With this choice, Struwe shows that for any smooth +u0 ∈ C∗ := +� +u ∈ H1(M, ¯g) | +� +M +f(x)e2u(x)dµ¯g(x) = 0, +� +M +e2u(x)dµ¯g(x) = 1 +� +there exists a unique, global smooth solution u of (1.9) satisfying u(t, ·) ∈ C∗ for all t > 0. Moreover, u(t, ·) → +u∞(·) in H2(M, ¯g) (and smoothly) as t → ∞ suitably, where u∞ + c∞ is a smooth solution of (1.2) for some +c∞ ∈ R. +In principle, the normalisation (1.11) could also be considered in the case χ(M) < 0, but then the flow is not +volume-preserving anymore, which results in a failure of uniform estimates for solutions of (1.9). Consequently, +we were not able to make use of the associated flow in this case. +The paper is organised as follows. In Section 2 we set up the framework for the new variant of the prescribed +Gauss curvature flow with additive normalisation, and we collect basic properties of it. In Section 3, we then +present our main result on the long-time existence and convergence of the flow (for suitable times tk → ∞) to +solutions of the corresponding static problem. In particular, our results show how sign changing functions of +the form fλ = f0 + λ arise depending on various assumptions on the shape of f0 and on the fixed volume A of +M with respect to the metric g(t). Before proving our results on the time-dependent problem, we first derive, +in Section 4, some results on the static problem with volume constraint. Most of these results will then be used +in Section 5, where the parabolic problem is studied in detail and the main results of the paper are proved. In +the appendix, we provide some regularity estimates and a variant of a maximum princple for a class of linear +evolution problems with H¨older continuous coefficients. +In the remainder of the paper, we will use the short form f, g(t), u(t), Kg(t), volg(t) := +� +M dµg(t) = +� +M e2u(t)dµ¯g, and so on instead of f(x), g(t, x), u(t, x), Kg(t,·)(x), +� +M dµg(t,·)(x) = +� +M e2u(t,x)dµ¯g(x), et cetera. +2. A New Flow Approach and Some of its Properties +Let f ∈ C∞(M) be a smooth function. We consider now the additive rescaled prescribed Gauss curvature flow +given by +∂tu(t) = f − Kg(t) − α(t) = f − e−2u(t)(∆¯gu(t) − ¯K) − α(t) +in (0, T) × M, +(2.1) +where α(t) is chosen such that the volume volg(t) of M with respect to g(t) = e2u(t)¯g remains constant along +the flow, that is, we require the condition +1 +2 +d +dt volg(t) = +� +M +∂tu(t)dµg(t) = +� +M +(f − Kg(t) − α(t))dµg(t) = 0. +(2.2) +Solving for α(t) then we find +α(t) = +1 +volg(t) +�� +M +fdµg(t) − ¯K +� +. + +4 +Franziska Borer, Peter Elbau, Tobias Weth +So, starting with +u0 ∈ Cp,A := +� +v ∈ W 2,p(M, ¯g) | +� +M +e2vdµ¯g = A +� +, +p > 2, +for a given A > 0, we have +volg(t) = volg(0) = volg0 = A, +for all t ≥ 0, +hence we can define +αA(t) = 1 +A +�� +M +fdµg(t) − ¯K +� +. +(2.3) +Therefore in the following we consider the flow +∂tu(t) = f − Kg(t) − αA(t) +in (0, T) × M; +(2.4) +u(0) = u0 ∈ Cp,A +on {0} × M, +(2.5) +with αA(t) is chosen like in (2.3). We can now state some first properties of the flow. +Proposition 2.1. Let u be a (sufficiently smooth) solution of (2.4), (2.5). Then +1. the volume volg(t) of (M, g(t)) is preserved along the flow, i.e., volg(t) ≡ volg0 = A for all t ≥ 0; +2. along this trajectory, we have a uniform bound for α given by +α(t) ≥ min +x∈M f(x) + | ¯K| +A =: α1 > −∞ +(2.6) +and +α(t) ≤ max +x∈M f(x) + | ¯K| +A =: α2 < ∞; +(2.7) +3. the flow is invariant under adding or subtracting a constant C > 0 to the function f; +4. and the energy Ef, defined in (1.5), is decreasing in time along the flow, so +Ef(u(t)) ≤ Ef(u0) +for all t ≥ 0. +Proof. The first statement directly follows by (2.2) and the choice of α in (2.3). +The second one we get since f is smooth and volg(t) = A. +To show the invariance of the flow, let C > 0 be a constant. We then replace f by f ± C in (2.4) and see that +f ± C − Kg(t) − 1 +A +�� +M +(f ± C)dµg(t) − ¯K +� += f − Kg(t) − 1 +A +�� +M +fdµg(t) − ¯K +� += ∂tu(t). +So, the flow (2.4) is left unchanged if we replace f by f ± C for a constant C > 0. +To see that the energy Ef is decreasing along the flow, we use (2.2) and get +d +dtEf(u(t)) = +� +M +(−∆¯gu(t) + ¯K − fe2u(t))∂tu(t)dµ¯g += +� +M +((−∆¯gu(t) + ¯K)e−2u(t) − f)e2u(t)∂tu(t)dµ¯g += +� +M +((−∆¯gu(t) + ¯K)e−2u(t) − f)∂tu(t)dµg(t) += +� +M +(Kg(t) − f)∂tu(t)dµg(t) = +� +M +(Kg(t) − f + α(t))∂tu(t)dµg(t) += − +� +M +|∂tu(t)|2dµg(t) ≤ 0. +(2.8) +Therefore on an interval [0, T], we have the uniform a-priori bound +Ef(u(T)) + +� T +0 +� +M +|∂tu(t)|2dµg(t)dt = Ef(u(0)) +(2.9) +for any T > 0. + +Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic +5 +3. Main Results +The following is our first main result. +Theorem 3.1. Let f ∈ C∞(M), p > 2, and u0 ∈ Cp,A for a given A > 0. Then the initial value problem (2.4), +(2.5) admits a unique global solution u ∈ C([0, ∞); C(M)) ∩ C([0, ∞); H1(M, ¯g)) ∩ C∞((0, ∞) × M) satisfying +the energy bound Ef(u(t)) ≤ Ef(u0) for all t. +Moreover, u is uniformly bounded in the sense that +sup +t>0 +∥u(t)∥L∞(M,¯g) < ∞. +Furthermore, as t → ∞ suitably, u converges to a function u∞ in H2(M, ¯g) solving the equation +− ∆¯gu + ¯K = fλe2u +in M, +(3.1) +where fλ := f + λ with +λ = 1 +A +� +¯K − +� +M +fe2u∞dµ¯g +� +. +(3.2) +In other words, u∞ induces a metric g∞ with Gauss curvature Kg∞ satisfying +Kg∞(x) = fλ(x) = f(x) + λ +for +x ∈ M. +(3.3) +Remark 3.2. For functions f < 0, the convergence of the flow (1.9) is shown in [9]. For the additive rescaled +flow (2.4) with initial data (2.5) we get convergence for arbitrary functions f ∈ C∞(M). In general we do not +have any information about λ and therefore no information about the sign of fλ in Theorem 3.1. On the other +hand, more information can be derived for certain functions f ∈ C∞(M) and certain values of A > 0. +(i) In the case where A ≤ − +¯ +K +∥f∥L∞(M,¯g) , it follows that +λ = 1 +A +� +¯K − +� +M +fe2udµ¯g +� +≤ +¯K +A + ∥f∥L∞(M,¯g) +A +� +M +e2udµ¯g = +¯K +A + ∥f∥L∞(M,¯g) ≤ 0 +for every solution u ∈ C2,A := +� +v ∈ H2(M, ¯g) | +� +M e2vdµ¯g = 0 +� +of the static problem (3.1), and therefore +this also applies to λ in Theorem 3.1 in this case. +(ii) The following theorems show that fλ in Theorem 3.1 may change sign if A > − +¯ +K +∥f∥L∞(M,¯g) , so in this case +we get a solution of the static problem (1.2) for sign-changing functions f ∈ C∞(M) by using the additive +rescaled prescribed Gauss curvature flow (2.4). +Theorem 3.3. Let p > 2. For every A > 0 and c > − +¯ +K +A there exists ε = ε(c, A, ¯K) > 0 with the following +property. +If u0 ≡ 1 +2 log(A) ∈ Cp,A and f ∈ C∞(M) with −c ≤ f ≤ 0 and ∥f + c∥L1(M,¯g) < ε is chosen in Theorem 3.1, +then the value λ defined in (3.2) is positive. +In particular, if f has zeros on M, then fλ in (3.3) is sign changing. +Under fairly general assumptions on f, we can prove that λ > 0 if A is sufficiently large and u0 ∈ Cp,A is +chosen suitably. +Theorem 3.4. Let f ∈ C∞(M) be nonconstant with maxx∈M f(x) = 0. Then there exists κ > 0 with the +property that for every A ≥ κ there exists u0 ∈ Cp,A such that the value λ defined in (3.2) is positive. +In fact we have even more information on the associated limit u∞ in this case, see Corollary 4.8 below. +It remains open how large λ can be depending on A and f. The only upper bound we have is +λ < − +� +M +fdµ¯g, +(3.4) +since we must have +¯fλ = +1 +vol¯g +� +M +fλdµ¯g = +� +M +fdµ¯g + λ +!< 0, +so that fλ fulfills the necessary condition (1.4) provided by Kazdan and Warner in [11]. + +6 +Franziska Borer, Peter Elbau, Tobias Weth +4. The static Minimisation Problem with Volume Constraint +To obtain additional information on the limiting function u∞ and the value λ ∈ R associated to it by (3.2) and +(3.3), we need to consider the associated static setting for the prescribed Gauss curvature problem with the +additional condition of prescribed volume. +Before going into the details of this static problem, we recall an important and highly useful estimate. +The following lemma (see e.g. [5, Corollary 1.7]) is a consequence of the Trudinger’s inequality [20] which was +improved by Moser in [15] (for more details see e.g. [19, Theorem 2.1 and Theorem 2.2]): +Lemma 4.1. For a two-dimensional, closed Riemannian manifold (M, ¯g) there are constants η > 0 and CMT > +0 such that +� +M +e(u−¯u)dµ¯g ≤ CMT exp +� +η∥∇¯gu∥2 +L2(M,¯g) +� +(4.1) +for all u ∈ H1(M, ¯g) where +¯u := +1 +vol¯g +� +M +u dµ¯g = +� +M +u dµ¯g, +in view of our assumption that vol¯g = 1. +As a consequence of Lemma 4.1, we have +� +M +epudµ¯g = ep¯u +� +M +e(pu− ¯ +pu)dµ¯g ≤ ep¯uCMT exp +� +η∥∇¯g(pu)∥2 +L2(M,¯g) +� +< ∞ +for every u ∈ H1(M, ¯g) and p > 0. Consequently, for a given A > 0, the set +C1,A := +� +u ∈ H1(M, ¯g) | V (u) := +� +M +e2udµ¯g = A +� +(4.2) +is well defined and coincides with the closure of C2,A with respect to the H1-norm. We also note that +¯u ≤ 1 +2 log(A) +for u ∈ C1,A, +(4.3) +since by Jensen’s inequality and our assumption that vol¯g = 1 we have +2¯u = − +� +M +2udµ¯g = +� +M +2udµ¯g ≤ log +� +− +� +e2udµ¯g +� += log(A) +for u ∈ C1,A. +Furthermore we want to recall the Gagliardo–Nirenberg–Ladyˇzhenskaya interpolation, see e.g. [4]. +Lemma 4.2 (Gagliardo–Nirenberg–Ladyˇzhenskaya inequality). There exists a constant CGNL > 0 such that +we have for every ζ ∈ H1(M, ¯g) the inequality +∥ζ∥4 +L4(M,¯g) ≤ CGNL∥ζ∥2 +L2(M,¯g)∥ζ∥2 +H1(M,¯g). +Now we enter the details of the static prescribed Gauss curvature problem with volume constraint. In this +problem, we wish to find, for given f ∈ C∞(M) and A > 0, critical points of the restriction of the functional +Ef defined in (1.5) to the set C1,A. A critical point u ∈ C1,A of this restriction is a solution of (3.1) for some +λ ∈ R, where, here and in the following, we put again fλ := f + λ ∈ C∞(M). In other words, such a critical +point induces, similarly as the limit u∞ in Theorem 3.1, a metric gu with Gauss curvature Kgu satisfying +Kgu(x) = fλ(x) = f(x) + λ. The unknown λ ∈ R arises in this context as a Lagrangian multiplier and is a +posteriori characterised again by +λ = 1 +A +� +¯K − +� +M +fe2udµ¯g +� +. +In the study of critical points of the restriction of Ef to C1,A, it is natural to consider the minimisation +problem first. For this we set +mf,A = +inf +u∈C1,A Ef(u). +We have the following estimates for mf,A: +Lemma 4.3. Let f ∈ C∞(M), A > 0. Then we have +mf,A ≤ 1 +2 +� +¯K log(A) − A +� +M +fdµ¯g +� +. +(4.4) +Moreover, if max f ≥ 0, then we have +lim sup +A→∞ +mf,A +A +≤ 0. +(4.5) + +Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic +7 +Proof. Let u0(A) ≡ 1 +2 log(A), so that +� +M e2u0(A)dµ¯g = A. Hence u0(A) is the (unique) constant function in +C1,A, and +mf,A ≤ Ef(u0(A)) = 1 +2 +� +M +(|∇¯gu0(A)|2 +¯g + 2 ¯Ku0(A) − fe2u0(A))dµ¯g += 1 +2 +� +M +( ¯K log(A) − fA)dµ¯g += 1 +2 +� +¯K log(A) − A +� +M +fdµ¯g +� +. +This shows (4.4). To show (4.5), we let ε > 0. Since f ∈ C∞(M) and max f ≥ 0 by assumption, there exists +an open set Ω ⊂ M with f ≥ −ε on Ω. Next, let ψ ∈ C∞(M), ψ ≥ 0, be a function supported in Ω and with +∥ψ∥L∞(M,¯g) = 2. Consequently, the set Ω′ := {x ∈ M | ψ > 1} is a nonempty open subset of Ω, and therefore +µ¯g(Ω′) > 0. +Next we consider the continuous function +h : [0, ∞) → [0, ∞); +h(τ) = +� +M +e2τψdµ¯g +and we note that h(0) = +� +M dµ¯g = 1, and that +h(τ) ≥ +� +Ω′ e2τψdµ¯g ≥ e2τµ¯g(Ω′) +for τ ≥ 0. +Hence for every A ≥ 1 there exists +0 ≤ τA ≤ 1 +2 +� +log(A) − log(µ¯g(Ω′)) +� +(4.6) +with h(τA) = A and therefore τAψ ∈ C1,A. Consequently, +mf,A ≤ Ef(τAψ) = 1 +2 +� +M +(|∇¯gτAψ|2 +¯g + 2 ¯KτAψ − fe2τAψ)dµ¯g += τ 2 +Ac1 − τAc2 − c3 − 1 +2 +� +Ω +fe2τAψdµ¯g +with +c1 = 1 +2 +� +M +|∇¯gψ|2 +¯gdµ¯g, +c2 = − ¯K +� +M +ψdµ¯g +and +c3 = 1 +2 +� +M\Ω +fdµ¯g. +Since f ≥ −ε on Ω, we thus deduce that +mf,A ≤ τ 2 +Ac1 − 2τAc2 + c3 + ε +2 +� +Ω +e2τAψdµ¯g ≤ τ 2 +Ac1 − 2τAc2 + c3 + εA +2 . +Since τA +A → 0 as A → ∞ by (4.6), we conclude that +lim sup +A→∞ +mf,A +A +≤ ε +2. +Since ε > 0 was chosen arbitrarily, (4.5) follows. +Lemma 4.4. Let f ∈ C∞(M) nonconstant with maxx∈M f(x) = 0. For every ε > 0 there exists κ0 > 0 with +the following property. If A ≥ κ0 and u ∈ C1,A is a solution of +− ∆¯gu + ¯K = (f + λ)e2u +(4.7) +for some λ ∈ R with Ef(u) < εA +2 , then we have λ < ε. +Proof. For given ε > 0, we may choose κ0 > 0 sufficiently large so that | ¯ +K| +2 +log(A) +|A| +< ε +2 for A ≥ κ0. +Now, let A ≥ κ0, and let u ∈ C1,A be a solution of (4.7) satisfying Ef(u) < εA +2 . Integrating (4.7) over M +with respect to µ¯g and using that vol¯g(M) = 1 and +� +M e2udµ¯g = A, we obtain +λ = 1 +A +� +¯K − +� +M +fe2udµ¯g +� +≤ − 1 +A +� +M +fe2udµ¯g += 1 +A +� +Ef(u) − 1 +2 +� +M +(|∇¯gu|2 +¯g + 2 ¯Ku)dµ¯g +� +≤ 1 +A +� +Ef(u) + | ¯K|¯u +� +≤ ε +2 + | ¯K| +2 +log(A) +A +< ε, +as claimed. Here we used (4.3) to estimate ¯u. + +8 +Franziska Borer, Peter Elbau, Tobias Weth +Proposition 4.5. Let f ∈ C∞(M) be a nonconstant function with maxx∈M f(x) = 0. Moreover, let λn → 0+ +for n → ∞, and let (un)n∈N be a sequence of solutions of +− ∆¯gun + ¯K = (f + λn)e2un +in M +(4.8) +which are weakly stable in the sense that +� +M +(|∇¯gh|2 +¯g − 2(f + λn)e2unh2)dµ¯g ≥ 0 +for all h ∈ H1(M). +(4.9) +Then un → u0 in C2(M), where u0 is the unique solution of +− ∆¯gu0 + ¯K = fe2u0 +in M. +(4.10) +Proof. We only need to show that +(un)n∈N is bounded in C2,α(M) for some α > 0. +(4.11) +Indeed, assuming this for the moment, we may complete the argument as follows. Suppose by contradiction +that there exists ε > 0 and a subsequence, also denoted by (un)n∈N, with the property that +∥un − u0∥C2(M) ≥ ε +for all n ∈ N. +(4.12) +By (4.11) and the compactness of the embedding C2,α(M) �→ C2(M), we may then pass to a subsequence, still +denoted by (un)n∈N, with un → u∗ in C2(M) for some u∗ ∈ C2(M). Passing to the limit in (4.8), we then see +that u∗ is a solution of (4.10), which by uniqueness implies that u∗ = u0. This contradicts (4.12), and thus the +claim follows. +The proof of (4.11) follows by similar arguments as in [7, p. 1063 f.]. Since the framework is slightly different, +we sketch the main steps here for the convenience of the reader. We first note that, by the same argument as +in [7, p. 1063 f.], there exists a constant C0 > 0 with +un ≥ −C0 +for all n. +(4.13) +Since {f < 0} is a nonempty open subset of M by assumption, we may fix a nonempty open subdomain +Ω ⊂⊂ {f < 0}. By [1, Appendix], there exists a constant C1 > 0 with +∥u+ +n ∥H1(Ω,¯g) ≤ C1 +for all n +and therefore +� +Ω +e2undµ¯g ≤ +� +Ω +e2u+ +n dµ¯g ≤ C2 +for all n +(4.14) +for some C2 > 0 by the Moser–Trudinger inequality. +Next, we consider a nontrivial, nonpositive function +h ∈ C∞ +c (Ω) ⊂ C∞(M) and the unique solution w ∈ C∞(M) of the equation +−∆¯gw + ¯K = he2w +in M. +Moreover, we let wn := un − w, and we note that wn satisfies +−∆¯gwn + he2w = (f + λn)e2un +in M. +Multiplying this equation by e2wn and integrating by parts, we obtain +� +M +(f + λn)e2(un+wn)dµ¯g = +� +M +� +−∆¯gwn + he2w� +e2wndµ¯g = +� +M +� +2e2wn|∇¯gwn|2 +¯g + he2(w+wn)� +dµ¯g += 2 +� +M +|∇¯gewn|2 +¯gdµ¯g + +� +Ω +he2undµ¯g. +(4.15) +Moreover, applying (4.9) to h = ewn gives +� +M +(f + λn)e2(un+wn)dµ¯g ≤ 1 +2 +� +M +|∇¯gewn|2 +¯gdµ¯g. +(4.16) +Combining (4.14), (4.15) and (4.16) yields +∥∇¯gewn∥2 +L2(M,¯g) ≤ −2 +3 +� +Ω +he2undµ¯g ≤ 2 +3∥h∥L∞(M,¯g)C2 +for all n. +(4.17) + +Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic +9 +Next we claim that also ∥ewn∥L2(M,¯g) remains uniformly bounded. Suppose by contradiction that +∥ewn∥L2(M,¯g) → ∞ +as n → ∞. +(4.18) +We then set vn := +ewn +∥ewn∥L2(M,¯g) , and we note that +∥vn∥L2(M,¯g) = 1 +for all n +and +∥∇¯gvn∥2 +L2(M,¯g) → 0 +as n → ∞ +(4.19) +by (4.17). Consequently, we may pass to a subsequence satisfying vn ⇀ v in H1(M, ¯g), where v is a constant +function with +∥v∥L2(M,¯g) = 1. +(4.20) +However, since +∥ewn∥L2(Ω,¯g) ≤ ∥eun∥L2(Ω,¯g)∥e−w∥L∞(Ω,¯g) ≤ +� +C2∥e−w∥L∞(Ω,¯g) +for all n ∈ N +by (4.14) and therefore +∥v∥L2(Ω,¯g) = lim +n→∞ ∥vn∥L2(Ω,¯g) = lim +n→∞ +∥ewn∥L2(Ω,¯g) +∥ewn∥L2(M,¯g) += 0 +by (4.18), we conclude that the constant function v must vanish identically, contradicting (4.20). +Consequently, ∥ewn∥L2(M,¯g) remains uniformly bounded, which by (4.17) implies that ewn remains bounded +in H1(M, ¯g) and therefore in Lp(M, ¯g) for any p < ∞. Since eun ≤ ∥ew∥L∞(M,¯g)ewn on M for all n ∈ N, it thus +follows that also eun remains bounded in Lp(M, ¯g) for any p < ∞. Moreover, by (4.13), the same applies to +the sequence un itself. Therefore, applying successively elliptic Lp and Schauder estimates to (4.8), we deduce +(4.11), as required. +Proposition 4.6. Let f ∈ C∞(M) be a nonconstant function with maxx∈M f(x) = 0. Then there exists λ♯ and +a C1-curve (−∞, λ♯] → C2(M); +λ �→ uλ with the following properties. +(i) If λ ≤ 0, then uλ is the unique solution of +− ∆¯gu + ¯K = fλe2u +in M +(4.21) +and a global minimum of Efλ. +(ii) If λ ∈ (0, λ♯], then uλ is the unique weakly stable solution of (4.21) in the sense of (4.9), and it is a local +minimum of Efλ. +(iii) The curve of functions λ �→ uλ is pointwisely strictly increasing on M, and so the volume function +(−∞, λ♯] → [0, ∞); +λ �→ V (λ) := +� +M +e2uλdµ¯g +(4.22) +is continuous and strictly increasing. +Proof. We already know that, for λ ≤ 0, the energy Efλ admits a strict global minimiser uλ which depends +smoothly on λ. Moreover, by [1, Proposition 2.4], the curve λ �→ uλ can be extended as a C1-curve to an +interval (−∞, λ♯] for some λ♯ > 0. We also know from [1, Proposition 2.4] that, for λ ∈ (−∞, λ♯], the solution +uλ is strongly stable in the sense that +Cλ := +inf +h∈H1(M,¯g) +1 +∥h∥2 +H1(M,¯g) +� +M +� +|∇¯gh|2 +¯g − 2fλe2uλh2� +dµ¯g > 0. +(4.23) +Here we note that the function λ �→ Cλ is continuous since uλ depends continuously on λ with respect to the +C2-norm. Next we prove that, after making λ♯ > 0 smaller if necessary, the function uλ is the unique weakly +stable solution of (4.21) for λ ∈ (0, λ♯]. Arguing by contradiction, we assume that there exists a sequence +λn → 0+ and corresponding weakly stable solutions (un)n∈N of +− ∆¯gun + ¯K = (f + λn)e2un +in M +(4.24) +with the property that un ̸= uλn for every n ∈ N. By Proposition 4.5, we know that un → u0 in C2(M). +Consequently, vn := un − uλn → 0 in C2(M) as n → ∞, whereas the functions vn solve +− ∆¯gvn = (f + λn) +� +e2un − e2uλn � += (f + λn)e2uλn � +e2vn − 1 +� +in M +for every n ∈ N. +(4.25) + +10 +Franziska Borer, Peter Elbau, Tobias Weth +Combining this fact with (4.23), we deduce that +∥vn∥2 +H1(M,¯g) ≤ 1 +Cλ +� +M +� +|∇¯gvn|2 +¯g − 2(f + λn)e2uλn v2 +n +� +dµ¯g += 1 +Cλ +� +M +(f + λn)e2uλn � +e2vn − 1 − 2vn +� +vndµ¯g. +Since vn → 0 in C2(M), there exists a constant C > 0 with |(e2vn − 1 − 2vn)vn| ≤ C|vn|3 on M for all n ∈ N, +which then implies with H¨older’s inequality and Lemma 4.2 that +∥vn∥2 +H1(M,¯g) ≤ C∥(f + λn)e2uλn ∥L∞(M,¯g)∥vn∥3 +L3(M,¯g) +≤ C +�� +M +|vn|3· 4 +3 dµ¯g +� 3 +4 += C∥vn∥3 +L4(M,¯g) ≤ C∥vn∥3 +H1(M,¯g) +with a constant C > 0 independent on M. This contradicts the fact that vn → 0 in H1(M) as n → ∞. The +claim thus follows. +It remains to prove that the curve of functions λ �→ uλ is pointwisely strictly increasing on M. This is a +consequence of the uniqueness of weakly stable solutions stated in (ii) and the fact that, as noted in [7], if uλ0 +is a solution for some λ0 ∈ (−∞, λ♯], it is possible to construct, via the method of sub- and supersolutions, for +every λ < λ0, a weakly stable solution uλ with uλ < uλ0 everywhere in M. +Corollary 4.7. Let f ∈ C∞(M) be nonconstant with maxx∈M f(x) = 0, and let λ♯ > 0 be given as in +Proposition 4.6. Then there exists κ1 > 0 with the following property. +If A ≥ κ1 and u ∈ C1,A is a solution of +− ∆¯gu + ¯K = (f + λ)e2u +(4.26) +for some λ ∈ R with Ef(u) < λ♯A +2 , then 0 < λ < λ♯, and u is not a weakly stable solution of (4.26), so u ̸= uλ. +Proof. Let κ0 > 0 be given as in Lemma 4.4 for ε = λ♯ > 0. Moreover, let +κ1 := max +� +κ0, V (uλ♯) +� +with V defined in (4.22). Next, let u ∈ C1,A be a solution of (4.26) for some λ ∈ R with Ef(u) < λ♯A +2 . From +Lemma 4.4, we then deduce that 0 < λ < λ♯, and by Proposition 4.6 (iii) we have u ̸= uλ. Since uλ is the +unique weakly stable solution of (4.26), it follows that u is not weakly stable. +Corollary 4.8. Let p > 2, f ∈ C∞(M) be nonconstant with maxx∈M f(x) = 0, and let λ♯ > 0 be given as in +Proposition 4.6. Then there exists κ > 0 with the property that for every A ≥ κ the set +˜C := +� +u0 ∈ C1,A ∩ W 2,p(M, ¯g) | Ef(u0) < λ♯A +2 +� +is nonempty, and for every u0 ∈ ˜C the global solution u ∈ C([0, ∞); C(M))∩C([0, ∞); H1(M, ¯g))∩C∞((0, ∞)× +M) of the initial value problem (2.4), (2.5) converges, as t → ∞ suitably, to a solution u∞ of the static problem +(4.26) for some λ ∈ (0, λ♯) which is not weakly stable and hence no local minimiser of Efλ. +Proof. Let κ1 > 0 be given by Corollary 4.7. By (4.5), there exists κ ≥ κ1 > 0 with mf,A < λ♯A +4 +for fixed +A > κ. Consequently, there exists u0 ∈ C1,A ∩ W 2,p(M, ¯g) with Ef(u0) < λ♯A +2 . By Theorem 3.1, the global +solution u ∈ C([0, ∞); C(M)) ∩ C([0, ∞); H1(M, ¯g)) ∩ C∞((0, ∞) × M) of the initial value problem (2.4), (2.5) +converges, as t → ∞ suitably, to a solution u∞ ∈ C1,A of the static problem (4.26) for some λ ∈ R, whereas +Ef(u∞) ≤ Ef(u0) < λ♯A +2 . Consequently, λ ∈ (0, λ♯) by Corollary 4.7, and u∞ is not weakly stable. +5. Proof of the Main Results +5.1. Notation and Some Regularity Results. In this chapter we summarise different kind of estimates +which will be useful later. In the following, for T > 0 we use the notation +Lp +t Lr +x := Lp([0, T]; Lr(M, ¯g)) +and +Lp +t Hq +x := Lp([0, T]; Hq(M, ¯g)). +A first regularity result is therefore given by Lemma 4.2. + +Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic +11 +Remark 5.1. We have +∥θ∥4 +Lp +t L4x ≤ CGNL∥θ∥2 +Lp +t L2x∥θ∥2 +Lp +t H1x +for θ ∈ Lp +t H1 +x with p ∈ [1, ∞]. +Lemma 5.2 (Sobolev inequality). There exists a constant CS > 0 such that for every ρ ∈ L∞ +t H1 +x, T ≤ 1, we +have +∥ρ∥2 +L4 +t L4x ≤ CS(∥ρ∥2 +L∞ +t L2x + ∥∇¯gρ∥2 +L2 +t L2x) < ∞. +(5.1) +Proof. With Lemma 4.2 there exists a constant CGNL > 0 such that we have for all T ≤ 1 +∥ρ∥4 +L4 +t L4x = +� T +0 +∥ρ(t)∥4 +L4(M,¯g)dt ≤ CGNL +� T +0 +∥ρ(t)∥2 +L2(M,¯g)∥ρ(t)∥2 +H1(M,¯g)dt +≤ CGNL∥ρ∥2 +L∞ +t L2x +� T +0 +(∥ρ(t)∥2 +L2(M,¯g) + ∥∇¯gρ(t)∥2 +L2(M,¯g))dt +≤ CGNL · T ∥ρ∥4 +L∞ +t L2x + CGNL∥ρ∥2 +L∞ +t L2x∥∇¯gρ∥2 +L2 +t L2x +≤ CGNL +� +∥ρ∥4 +L∞ +t L2x + ∥ρ∥2 +L∞ +t L2x∥∇¯gρ∥2 +L2 +t L2x +� +. +By using Young’s inequality we have +∥ρ∥L∞ +t L2x∥∇¯gρ∥L2 +t L2x ≤ 1 +2 +� +∥ρ∥2 +L∞ +t L2x + ∥∇¯gρ∥2 +L2 +t L2x +� +and therefore +∥ρ∥2 +L4 +t L4x ≤ C +1 +2 +GNL +� +∥ρ∥4 +L∞ +t L2x + 1 +4(∥ρ∥2 +L∞ +t L2x + ∥∇¯gρ∥2 +L2 +t L2x)2 +≤ C +1 +2 +GNL(∥ρ∥2 +L∞ +t L2x + 1 +2∥ρ∥2 +L∞ +t L2x + 1 +2∥∇¯gρ∥2 +L2 +t L2x) +≤ 3 +2C +1 +2 +GNL(∥ρ∥2 +L∞ +t L2x + ∥∇¯gρ∥2 +L2 +t L2x) +=: CS(∥ρ∥2 +L∞ +t L2x + ∥∇¯gρ∥2 +L2 +t L2x). +Since T is finite, ρ ∈ L∞ +t H1 +x implies that ρ ∈ Lp +t H1 +x for all p ∈ [1, ∞] which shows that the upper bound is +finite. +Furthermore, since T < ∞ and vol¯g = 1, with Lemma 4.1 we also have for every p, s ∈ [1, ∞] that Lq +tLr +x ⊂ +Ls +tLp +x for q ≥ s, r ≥ p. +Since we will often use it in the following, we recall that for v ∈ CtCx := C([0, T], C(M)) we have +∥1 − ev∥2 +L∞ +t L∞ +x ≤ e2∥v∥L∞ +t +L∞ +x ∥v∥2 +L∞ +t L∞ +x +(5.2) +since for x ∈ R we get with the Taylor expansion +|ex − 1| = |1 − ex| ≤ |x|e|x|. +(5.3) +Lemma 5.3. With Lemma 4.1 we get the following statements: +1. For a (sufficiently smooth) solution u of (2.4), (2.5) we have +¯u(t) ≥ 1 +2 log +� A +Cup +� +=: m0(A, Ef(u0), f, CMT, η1), +(5.4) +with Cup = CMT exp(4η1(2Ef(u0) + | ¯K| log(A) + A maxx∈M f(x))) where η1 is a number determined by +Lemma 4.1. So, especially for a solution u of (2.4), (2.5) we have the uniform bound +m0 ≤ ¯u(t) ≤ 1 +2 log(A), +(5.5) +where we used (4.3) and the volume preserving property to get the upper bound of ¯u(t). +2. For a solution u of (2.4), (2.5) we have for all p ∈ R that +� +M +e2pu(t)dµ¯g ≤ Cint(A, CMT, Ef(u0), f, ¯K, η1, η2, p), +(5.6) +where again, η1, η2 are numbers determined by Lemma 4.1. + +12 +Franziska Borer, Peter Elbau, Tobias Weth +3. For this part we choose f = f0 where f0 ≤ 0 is a nonconstant, smooth function with maxx∈M f0(x) = 0. +Then there exists a constant Clow = Clow(Cint, f0) > 0 such that +� +M +|f0|dµg(t) ≥ Clow. +(5.7) +Proof. +1. Let u be a solution of (2.4), (2.5). We then know that u(t) ∈ CA. So, with (2.9) we have for all +t ≥ 0 that +∥∇¯gu(t)∥2 +L2(M,¯g) = 2Ef(u(t)) − +� +M +(2 ¯Ku(t) − fe2u(t))dµ¯g += 2Ef(u(t)) + +� +M +(2| ¯K|u(t) + fe2u(t))dµ¯g +≤ 2Ef(u0) + | ¯K| log(A) + A max +x∈M f(x), +(5.8) +where we used the fact that +� +M 2u(t)dµ¯g ≤ log(A) by (4.3) and since +� +M e2u(t)dµ¯g ≡ A. With this and +Lemma 4.1 we can now estimate +A = +� +M +e2u(t)dµ¯g = e2¯u(t) +� +M +e2(u(t)−¯u(t))dµ¯g +≤ e2¯u(t)CMT exp(η1∥∇¯g(2u(t))∥2 +L2(M,¯g)) +≤ e2¯u(t)CMT exp(4η1(2Ef(u0) + | ¯K| log(A) + A max +x∈M f(x))) +=: Cupe2¯u(t), +with Cup = Cup(A, CMT, Ef(u0), f, ¯K, η1) > 0 and therefore +¯u(t) ≥ 1 +2 log +� A +Cup +� +=: m0(A, CMT, Ef(u0), f, ¯K, η1) ∈ R. +So, for a solution u(t) ∈ CA of (2.4), (2.5) we get the uniform bound +m0 ≤ ¯u(t) ≤ 1 +2 log(A). +2. Let u be a solution of (2.4), (2.5). So, u(t) ∈ CA. With Lemma 4.1, (5.5), and (5.8) we directly get for +any p ∈ R that +� +M +e2pu(t)dµ¯g = e2p¯u(t) +� +M +e2p(u(t)−¯u(t))dµ¯g +≤ e2p¯u(t)CMT exp(4η2p2∥∇¯gu(t)∥2 +L2(M,¯g)) +≤ Cint, +(5.9) +where Cint = Cint(A, CMT, Ef(u0), f, ¯K, η1, η2, p) > 0. +3. Similar to [19, Lemma 2.3] we see by the choice of f0, H¨older’s inequality, and (5.9) that +0 < +���� +� +M +� +|f0|dµ¯g +���� +2 +≤ +� +M +|f0|e2u(t)dµ¯g +� +M +e−2u(t)dµ¯g ≤ Cint +� +M +|f0|e2u(t)dµ¯g +(5.10) +which shows the claim. +So, Lemma 5.3 is proven. +Now we can turn to the proofs of the main results. +5.2. Short-Time Existence. Let A > 0. We are looking for a short-time solution of (2.4) with initial data +(2.5). Using the Gauss equation (1.1) we can rewrite (2.4), (2.5) in the following way: +∂tu(t) = f − Kg(t) − αA(t) += e−2u(t)∆¯gu(t) + ¯K +� 1 +A − e−2u(t) +� ++ f − 1 +A +� +M +fe2u(t)dµ¯g; +(5.11) +u(0) = u0 ∈ Cp,A := +� +u ∈ W 2,p(M, ¯g) | +� +M +e2u = A +� +, +(5.12) + +Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic +13 +with p > 2, where +αA(t) = 1 +A +�� +M +fdµg(t) − ¯K +� +. +To find a solution of (5.11), (5.12), we consider the linear equation +∂tu(t) = e−2v(t)∆¯gu(t) + ¯K +� 1 +A − e−2v(t) +� ++ f − 1 +A +� +M +fe2v(t)dµ¯g; +(5.13) +u(0) = u0 ∈ Cp,A, +(5.14) +and use a fixed point argument in the space (X, ∥ · ∥X) := (CtCx, ∥ · ∥CtCx). First we observe that for v ∈ CtCx, +equation (5.13) is strongly parabolic. Furthermoren, with p > 2 and the fact that M is compact, we have +u0 ∈ Cp,A ⊂ H2(M, ¯g), and therefore u0 ∈ L∞(M, ¯g). +For the fixed point argument we fix R = R(u0) := ∥u0∥L∞(M,¯g) + 1. For fixed T > 0, let +X = CtCx = C([0, T], C(M, ¯g)) �→ L∞ +t L∞ +x +with +∥u∥X = +max +t∈[0,T ], x∈M |u(x, t)|. +For v ∈ X, by [14, Theorem 7.32] and the appendix, we get a unique solution uv ∈ W 2,1 +p += W 1,p +t +Lp +x ∩Lp +t W 2,p +x +of (5.13), (5.14) for t ∈ [0, T], x ∈ M. On XR = {U ∈ X | ∥U∥X ≤ R}, we now define the function Φ as follows: +for v ∈ XR, let Φ(v) =: uv be the unique solution of (5.13), (5.14). First, we want to show that Φ : XR → XR +if T > 0 is chosen small enough. +Lemma 5.4. If T > 0 is fixed with +T ≤ +� +| ¯K|e2(∥u0∥L∞(M,¯g)+1) + ∥f∥L∞(M,¯g) +� +1 + e2(∥u0∥L∞(M,¯g)+1) +A +��−1 +(5.15) +and v ∈ XR, then Φ(v) ∈ XR. +Proof. With Proposition 6.3 (ii) we directly get +∥Φ(v)∥L∞ +t L∞ +x = ∥uv∥L∞ +t L∞ +x ≤ ∥u+ +0 ∥L∞(M,¯g) + TdT +(5.16) +where +dT ≤ | ¯K|e2∥v∥L∞ +t +L∞ +x + ∥f∥L∞(M,¯g) + ∥f∥L∞(M,¯g)e2∥v∥L∞ +t +L∞ +x +A +≤ | ¯K|e2R + ∥f∥L∞(M,¯g) +� +1 + e2R +A +� +, +hence +∥Φ(v)∥L∞ +t L∞ +x ≤ T +� +| ¯K|e2R + ∥f∥L∞(M,¯g) +� +1 + e2R +A +�� ++ ∥u+ +0 ∥L∞(M,¯g) +≤ 1 + ∥u0∥L∞(M,¯g) = R, +by (5.15) and since R = ∥u0∥L∞(M,¯g) + 1, which shows the claim. +We now use Schauder’s fixed point Theorem [17] to show the following proposition. +Proposition 5.5. If u0 ∈ Cp,A ⊂ W 2,p(M, ¯g) and T > 0 is fixed with (5.15), then there exists a short-time +solution u ∈ X ∩ C∞(M × (0, T)) of (5.11), (5.12). +Moreover, any such solution satisfies u ∈ C([0, T), H1(M, ¯g)). +Proof. Step 1: First we recall Schauder’s Theorem: It asserts that if H is a nonempty, convex, and closed +subset of a Banach space B and F is a continuous mapping of H into itself such that F(H) is a relatively +compact subset of H, then F has a fixed point. +In our case, B ˆ=X = C([0, T]; C(M, ¯g)), H ˆ=XR = {u ∈ X | ∥u∥X = ∥u∥CtCx ≤ R}, and F ˆ=Φ. So to show +the existence of a fixed point of Φ in XR, it remains to show that +1. Φ : XR → XR ist continuous and + +14 +Franziska Borer, Peter Elbau, Tobias Weth +2. Φ(XR) ⊂ XR is relatively compact. +In a first step we show that Φ : XR → XR ist continuous. For this, let (vn)n∈N ⊂ XR be a sequence with +∥vn − v∥X → 0 for n → ∞ with v ∈ XR. With Proposition 6.1 we know that for all vn there exists un ∈ W 2,1 +p +, +p > 2, which is the unique solution of (5.13), (5.14) such that +∥un∥W 2,1 +p +≤ C(∥u0∥W 2,p(M,¯g) + ∥dn∥Lp +t Lp +x) +with +dn(t) := ¯K +� 1 +A − e−2vn(t) +� ++ f − 1 +A +� +M +fe2vn(t)dµ¯g. +Since vn → v in CtCx and therefore vn → v in L∞ +t L∞ +x , we know that vn → v in Lp +t Lp +x for all p. Furthermore, +since the exponential map is continuous, we have e±2vn → e±2v in Lp +t Lp +x for all p, and therefore dn → d in Lp +t Lp +x +for all p. +Hence, for every ε > 0 there exist NV , Nd ∈ N such that +∥vn − v∥Lp +t Lp +x < ε +for all n ≥ N +and +∥dn − d∥Lp +t Lp +x < ε +for all n ≥ N, +with N := max{NV , Nd}. +Furthermore we have the estimate +∥e2vn − e2v∥L∞ +t L∞ +x = ∥(e2vn−2v − 1)e2v∥L∞ +t L∞ +x ≤ ∥e2vn−2v − 1∥L∞ +t L∞ +x ∥e2v∥L∞ +t L∞ +x +≤ ∥2vn − 2v∥e∥2Vn−2V ∥L∞ +t +L∞ +x ∥e2v∥L∞ +t L∞ +x < 2εe2εe2R, +and similarly ∥e−2vn − e−2v∥L∞ +t L∞ +x < 2εe2εe2R. +Considering now the difference un − u, where un = Φ(vn) and u = Φ(v), we see that un − u fulfils the +equation +∂t(un − u)(t) = e−2vn(t)∆¯gun(t) + dn(t) − e−2v(t)∆¯gu(t) − d(t) += e−2vn(t)∆¯g(un − u)(t) + (e−2vn(t) − e−2v(t))∆¯gu(t) + dn(t) − d(t) +with +∥un − u∥W 2,1 +p +≤ C∥(e−2vn − e−2v)∆¯gu + dn − d∥Lp +t Lp +x +≤ C +� +∥e−2vn − e−2v∥L∞ +t L∞ +x ∥∆¯gu∥Lp +t Lp +x + ∥dn − d∥Lp +t Lp +x +� +≤ C(2εe2εe2R∥∆¯gu∥Lp +t Lp +x + ε) +for n ≥ N. +Since ∥∆¯gu∥Lp +t Lp +x is finite and ε > 0 was arbitrary, we see that ∥Φ(vn) − Φ(v)∥W 2,1 +p +→ 0 for n → ∞. So, we +get +∥Φ(vn) − Φ(v)∥X ≤ C∥Φ(vn) − Φ(v)∥Cα ≤ C∥Φ(vn) − Φ(v)∥W 2,1 +p +→ 0 +for n → ∞ +which shows the continuity of Φ : XR → XR. +In a second step we show that Φ(XR) is relatively compact. For this let (un)n∈N ⊂ Φ(XR) be an arbitrary +sequence in the image of Φ. So, again with Proposition 6.1, we see that for every un ∈ Φ(XR) there exists a +vn ∈ XR with Φ(vn) = un such that +∥un∥W 2,1 +p +≤ C(∥u0∥W 2,p(M,¯g) + ∥dn∥Lp +t Lp +x) +≤ C +� +∥u0∥W 2,p(M,¯g) + T| ¯K| +A ++ ∥ ¯Ke−2vn∥Lp +t Lp +x + ∥f∥Lp +t Lp +x + +���� +1 +A +� +M +fe2vndµ¯g +���� +Lp +t Lp +x +� +≤ C +� +∥u0∥W 2,p(M,¯g) + T| ¯K| +A ++ | ¯K|e2R + T∥f∥L∞(M,¯g) + T +A∥f∥L∞(M,¯g)e2R +� +≤ C(A, f, ¯K, R, T, u0) =: Cd. +So, (un)n∈N is uniformly bounded in W 2,1 +p +((0, T) × M). Using now that W 2,1 +p +((0, T) × M) is continuously +embedded in Cα([0, T] × M) for some 0 < α < 1 and this on the other hand is compactly embedded in +Cβ([0, T] × M) for some 0 < β < α < 1 we can conclude the claim. +We have thus proved that Φ has a fixed point u in XR, which then is a (strong) solution u ∈ W 2,1 +p +((0, T) × M) +of (5.11), (5.12). +Step 2: We now show that u ∈ C∞(M × (0, T)). +To see this, we first note the trivial fact that u ∈ +W 2,1 +p +((0, T)×M) is a strong solution of (5.13), (5.14) with v = u. Since then v ∈ W 2,1 +p +((0, T)×M) ⊂ Cα([0, T]× + +Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic +15 +M), [14, Theorems 5.9 and 5.10] imply the existence of a classical solution ˜u ∈ X ∩ C2+α′,1+α′ +loc +((0, T) × M) +of (5.13), (5.14) with v = u for some α′ > 0. Here C2+α′,1+α′ +loc +((0, T) × M) denotes the space of functions +f ∈ C2,1((0, T) × M) with the property that ∂tf and all derivatives up to second order of f with respect to +x ∈ M are locally α′-H¨older continuous. In particular, ˜u ∈ W 2,1 +p +((ε, T − ε) × M) for ε ∈ (0, T). The function +w := u − ˜u ∈ W 2,1 +p +((ε, T − ε) × M) is then a strong solution of the initial value problem +∂tw(t) = e−2v(t)∆¯gw(t) +for t ∈ (ε, T − ε), +w(ε) = u(ε, ·) − ˜u(ε, ·). +By Proposition 6.3 (ii) we then have |w| ≤ ∥u(ε, ·) − ˜u(ε, ·)∥L∞(M,¯g) on (ε, T − ε) × M, whereas ∥u(ε, ·) − +˜u(ε, ·)∥L∞(M,¯g) → 0 as ε → 0 by the continuity of u and ˜u. It thus follows that u ≡ ˜u on (0, T) × M), and +therefore u ∈ C2+α′,1+α′ +loc +((0, T) × M). Since u solves (5.13), (5.14) with v = u ∈ C2+α′,1+α′ +loc +((0, T) × M), +we can apply [14, Theorems 5.9] and the above argument again to get u ∈ C4+α′′,2+α′′ +loc +((0, T) × M) for some +α′′ > 0. +Repeating this argument inductively, we get u ∈ C +k, k +2 +loc ((0, T) × M) for every k > 0, and hence +u ∈ C∞(M × (0, T)). +Step 3: It remains to show that any solution u ∈ X ∩ C∞((0, T) × M) of (5.11), (5.12) also satisfies u ∈ +C([0, T), H1(M, ¯g)). Since u ∈ C∞((0, T) × M), only the continuity in t = 0 needs to be proved. Setting +φ(t) = ∥u(t)∥2 +H1(M,¯g) for t ∈ (0, T), we see that +1 +2(φ(t2) − φ(t1)) = 1 +2 +� t2 +t1 +∂t∥u(t)∥2 +H1(M,¯g) dt = +� t2 +t1 +� +M +� +u(t)∂tu(t) + ∇u(t)∇∂tu(t) +� +dµ¯gdt += +� t2 +t1 +� +M +� +u(t)∂tu(t) − [∆u(t)]∂tu(t) +� +dµ¯gdt +and therefore, by H¨older’s inequality, +1 +2|φ(t2) − φ(t1)| ≤ +� t2 +t1 +� +M +� +|u||∂tu| + |∆u||∂tu| +� +dµ¯gdt +≤ C∥∂tu∥Lp((0,T )×M) +� +∥u∥Lp((0,T )×M) + ∥∆u∥Lp((0,T )×M) +� +(t2 − t1)β +≤ C∥u∥W 1,2 +p +((0,T )×M)(t2 − t1)β, +for 0 < t1 < t2 < T with some β > 0 depending on p > 2, which implies that the function φ is uniformly +continuous and therefore bounded on (0, T). +We now assume by contradiction that u is not continuous at t = 0 with respect to the H1(M, ¯g)-norm. Then +there exists a sequence (tn)n∈N in (0, T) and ε > 0 with tn → 0+ as n → ∞ and +∥u(tn) − u0∥H1(M,¯g) ≥ ε +for all n ∈ N. +(5.17) +Since ∥u(tn)∥2 +H1(M,¯g) = φ(tn) remains bounded as n → ∞, we conclude that, passing to a subsequence, the +sequence u(tn) converges weakly in H1(M, ¯g) and therefore strongly in L2(M, ¯g). Since the strong L2-limit +of u(tn) must be u0 = u(0) as a consequence of the fact that u ∈ X, we deduce that u(tn) ⇀ u0 weakly in +H1(M, ¯g) as n → ∞. Combining this information with Proposition 6.1 from the appendix, we deduce that +lim sup +n→∞ ∥u(tn)∥2 +H1(M,¯g) ≤ ∥u0∥2 +H1(M,¯g) ≤ lim inf +n→∞ ∥u(tn)∥2 +H1(M,¯g) +(5.18) +and therefore ∥u(tn)∥H1(M,¯g) → ∥u0∥H1(M,¯g). Note here that this part of Proposition 6.1 applies since u solves +(5.13), (5.14) with v = u ∈ W 2,1 +p +((0, T) × M) ⊂ Cα([0, T] × M) for some α > 0. From (5.18) and the uniform +convexity of the Hilbert space H1(M, ¯g), we conclude that u(tn) → u0 strongly in H1(M, ¯g), contrary to +(5.17). +5.3. Uniqueness. We now show that the solution from Proposition 5.5 is unique. +Lemma 5.6. Let u0 ∈ W 2,p(M, ¯g), p > 2, and T > 0 be fixed with (5.15). Then the short-time solution of +u ∈ X ∩ C∞(M × (0, T)) of (5.11), (5.12) given by Proposition 5.5 is unique. +Proof. Let u1, u2 ∈ X ∩ C∞(M × (0, T)) be two solutions of (5.11), (5.12). The difference u := u1 − u2 ∈ +X ∩ C∞(M × (0, T)) then fulfils +∂tu(t) = e−2u1(t)∆¯gu1(t) − e−2u2(t)∆¯gu2(t) +− ¯K(e−2u1(t) − e−2u2(t)) − 1 +A +� +M +f(e2u1(t) − e2u2(t))dµ¯g += e−2u1(t)∆¯gu(t) + ∆¯gu2(t) +� +e−2u1(t) − e−2u2(t)� +− ¯K(e−2u1(t) − e−2u2(t)) − 1 +A +� +M +f(e2u1(t) − e2u2(t))dµ¯g +for t ∈ (0, T). +(5.19) + +16 +Franziska Borer, Peter Elbau, Tobias Weth +In the following, the letter C denotes different positive constants. Multiplying (5.19) with 2u and integrating +over M gives +d +dt∥u(t)∥2 +L2(M,¯g) = 2 +� +M +u(t)∂tu(t)dµ¯g += 2 +� +M +e−2u1(t)u(t)∆¯gu(t)dµ¯g + 2 +� +M +u(t)∆¯gu2(t) +� +e−2u1(t) − e−2u2(t)� +dµ¯g +(5.20) +− 2 +� +M +¯Ku(t)(e−2u1(t) − e−2u2(t))dµ¯g − 2 +A +� +M +f(e2u1(t) − e2u2(t))dµ¯g +� +M +u(t)dµ¯g +≤ 2 +� +M +e−2u1(t)u(t)∆¯gu(t) + 2 +� +M +V (t, x)u2(t) + 2ρ(t)∥u(t)∥L2(M,¯g) +� +M +|u(t)|dµ¯g +≤ 2 +� +− +� +M +e−2u1(t)|∇¯gu(t)|2 +¯g + 2 +� +M +e−2u1(t)u(t)⟨∇¯gu1(t), ∇¯gu(t)⟩¯gdµ¯g +� ++ 2∥V (t, ·)∥Lp(M,¯g)∥u(t)∥2 +L2p′(M,¯g) + C∥u(t)∥2 +L2(M,¯g) +≤ C∥∇¯gu1(t)∥L4(M,¯g)∥u(t)∥L4(M,¯g)∥∇¯gu(t)∥L2(M,¯g) ++ 2∥V (t, ·)∥Lp(M,¯g)∥u(t)∥2 +L2p′(M,¯g) + C∥u(t)∥2 +L2(M,¯g) +≤ C +� +∥u1(t)∥H2(M,¯g)∥u(t)∥2 +H1(M,¯g) + 2∥V (t, ·)∥Lp(M,¯g)∥u(t)∥2 +H1(M,¯g) + ∥u(t)∥2 +L2(M,¯g) +� +≤ C +� +∥u1(t)∥H2(M,¯g) + 2∥V (t, ·)∥Lp(M,¯g) + 1 +� +∥u∥2 +H1(M,¯g), +(5.21) +with functions V ∈ Lp((0, T) × M) ∩ C∞((0, T) × M) and ρ ∈ L∞(0, T). Here we used the Sobolev embeddings +H1(M, ¯g) �→ Lρ(M) for ρ ∈ [1, ∞). Multiplying (5.19) with −2∆u and integrating over M yields +d +dt∥∇gu(t)∥2 +L2(M,¯g) = 2 +� +M +∇u(t)∇∂tu(t)dµ¯g = −2 +� +M +∆gu(t)∂tu(t)dµ¯g +≤ −2 +� +M +e−2u1(t)|∆¯gu(t)|2dµ¯g + 2 +� +M +V (x, t)|u(t)||∆u(t)|dµ¯g +≤ −κ∥∆¯gu(t)∥2 +L2(M,¯g) + 2∥V (t, ·)∥Lp(M,¯g)∥u∥Lα(M,¯g)∥∆gu∥L2(M,¯g) +≤ −κ∥∆¯gu(t)∥2 +L2(M,¯g) + 1 +κ∥V (t, ·)∥2 +Lp(M,¯g)∥u∥2 +Lα(M,¯g) + κ∥∆gu∥2 +L2(M,¯g) += 1 +κ∥V (t, ·)∥2 +Lp(M,¯g)∥u∥2 +Lα(M,¯g) ≤ C∥V (t, ·)∥2 +Lp(M,¯g)∥u∥2 +H1(M,¯g), +(5.22) +where we used first H¨older’s inequality with α = +2p +p−2, then Young’s inequality and finally Sobolev embeddings +again. Here we note that, by making C > 0 larger if necessary, we may assume that the constants are the same +in (5.21) and (5.22). Combining these estimates gives +d +dt∥u(t)∥2 +H1(M,¯g) ≤ g(t)∥u(t)∥2 +H1(M,¯g) +for t ∈ (0, T) +(5.23) +with the function g ∈ L1(0, T) given by g1(t) = C +� +∥u1(t)∥H2(M,¯g) + 3∥V (t, ·)∥Lp(M,¯g) + 1 +� +. Integrating and +using the fact that u ∈ C([0, T), H1(M, ¯g)) by Proposition 5.5 with u(0) = u1(0) − u2(0) = 0, we see that +∥u(t)∥2 +H1(M,¯g) ≤ +� t +0 +g(s)∥u(s)∥2 +H1(M,¯g) ds +for t ∈ [0, T). +It then follows from Gronwall’s inequality [3] that ∥u(t)∥2 +H1(M,¯g) ≡ 0 on [0, T), hence u1 ≡ u2. +5.4. Global Existence. From Section 5.2 and Section 5.3 we know that there exists a unique solution +u ∈ C([0, T], C(M)) ∩ C([0, 1], H1(M, ¯g)) ∩ C∞((0, T) × M), +of the initial value problem (5.11), (5.12). In particular we know that u ∈ L∞ +t L∞ +x for t ∈ [0, T], where T > 0 +is given by (5.15). In this section we want to show that u posses an L∞-a-priori bound on any time interval +[0, T], T < ∞, and therefore, u is the unique global solution of (5.11), (5.12). For this we partially follow the +idea of [2, Chapter 6]. +Lemma 5.7. For every T > 0, there exists M(T) > 0 such that we have +sup +t∈[0,T ] +∥u(t)∥L∞(M,¯g) ≤ M(T). + +Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic +17 +Proof. Let +I := +� +t ≥ 0 +��� u is a solution of (5.11) on (0, t] × M +with initial data u(0) ∈ Cp,A +� +, +Tmax := sup I, and Tk ⊂ I a sequence in I such that Tk → Tmax for k → ∞. +For any t ∈ [0, Tk] and any xmax(t) ∈ M where +u(t, xmax(t)) = max +x∈M u(t, x) ≥ 0 +we have with ∂tu(t) = ∆g(t)u(t) − e−2u(t) ¯K + f − α(t) and the upper bound for |α| which is given by +α0 := max{|α1|, |α2|}, +(5.24) +that +d +dt [u(t, xmax(t))] = ∂tu(t, xmax(t)) ≤ | ¯K|e−2u(t,xmax(t)) + f(xmax(t)) + α0 +≤ | ¯K| + ∥f∥L∞(M,¯g) + α0, +(5.25) +where we used that ∇¯gu(t, xmax(t)) = 0 and therefore +d +dt [u(t, xmax(t)] = ∂tu(t, xmax(t)) + ∇¯gu(t, xmax(t)) ˙xmax(t) = ∂tu(t, xmax(t)). +Integrating (5.25) on both side with respect to t and taking the supremum over t yields (together with the +fact that u(0) = u0 ∈ Cp,A) +sup +t∈[0,Tk] +x∈M +u(t, x) ≤ Tk(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup +x∈M +u0(x) +→ Tmax(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup +x∈M +u0(x) =: M1(Tmax) < ∞ +(5.26) +for k → ∞ which shows the upper bound for u. +Analogously, at any point xmin(t) ∈ M where +u(t, xmin(t)) = min +x∈M u(t, x) ≤ 0 +we have with ∂tu(t) = ∆g(t)u(t) − e−2u(t) ¯K + f − α(t), the fact that ¯K < 0, and the upper bound for |α| given +by α0 that +d +dt [u(t, xmin(t))] = ∂tu(t, xmin(t)) ≥ −∥f∥L∞(M,¯g) − α0. +(5.27) +Integrating (5.27) on both side with respect to t and taking the infimum over t yields (together with the fact +that u(0) = u0 ∈ Cp,A) +inf +t∈[0,Tk] +x∈M +u(t, x) ≥ −Tk(∥f∥L∞(M,¯g) + α0) + inf +x∈M u0(x) +→ −Tmax(∥f∥L∞(M,¯g) + α0) + inf +x∈M u0(x) =: M2(Tmax) > −∞ +(5.28) +for k → ∞ which shows the lower bound for u. +So, we get +sup +t∈[0,T ] +x∈M +|u(t, x)| ≤ max{|M1(T)|, |M2(T)|} +≤ T(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup +x∈M +|u0(x)| =: M(T) +(5.29) +which shows the claim. +In fact, with the help of (2.9) we can turn (5.29) into a uniform estimate for all time. +Lemma 5.8. Let u be the global, smooth solution of (5.11) with u(0) = u0 ∈ Cp,A. +Then we have that +supt>0 ∥u(t)∥L∞(M,¯g) ≤ Cuni < ∞. + +18 +Franziska Borer, Peter Elbau, Tobias Weth +Proof. We follow the proof of [19, Lemma 2.5]. +By using the fact that u(t) is a volume preserving solution of (5.11) with u(0) = u0 ∈ Cp,A and therefore +� +M e2u(t)dµ¯g ≡ A, we get with (4.3) and the fact that ¯K < 0 that +Ef(u(t)) = 1 +2∥∇¯gu(t)∥2 +L2(M,¯g) + +� +M +¯Ku(t)dµ¯g − 1 +2 +� +M +fe2u(t)dµ¯g +≥ +¯K +2 +� +M +2u(t)dµ¯g − 1 +2 +� +M +fe2u(t)dµ¯g +≥ +¯K +2 log(A) − A +2 ∥f∥L∞(M,¯g) > −∞. +(5.30) +Defining +F(t) := +� +M +|∂tu(t)|2dµg(t) = +� +M +|∂tu(t)|2e2u(t)dµ¯g +and using the uniform lower bound of Ef given by (5.30), we then get from (2.8) or (2.9), respectively, the +estimate +� ∞ +0 +F(t)dt = +� ∞ +0 +� +M +|∂tu(t)|2dµg(t)dt ≤ Ef(u0) + | ¯K| +2 | log(A)| + A +2 ∥f∥L∞(M,¯g). +(5.31) +Hence, for any T > 0 we find tT ∈ [T, T + 1] such that +F(tT ) = +inf +t∈(T,T +1) F(t) ≤ Ef(u0) + | ¯K| +2 | log(A)| + A +2 ∥f∥L∞(M,¯g). +(5.32) +So, at time tT we get with (2.1), H¨olders inequality, (5.6), and (5.32) that +∥∆¯gu(tT )∥L +3 +2 (M,¯g) +≤ ∥e2u(tT )∂tu(tT )∥L +3 +2 (M,¯g) + ∥ ¯K∥L +3 +2 (M,¯g) + ∥e2u(tT )f∥L +3 +2 (M,¯g) + ∥e2u(tT )α(tT )∥L +3 +2 (M,¯g) +≤ ∥eu(tT )∥L6(M,¯g)F(tT ) +1 +2 + | ¯K| + +�� +M +e3u(tT )|f| +3 +2 dµ¯g +� 2 +3 ++ +�� +M +e3u(tT )|α(tT )| +3 +2 dµ¯g +� 2 +3 +≤ C +1 +6 +int(A, Ef(u0), f, ¯K, η1, η2, 3) +� +Ef(u0) + | ¯K| +2 | log(A)| + A +2 ∥f∥L∞(M,¯g) +� 1 +2 ++ | ¯K| ++ C +2 +3 +int +� +A, Ef(u0), f, ¯K, η1, η2, 3 +2 +� +(∥f∥L∞(M,¯g) + α0) +=: C10 +� +A, Ef(u0), f, ¯K, η1, η2, 3 +2, 3 +� +. +(5.33) +Furthermore, with Sobolev’s embedding theorem we have W 2, 3 +2 ⊂ C0, 2 +3 . Therefore we get with Poincar´e’s +inequality, the Calder´on–Zygmund inequality for closed surfaces, and with (5.33) that +∥u(tT ) − ¯u(tT )∥ +3 +2 +L∞(M,¯g) ≤ C∥u(tT ) − ¯u(tT )∥ +3 +2 +W 2, 3 +2 (M,¯g) ≤ C∥∇2 +¯gu(tT )∥ +3 +2 +L +3 +2 (M,¯g) +≤ C∥∆¯gu(tT )∥ +3 +2 +L +3 +2 (M,¯g) ≤ CC +3 +2 +10, +(5.34) +and therefore with (5.5) we obtain the uniform bound +∥u(tT )∥L∞(M,¯g) ≤ CC10 + max +� +|m0|, 1 +2| log(A)| +� +. +(5.35) +Upon shifting time by tT , from (5.29) we now get +sup +s∈[T +1,T +2] +∥u(s)∥L∞(M,¯g) ≤ +sup +s∈[tT ,T +2] +∥u(s)∥L∞(M,¯g) +≤ 2(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup +x∈M +|u(tT , x)| +≤ 2(| ¯K| + ∥f∥L∞(M,¯g) + α0) + CC10 + max +� +|m0|, 1 +2| log(A)| +� +. +(5.36) +Since T > 0 is arbitrary, the claim follows. + +Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic +19 +5.5. Convergence of the Flow. Let f0 ≤ 0 be a smooth, nonconstant function withmaxx∈M f0(x) = 0. +Following here the argumentation of [19], and using (5.31), we know that for a suitable sequence tl → ∞, +l → ∞, with associated metrics gl = g(tl) we obtain convergence +� +M +|∂tu(tl)|2dµg(tl) = +� +M +|f0 − Kgl − α(tl)|2dµg(tl) → 0 +for l → ∞. +(5.37) +Provided that we can also show convergence of the associated sequence of metrics g(tl) to a limit metric +g∞ +A = e2u∞ +A ¯g with Gauss curvature Kg∞ +A , it then follows that Kg∞ +A = f0 − α∞ +A for a constant α∞ +A . Later we will +have a closer look at this constant α∞ +A . +Lemma 5.9. For F(t) = +� +M |∂tu(t)|2dµg(t) as above, we have F(t) → 0 for t → ∞. +Proof. First we consider the evolution equation of the curvature Kg(t) and of α(t). From the Gauss equation +(1.1) we get for the curvature that +∂tKg(t) = ∂t(−e−2u(t)∆¯gu(t) + e−2u(t) ¯K) += −2∂tu(t)Kg(t) − ∆g(t)∂tu(t) += 2Kg(t)(Kg(t) − f0 + α(t)) + ∆g(t)(Kg(t) − f0 + α(t)) += 2(Kg(t) − f0 + α(t))2 + 2(f0 − α(t))(Kg(t) − f0 + α(t)) + ∆g(t)(Kg(t) − f0 + α(t)). +(5.38) +With (2.3) we get for the evolution equation for α(t): +d +dtα(t) = 2 +A +� +M +f0e2u(t)∂tu(t)dµ¯g = 2 +A +� +M +f0(f0 − Kg(t) − α(t))dµg(t). +(5.39) +So, with (5.38) and (5.39) we arrive at +∂t(Kg(t) − f0 − α(t)) − ∆g(t)(Kg(t) − f0 + α(t)) += 2(Kg(t) − f0 + α(t))2 + 2(f0 − α(t))(Kg(t) − f0 + α(t)) ++ 2 +A +� +M +f0(Kg(t) − f0 + α(t))dµg(t). +(5.40) +Following the proof of Lemma 3.1 in [19] we therefore get +1 +2 +d +dt +� +M +|f0 − Kg(t) − α(t)|2dµg(t) += +� +M +�� +∂tKg(t) + +� d +dtα(t) +�� +(Kg(t) − f0 + α(t)) − (Kg(t) − f0 − α(t))3 +� +dµg(t) += − +� +M +|∇g(t)(Kg(t) − f0 + α(t))|2 +g(t)dµg(t) + 2 +� +M +(f0 − α(t))(Kg(t) − f0 + α(t))2dµg(t) ++ +� +M +(Kg(t) − f0 + α(t))3dµg(t), +(5.41) +where we used in the second step the fact that +� d +dtα(t) +� � +M +(Kg(t) − f0 + α(t))dµg(t) = 0 +by (2.2). +With H¨older’s inequality we can estimate +� +M +(Kg(t) − f0 + α(t))3dµg(t) ≤ ∥∂tu(t)∥3 +L3(M,g(t)) ≤ ∥∂tu(t)∥L2(M,g(t))∥∂tu(t)∥2 +L4(M,g(t)) +(5.42) + +20 +Franziska Borer, Peter Elbau, Tobias Weth +and by Lemma 4.2 we further get with the uniform bound for u ∈ CtCx that +∥∂tu(t)∥2 +L4(M,g(t)) += +�� +M +|∂tu(t)|4e2u(t)dµ¯g +� 1 +2 +≤ e∥u∥L∞ +t +L∞ +x ∥∂tu(t)∥2 +L4(M,¯g) +≤ e∥u∥L∞ +t +L∞ +x � +CGNL∥∂tu(t)∥L2(M,¯g)∥∂tu(t)∥H1(M,¯g) += e∥u∥L∞ +t +L∞ +x � +CGNL +�� +M +|∂tu(t)|2e2u(t)e−2u(t)dµ¯g +� 1 +2 �� +M +|∂tu(t)|2e2u(t)e−2u(t)dµ¯g + +� +M +|∇¯g∂tu(t)|2 +¯gdµ¯g +� 1 +2 += e∥u∥L∞ +t +L∞ +x � +CGNL +�� +M +|∂tu(t)|2e−2u(t)dµg(t) +� 1 +2 �� +M +|∂tu(t)|2e−2u(t)dµg(t) + +� +M +|∇g(t)∂tu(t)|2 +g(t)dµg(t) +� 1 +2 +≤ e∥u∥L∞ +t +L∞ +x max{e∥u∥L∞ +t +L∞ +x , e2∥u∥L∞ +t +L∞ +x } +� +CGNL∥∂tu(t)∥L2(M,g(t))∥∂tu(t)∥H1(M,g(t)) +=: ˜C2∥∂tu(t)∥L2(M,g(t))∥∂tu(t)∥H1(M,g(t)), +(5.43) +where we used the fact that +� +M +|∇¯g∂tu(t)|2 +¯gdµ¯g = +� +M +|∇g(t)∂tu(t)|2 +g(t)dµg(t) =: G(t). +Plugging in (5.43) into (5.42) we arrive at +� +M +(Kg(t) − f0 + α(t))3dµg(t) ≤ ˜C2∥∂tu(t)∥2 +L2(M,g(t))∥∂tu(t)∥H1(M,g(t)) +≤ +˜C2 +2 +2 ∥∂tu(t)∥4 +L2(M,g(t)) + 1 +2∥∂tu(t)∥2 +H1(M,g(t)) +≤ ˜C2 +2F 2(t) + 1 +2(F(t) + G(t)), +(5.44) +where we used Young’s inequality in the second step. +With α0 = max{|α1|, |α2|} > 0 we furthermore have that +2 +� +M +(f0 − α(t))(Kg(t) − f0 + α(t))2dµg(t) ≤ 2(∥f0∥L∞(M,¯g) + α0)F(t) =: ˜C3(α0, f0)F(t) +So, (5.41) yields +d +dtF(t) + G(t) ≤ 2 +� +˜C3F(t) + ˜C2 +2F 2(t) + 1 +2F(t) +� += (2 ˜C3 + 1)F(t) + 2 ˜C2 +2F 2(t) +=: ˜C4F(t) + 2 ˜C2 +2F 2(t). +(5.45) +We recall that with (5.31) we have lim inft→∞ F(t) = 0 and therefore we know that there exist tl → ∞ with +F(tl) → 0 as l → ∞, see (5.37). +By integrating (5.45) over (tl, t) ⊂ (tl, T) and taking the supremum over (tl, T) we get with +� T +tl G(t)dt ≥ 0 +that +sup +t∈(tl,T ) +F(t) ≤ F(tl) + ˜C4 +� T +tl +F(t)dt + 2 ˜C2 +2 +� T +tl +F 2(t)dt +≤ F(tl) + ˜C4 +� T +tl +F(t)dt + 2 ˜C2 +2 +sup +t∈(tl,T ) +F(t) +� T +tl +F(t)dt +≤ F(tl) + ˜C4 +� T +tl +F(t)dt + 2 ˜C2 +2 +sup +t∈(tl,T ) +F(t) +� ∞ +tl +F(t)dt. +With (5.31) we also have +� ∞ +tl F(t)dt → 0 for l → ∞. So, for T > 0 big enough such that for tl < T big +enough we have that 2 ˜C2 +2 +� ∞ +tl F(t)dt is small enough to guarantee that 1 − 2 ˜C2 +2 +� ∞ +tl F(t)dt > 0 and therefore the +term 2 ˜C2 +2 supt∈(tl,T ) F(t) +� ∞ +tl F(t)dt can be absorbed on the left hand side. So, we get +sup +t∈(tl,T ) +F(t) ≤ +1 +� +1 − 2 ˜C2 +2 +� ∞ +tl F(t)dt +� +� +F(tl) + ˜C4 +� T +tl +F(t)dt +� +. + +Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic +21 +Letting T → ∞ yields +sup +t∈(tl,∞) +F(t) ≤ +1 +� +1 − ˜C2 +2 +� ∞ +tl F(t)dt +� +� +F(tl) + ˜C4 +� ∞ +tl +F(t)dt +� +→ 0 +as l → ∞ +which shows the claim. +To prove now the convergence of the flow, let A > 0 and u0 ∈ Cp,A, p > 2. Furthermore let f ∈ C∞(M) be +a smooth, nonconstant function, and (f0, λ) ∈ C∞(M) × R the unique pair such that +f = f0 + λ +with f0 ≤ 0, f0 nonconstant, and maxx∈M f0(x) = 0. Since by Proposition 2.1 the additive rescaled prescribed +Gauss curvature flow (2.4) is invariant under adding or subtracting a constant C > 0 to the function f, for all +functions +f ∈ {f0 + λ | λ ∈ R} +we consider the same flow given by +∂tu(t) = f0 − Kg(t) − αA(t) +in (0, T) × M, +(5.46) +which is (2.4) with f replaced by f0. +With (2.9) we know that +1 +2 +� +M +(|∇¯gu(T)|2 +¯g + 2 ¯Ku(T) − f0e2u(T ))dµ¯g = Ef0(u(T)) ≤ Ef0(u(0)). +So, we get with (4.3) that +1 +2 +� +M +|∇¯gu(T)|2 +¯gdµ¯g = Ef0(u(T)) − +� +M +¯Ku(T)dµ¯g + 1 +2 +� +M +f0e2u(T )dµ¯g +≤ Ef0(u(T)) + | ¯K| +� +M +u(T)dµ¯g +≤ Ef0(u(0)) + | ¯K| +2 | log(A)|. +So, u is uniformly (in T) bounded in H1(M, ¯g), i.e., ∥u∥L∞ +t H1x ≤ C. +We now consider ul := u(tl) for a suitable sequence tl → ∞. By the Theorem of Banach-Alao˘glu we know +that (ul)l is weak∗ relatively compact in H1(M, ¯g) and therefore (since H1 is reflexive) also weak relatively +compact. This means that that there exists a subsequence ulk which we again call ul such that ul → u∞ +A weakly +in H1(M, ¯g) and therefore strongly in L2(M, ¯g) (by a direct consequence of the Rellich–Kondrachov embedding +Theorem). Furthermore with (2.6) and (2.7) we know that αl := α(tl) → α∞ +A as l → ∞. Moreover we have +e±ul → e±u∞ +A (as l → ∞) in Lp(M, ¯g) for any 2 ≤ p < ∞. Indeed, with Lemma 5.8 and (5.3) we have +∥eul − eu∞ +A ∥p +Lp(M,¯g) = +� +M +epul|1 − eu∞ +A −ul|pdµ¯g ≤ epCuni +� +M +|1 − eu∞ +A −ul|pdµ¯g +≤ epCuni +� +M +|u∞ +A − ul|pep|u∞ +A −ul||dµ¯g +≤ epCunie2pCuni +� +M +|u∞ +A − ul|p−2|u∞ +A − ul|2dµ¯g +≤ e3pCuni(2Cuni)p−2∥u∞ +A − ul∥2 +L2(M,¯g) → 0 +as l → ∞. +Replacing ul by −ul we get also e−ul → e−u∞ +A in Lp(M, ¯g) as l → ∞ for any p < ∞. Moreover, with Lemma 5.8 +and Lemma 5.9 we also have e2ul∂tul → 0 in L2(M, ¯g) as l → ∞. Furthermore we have +∥e2ulαl − e2u∞ +A α∞ +A ∥L2(M,¯g) ≤ ∥e2ul(αl − α∞ +A )∥L2(M,¯g) + ∥α∞ +A (e2ul − e2u∞ +A )∥L2(M,¯g) +≤ ∥e2ul∥L∞(M,¯g)|αl − α∞ +A |A +1 +2 + |α∞ +A |∥e2ul − e2u∞ +A ∥L2(M,¯g) +→ 0 +for l → ∞. +So, considering our evolution equation (5.11), we therefore get +∆¯gul = e2ul∂tul + ¯K − e2ulf0 + e2ulαl +→ ¯K − e2u∞ +A f0 + e2u∞ +A α∞ +A =: (∆¯gu)∞ +A + +22 +Franziska Borer, Peter Elbau, Tobias Weth +in L2(M, ¯g). +Since the Laplace operator ∆¯g is closed we know that (∆¯gu)∞ +A = ∆¯gu∞ +A . +Hence ∥∆¯g(ul − +u∞ +A )∥L2(M,¯g) → 0 as l → ∞. So, we even have strong convergence ul → u∞ +A in H2(M, ¯g) and uniformly. +Thus, passing to the limit l → ∞ in the equation +e2ul∂tul − ∆¯gul = − ¯K + e2ulf0 − e2ulαl +we get the identity +−∆¯gu∞ +A = − ¯K + e2u∞ +A f0 − e2u∞ +A α∞ +A +and therefore +Kg∞ +A = f0 − α∞ +A = f0 + 1 +A +� +¯K + +� +M +|f0|dµg∞ +A +� +which shows the convergence of the flow. +5.6. The Sign of the Constant α∞ +A . In this subsection we prove Theorem 3.3 and Theorem 3.4, with other +words, under certain assumptions we can now further estimate the expression +1 +A +� +¯K + +� +M +|f0|dµg∞ +A +� +to show that it is positive. +The proof of Theorem 3.4 is already covered by the proof of Corollary 4.8. So we can turn to Theorem 3.3. +Proof of Theorem 3.3. We have seen in Lemma 5.7 that in the case where u0 ≡ 1 +2 log(A) ∈ Cp,A, the uniform +L∞-bound on the global solution of the initial value problem (5.11), (5.12) only depends on A and an upper +bound on ∥f∥L∞(M,¯g). In other words, if A > 0 and c > 0 are fixed, then there exists τ > 0 with the property +that +sup +t>0 +∥u(t)∥L∞(M,¯g) ≤ τ +for every f ∈ C∞(M) with ∥f∥L∞(M,¯g) ≤ c and the corresponding solution u of the initial value problem (5.11), +(5.12) with u0 ≡ 1 +2 log(A) ∈ Cp,A. Consequently, we also have ∥u∞∥L∞(M,¯g) ≤ τ under the current assumptions +on f, which implies that +λ = 1 +A +� +¯K − +� +M +fe2u∞dµ¯g +� += 1 +A +� +¯K + cA − +� +M +(f + c)e2u∞dµ¯g +� +≥ c + +¯K +A − ∥f + c∥L1(M,¯g)∥e2u∞∥L∞(M,¯g) ≥ c + +¯K +A − ∥f + c∥L1(M,¯g)e2τ. +Hence, if ∥f + c∥L1(M,¯g) < ε := c+ ¯ +K +A +e2τ , we have λ > 0. +6. Appendix +As before, let (M, ¯g) be a two-dimensional, smooth, closed, connected, oriented Riemann manifold endowed +with a smooth background metric ¯g. For a domain Ω ⊂ M × R and p ≥ 1, we let W 2,1 +p +(Ω) denote the space of +functions u ∈ Lp(Ω) which have weak derivatives Du, D2u and ∂tu in Lp(Ω). In the following, we fix p > 2, +which implies that +W 2,1 +p +(Ω) is continuously embedded in Cα(Ω) for some α = α(p) > 0, +(6.1) +see e.g. [13, Lemma 3.3]. We consider the linear parabolic problem +∂tu(x, t) = a(x, t)∆¯gu(x, t) + c(x, t)u(x, t) + d(x, t), +(6.2) +with a, c, d ∈ C(Ω) and d ∈ Lp(Ω). We say that a function u ∈ W 2,1 +p +(Ω) is a (strong) solution of (6.2) in Ω if +(6.2) holds almost everywhere in Ω. Specifically, we consider (6.2) on the cylindrical domains ΩT = M × (0, T) +and �ΩT = M × (−∞, T) in the following. +In particular, we consider strong solutions of (6.2) together with the initial condition +u(0, x) = u0(x) +in M +(6.3) +with u0 ∈ W 2,p(M, ¯g), which is supposed to hold in the (initial) trace sense. + +Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic +23 +Proposition 6.1. Let T > 0, a, c ∈ C(ΩT ) with aT := +min +(x,t)∈ΩT +a(x, t) > 0, let d ∈ Lp(ΩT ) for some p > 2, and +let u0 ∈ W 2,p(M, ¯g). +Then the initial value problem (6.2), (6.3) has a unique strong solution u ∈ W 2,1 +p +(ΩT ). Moreover, u satisfies +the estimate +∥u∥W 2,1 +p +(ΩT ) ≤ C +� +∥u0∥W 2,p(M,¯g) + ∥d∥Lp(ΩT ) +� +(6.4) +with a constant C > 0 depending only on ∥a∥L∞(ΩT ), ∥c∥L∞(ΩT ) and aT . Moreover, C does not increase after +making T smaller. +If, moreover, a, c, d ∈ Cα(ΩT ) for some α > 0, then u ∈ C(ΩT )∩C2,1(ΩT ) is a classical solution of (6.2), (6.3), +and we have the inequality +∥u0∥H1(M,¯g) ≥ lim sup +t→0+ ∥u(t)∥H1(M,¯g) +(6.5) +Proof. In the following, the letter C stands for various positive constants depending only on ∥a∥L∞(ΩT ), +∥c∥L∞(ΩT ), and aT , and which do not increase after making T smaller. +Step 1: We first assume that we are given a strong solution u ∈ W 2,1 +p +(ΩT ) of (6.2), (6.3) with u0 ≡ 0 ∈ +W 2,p(M, ¯g). We then define v : �ΩT → R by +v(x, t) = +� +u(x, t), +for t > 0; +0, +for t ≤ 0. +Then v ∈ W 2,1 +p +(�ΩT ) solves (6.2) with a, c, d replaced by suitable extensions ˜a, ˜c, ∈ L∞(�ΩT ), ˜d ∈ Lp(�ΩT ) satisfying +˜a(x, t) = a(x, 0), ˜c(x, t) = c(x, 0) and ˜d(x, t) = 0 for t ≤ 0, x ∈ M. +Therefore, [14, Theorem 7.22] gives rise to the uniform bound +∥D2v∥Lp(�ΩT ) + ∥∂tv∥Lp(�ΩT ) ≤ C +� +∥ ˜d∥Lp(�ΩT ) + ∥v∥Lp(�ΩT ) +� +. +(6.6) +This translates into the estimate +∥D2u∥Lp(ΩT ) + ∥∂tu∥Lp(ΩT ) ≤ C +� +∥d∥Lp(ΩT ) + ∥u∥Lp(ΩT ) +� +. +(6.7) +Moreover, setting V (t) := ∥u(t)∥p +Lp(M,¯g) for t ∈ R, we have V (0) = 0 and +˙V (t) = p +� +M +|u(t)|p−2u(t)∂tu(t)dµ¯g ≤ pV (t) +1 +p′ ∥∂tu(t)∥Lp(M,¯g) +≤ p +� +V (t) +p′ ++ +∥∂tu(t)∥p +Lp(M,¯g) +p +� += p +p′ V (t) + ∥∂tu(t)∥p +Lp(M,¯g) +for t ∈ (0, T), therefore +V (t) = +� t +0 +˙V (s) ds ≤ p +p′ +� t +0 +V (s) ds + ∥∂tu∥p +Lp(Ωt) +≤ p +p′ +� t +0 +V (s) ds + C +� +∥d∥p +Lp(Ωt) + ∥u∥p +Lp(Ωt) +� +≤ C +�� t +0 +V (s) ds + ∥d∥p +Lp(Ωt) +� +. +By Gronwall’s inequality we get V (t) ≤ C∥d∥p +Lp(Ωt) and thus +∥u(t)∥Lp(M,¯g) ≤ C∥d∥Lp(Ωt) +for t ∈ [0, T]. +(6.8) +This already implies the uniqueness of strong solutions of (6.2), (6.3), since the difference u of two solutions +u1, u2 ∈ W 2,1 +p +(ΩT ) of (6.2), (6.3) satisfies (6.2), (6.3) with u0 = 0 and d = 0. Moreover, if u ∈ W 2,1 +p +(ΩT ) is a +strong solution of (6.2), (6.3), then the function ˆu ∈ W 2,1 +p +(ΩT ) given by ˆu(x, t) := u(x, t) − u0(x) safisfies (6.2), +(6.3) with u0 = 0 and d replaced by ˆd given by +ˆd(x, t) = d(x, t) + a(x, t)∆¯gu0(x) + c(x, t)u0(x). +Consequently, combining (6.7) and (6.8), and using an interpolation estimate for Du, we find that +∥u∥W 2,1 +p +(ΩT ) ≤ ∥ˆu∥W 2,1 +p +(ΩT ) + ∥u0∥W 2,p(M,¯g) ≤ C +� +∥ ˆd∥Lp(ΩT ) + ∥ˆu∥Lp(ΩT ) +� ++ ∥u0∥W 2,p(M,¯g) +≤ C∥ ˆd∥Lp(ΩT ) + ∥u0∥W 2,p(M,¯g) ≤ C +� +∥d∥Lp(ΩT ) + ∥u0∥W 2,p(M,¯g) +� +, + +24 +Franziska Borer, Peter Elbau, Tobias Weth +as claimed in (6.4). +Step 2 (Existence): In the case where a, c, d ∈ Cα(ΩT ) and u0 ∈ C2+α(M), the existence of a classical +solution u ∈ C(ΩT ) ∩ C2,1(ΩT ) of (6.2), (6.3) follows as in [14, Theorem 5.14]. +In the general case we consider (6.2), (6.3) with coefficients an, cn, dn ∈ Cα(ΩT ), u0,n ∈ C2+α(M), in place +of a, c, d, u0 with the property that an → a, cn → c in L∞(ΩT ), dn → d ∈ Lp(ΩT ) as well as u0,n → u0 in +W 2,p. The associated unique solutions un ∈ C(ΩT ) ∩ C2,1(ΩT ) are uniformly bounded in W 2,1 +p +(ΩT ) by (6.4), +and therefore we have un ⇀ u in W 2,1 +p +(ΩT ) after passing to a subsequence. For every φ ∈ C∞ +c (ΩT ), we then +have +� +ΩT +� +∂tu(x, t) − a(x, t)∆¯gu(t.x) − c(x, t)u(x, t) − d(x, t) +� +φ(x, t)dµ¯g(x)dt += lim +n→∞ +� +ΩT +� +∂tun(x, t) − an(x, t)∆¯gun(x, t) − cn(x, t)un(x, t) − dn(x, t) +� +φ(x, t)dµ¯g(x)dt = 0, +and from this we deduce that ∂tu(x, t) − a(x, t)∆¯gu(x, t) − c(x, t)u(x, t) − d(x, t) = 0 almost everywhere in ΩT , +so u is a strong solution of (6.2). +Step 3: It remains to show the inequality (6.5) in the case where a, c, d ∈ Cα(ΩT ) for some α > 0. Since +u ∈ C(ΩT ) ∩ C2,1(ΩT ) in this case and therefore +∥u0∥L2(M,¯g) = lim +t→0+ ∥u(t)∥L2(M,¯g), +it suffices to show that +∥∇u0∥L2(M,¯g) ≥ lim sup +t→0+ ∥∇u(t)∥L2(M,¯g). +(6.9) +If u0 ∈ C2+α(M) for some α > 0, this follows by [14, Theorem 5.14] with lim in place of lim sup, since the +function t �→ u(t) is continuous from [0, T) → C2+α(M) in this case. Moreover, in this case we have, by H¨older’s +and Young’s inequality, +d +dt∥∇u(t)∥2 +L2(M,¯g) = − +� +M +∂tu(t)∆u(t)dµ¯g += − +� +M +� +a(t)|∆u(t)|2 + c(t)u(t)∆u(t) + d(t)∆u(t) +� +dµ¯g +≤ −aT ∥∆¯gu(t)∥2 +L2(M,¯g) + ∥c(t)u(t) + d(t)∥L2(M,¯g)∥∆¯gu(t)∥L2(M,¯g) +≤ −aT ∥∆¯gu(t)∥2 +L2(M,¯g) + aT ∥∆¯gu(t)∥2 +L2(M,¯g) + +1 +4aT +∥c(t)u(t) + d(t)∥2 +L2(M,¯g) += +1 +4aT +∥c(t)u(t) + d(t)∥2 +L2(M,¯g), +and therefore +∥∇u(t)∥2 +L2(M,¯g) ≤ ∥∇u(0)∥2 +L2(M,¯g) + +1 +4aT +� t +0 +∥c(s)u(s) + d(s)∥2 +L2(M,¯g) ds +for t > 0. +(6.10) +In the general case, we consider (6.2), (6.3) with a sequence of initial conditions un,0 in place of u0, where +un,0 → u0 in H2(M). +The associated unique solutions un ∈ C(ΩT ) ∩ C2,1(ΩT ) are uniformly bounded in +W 2,1 +p +(ΩT ) by (6.4), and they are also uniformly bounded in C2,1([ε, T] × M) by [14, Theorem 5.15] for every +ε ∈ (0, T). Fix t ∈ (0, T). Passing to a subsequence, we may assume that un ⇀ u in W 2,1 +p +(ΩT ), un → u +strongly in C0(ΩT ) and un(t) → u(t) strongly in C1(M). As in Step 2, we see, by testing with φ ∈ C∞ +c (ΩT ), +that ∂tu(x, t) − a(x, t)∆¯gu(x, t) − c(x, t)u(x, t) − d(x, t) = 0 almost everywhere in ΩT , so u is the unique strong +solution of (6.2), (6.3). Moreover, by (6.10) we have +∥∇u(t)∥2 +L2(M,¯g) = lim +n→∞ ∥∇un(t)∥2 +L2(M,¯g) +≤ lim +n→∞ +� +∥∇un(0)∥2 +L2(M) + +1 +4aT +� t +0 +∥c(s)un(s) + d(s)∥2 +L2(M,¯g) ds +� += ∥∇u(0)∥2 +L2(M,¯g) + +1 +4aT +� t +0 +∥c(s)u(s) + d(s)∥2 +L2(M,¯g) ds. +It thus follows that +∥∇u(t)∥2 +L2(M,¯g) − ∥∇u(0)∥2 +L2(M,¯g) ≤ +1 +4aT +� t +0 +∥c(s)u(s) + d(s)∥2 +L2(M,¯g) ds + +Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic +25 +and therefore +lim sup +t→0 +� +∥∇u(t)∥2 +L2(M,¯g) − ∥∇u(0)∥2 +L2(M,¯g) +� +≤ +1 +4aT +lim +t→0+ +� t +0 +∥c(s)u(s) + d(s)∥2 +L2(M,¯g) ds = 0, +as claimed in (6.9). +Next we prove a maximum principle for solutions of (6.2), (6.3). We need the following preliminary lemma. +Lemma 6.2. Let T > 0. +(i) For any function u ∈ C2(M) we have +� +{x∈M|u(x)>0} +∆¯gudµ¯g ≤ 0. +(ii) Let u, ρ ∈ C1([0, T]) be functions with u(0) ≤ 0 and ρ(T) ≥ 0. Then +� +{t∈[0,T ]|u(t)>0} +� +ρ(t)∂tu(t) + κu(t) +� +dt ≥ 0 +with +κ := +sup +s∈(0,T ) +∂tρ(s). +(6.11) +(iii) Let u ∈ C2,1(ΩT ) ∩ C0,1(ΩT ), ρ ∈ C0,1(ΩT ) be functions with u ≤ 0 on M × {0} and ρ ≥ 0 on M × {T}. +Then we have +� +{(x,t)∈M×[0,T ]|u(x,t)>0} +(ρ(x, t)∂tu(x, t) + κu(x, t) − ∆¯gu(x, t))dµ¯g(x)dt ≥ 0 +with +κ := +sup +(s,x)∈M×(0,T ) +∂tρ(s, x). +(6.12) +Proof. (i) By Lebesgue’s theorem, it suffices to prove +� +{x∈M|u(x)>εn} +∆¯gudµ¯g ≤ 0 +(6.13) +for a sequence εn → 0+. By Sard’s Lemma, we may choose this sequence such that Ωε := {x ∈ M | u(x) > εn} +is an open set of class C1, whereas the outer unit vector field of Ωε is given by (x, t) �→ − +∇¯gu(x,t) +|∇¯gu(x,t)|¯g . Hence +(6.13) follows from the divergence theorem. +(ii) The set {t ∈ [0, T] | u(t) > 0} is a union of at most countably many open intervals Ij, j ∈ N. For any such +interval, partial integration gives +� +Ij +� +ρ(t)∂tu(t) + ∂tρ(t)u(t) +� +dt = +� +0, +if T ̸∈ Ij; +ρ(T)u(T) ≥ 0, +if T ∈ Ij. +Consequently, +� +{t∈[0,T ]|u(t)>0} +ρ(t)∂tu(t) dt ≥ − +� +{t∈[0,T ]|u(t)>0} +∂tρ(t)u(t) dt ≥ − +� +{t∈[0,T ]|u(t)>0} +κu(t) dt +with κ given in (6.11). This shows the claim. +(iii) This is a direct consequence of (i), (ii) and Fubini’s theorem. +Proposition 6.3. (Maximum principle) +Let T > 0, a, c ∈ C(ΩT ) with aT := +min +(x,t)∈ΩT +a(x, t) > 0, let d ∈ Lp(ΩT ) for some p > 2 with dT := +sup(x,t)∈ΩT d(x, t) < ∞, and let u0 ∈ W 2,p(M, ¯g). +Moreover, let u ∈ W 2,1 +p +(ΩT ) be the unique solution of +(6.2), (6.3). +(i) If u0 ≤ 0 on M and dT ≤ 0, then u ≤ 0 on ΩT . +(ii) If c ≡ 0 on ΩT , then +u(x, t) ≤ ∥u+ +0 ∥L∞(M,¯g) + tdT +for t ∈ [0, T], x ∈ M. +(6.14) + +26 +Franziska Borer, Peter Elbau, Tobias Weth +Proof. (i) Step 1: We consider the special case a ∈ C0,1(ΩT ), u0 ≤ 0 and dT ≤ −ε for some ε > 0. We put +ρ := 1 +a ∈ C0,1(ΩT ) and κ := +sup +(s,x)∈M×(0,T ) +∂tρ(s, x) as in (6.12). Moreover, we consider the function +˘u ∈ W 2,1 +p +(ΩT ), +˘u(x, t) = e−˘κtu(x, t) +with ˘κ = +|κ| +min(x,t)∈ΩT ρ(x,t) + ∥c∥L∞(ΩT ), noting that ˘u satisfies +ρ(x, t)∂t˘u(x, t) − ∆¯g˘u(x, t) + κ˘u(x, t) += e−˘κt� +u(x, t)(ρ(x, t)c(x, t) − ρ(x, t)˘κ + κ) + ρ(x, t)d(x, t) +� +≤ −ρ(x, t)εe−˘κt +almost everywhere in {(x, t) ∈ ΩT | ˘u(x, t) > 0}. +(6.15) +We now let (un)n∈N be a sequence in C2,1(ΩT ) ∩ C0,1(ΩT ) with un(x, 0) ≤ 0 and un → ˘u in W 2,1 +p +(ΩT ). +Since the functions gn := 1{(x,t)∈M×[0,T ]|un(x,t)>0} are bounded in Lp′(ΩT ), we may pass to a subsequence such +that gn ⇀ g in Lp′(ΩT ), where g ≥ 0 and g ≡ 1 in {(x, t) ∈ M × [0, T] | ˘u(x, t) > 0}, since un → ˘u uniformly +as a consequence of (6.1) and therefore gn → 1 pointwisely on {(x, t) ∈ M × [0, T] | ˘u(x, t) > 0}. Applying +Lemma 6.2 (iii) to un, we find that +0 ≤ +� +{(x,t)∈M×[0,T ]|un(x,t)>0} +� +ρ(x, t)∂tun(t) − ∆¯gun(x, t) + κun(x, t) +� +dµ¯g(x)dt += +� +M×(0,T ) +gn(x, t) +� +ρ(x, t)∂tun(x, t) − ∆¯gun(x, t) + κun(x, t) +� +dµ¯g(x)dt +for all n ∈ N and therefore +0 ≤ lim +n→∞ +� +M×(0,T ) +gn(x, t) +� +ρ(x, t)∂tun(x, t) − ∆¯gun(x, t) + κun(x, t) +� +dµ¯g(x)dt += +� +M×(0,T ) +g(x, t) +� +ρ(x, t)∂t˘u(x, t) − ∆¯g˘u(x, t) + κ˘u(x, t) +� +dµ¯gdt +≤ − +� +M×(0,T ) +g(x, t)ρ(x, t)εe−˘κtdµ¯g(x)dt ≤ − +� +{(x,t)∈M×(0,T )|˘u(x,t)>0} +ρ(x, t)εe−˘κtdµ¯g(x)dt. +We thus conclude that {(x, t) ∈ M × (0, T) | ˘u(x, t) > 0} = {(x, t) ∈ M × (0, T) | u(x, t) > 0} = ∅ and therefore +u ≤ 0 in M × (0, T). +Step 2: In the special case where a ∈ C0,1(ΩT ), u0 ≤ 0 and dT ≤ 0, we may apply Step 1 to the functions +uε ∈ W 2,1 +p +(ΩT ) defined by uε(x, t) = u(x, t) − εt, which yields that uε ≤ 0 for every ε > 0 and therefore u ≤ 0 +in ΩT . +Step 3: In the general case, we consider a sequence an ∈ C0,1(ΩT ) with an → a in C(ΩT ), and we let un denote +the associated solutions of (6.2), (6.3) with a replaced by an. 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Mech. 17 (1967), +pp. 473–484. + diff --git a/LtFLT4oBgHgl3EQfMS8S/content/tmp_files/load_file.txt b/LtFLT4oBgHgl3EQfMS8S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf555cbf2cd21a64f714c1eade861bd70f38778b --- /dev/null +++ b/LtFLT4oBgHgl3EQfMS8S/content/tmp_files/load_file.txt @@ -0,0 +1,1326 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf,len=1325 +page_content='A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic Franziska Borer∗ Peter Elbau† Tobias Weth‡ Abstract On a closed Riemannian surface (M, ¯g) with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume A > 0 and the property that their Gauss curvatures fλ = f + λ are given as the sum of a prescribed function f ∈ C∞(M) and an additive constant λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Our main tool in this study is a new variant of the prescribed Gauss curvature flow, for which we establish local well-posedness and global compactness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In contrast to previous work, our approach does not require any sign conditions on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, we exhibit conditions under which the function fλ is sign changing and the standard prescribed Gauss curvature flow is not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Acknowledgment This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project 408275461 (Smoothing and Non-Smoothing via Ricci Flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We would like to thank Esther Cabezas–Rivas for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Introduction Let (M, ¯g) be a two-dimensional, smooth, closed, connected, oriented Riemann manifold endowed with a smooth background metric ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A classical problem raised by Kazdan and Warner in [11] and [10] is the question which smooth functions f : M → R arise as the Gauss curvature Kg of a conformal metric g(x) = e2u(x)¯g(x) on M and to characterise the set of all such metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For a constant function f, this prescribed Gauss curvature problem is exactly the statement of the Uni- formisation Theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [16], [12]): There exists a metric g which is pointwise conformal to ¯g and has constant Gauss curvature Kg ≡ ¯K ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We now use this statement to assume in the following without loss of generality that the background metric ¯g itself has constant Gauss curvature K¯g ≡ ¯K ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore we can normalise the volume of (M, ¯g) to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We recall that the Gauss curvature of a conformal metric g(x) = e2u(x)¯g(x) on M is given by the Gauss equation Kg(x) = e−2u(x)(−∆¯gu(x) + ¯K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) Therefore the problem reduces to the question for which functions f there exists a conformal factor u solving the equation − ∆¯gu(x) + ¯K = f(x)e2u(x) in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) Given a solution u, we may integrate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) with respect to the measure µ¯g on M induced by the Riemannian volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Using the Gauss–Bonnet Theorem, we then obtain the identity � M f(x)dµg(x) = � M ¯Kdµ¯g(x) = ¯K vol¯g = ¯K = 2πχ(M), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) where dµg(x) = e2u(x)dµ¯g(x) is the element of area in the metric g(x) = e2u(x)¯g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We note that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) immediately yields necessary conditions on f for the solvability of the prescribed Gauss curvature problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular, if ±χ(M) > 0, then ±f must be positive somewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, if χ(M) = 0, then f must change sign or must be identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ∗Technical University of Berlin, Faculty II—Mathematics and Natural Sciences, Straße des 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Juni 136, 10623 Berlin, Germany email: borer@tu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='de †Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria email: peter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='elbau@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='at ‡Goethe University Frankfurt, Institut f¨ur Mathematik, Robert-Mayer-Straße 10, 60629 Frankfurt, Germany email: weth@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='uni-frankfurt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='de 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12015v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='AP] 27 Jan 2023 2 Franziska Borer, Peter Elbau, Tobias Weth In the present paper we focus on the case χ(M) < 0, so M is a surface of genus greater than one and ¯K < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The complementary cases χ(M) ≥ 0—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=', the cases where M = S2 or M = T, the 2-torus—will be discussed briefly at the end of this introduction, and we also refer the reader to [18, 19, 2, 8] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Multiplying equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) with the factor e−2u and integrating over M with respect to the measure µ¯g, we get the following necessary condition—already mentioned by Kazdan and Warner in [11]—for the average ¯f := 1 vol¯g � M f(x)dµ¯g(x), with vol¯g := � M dµ¯g(x): ¯f = 1 vol¯g � M f(x)dµ¯g(x) = � M (−∆¯gu(x) + ¯K)e−2u(x)dµ¯g(x) = � M (−2|∇¯gu(x)|2 ¯g + ¯K)e−2u(x)dµ¯g(x) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) This condition is not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Indeed, it has already been pointed out in [11, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5] that in the case χ(M) < 0 there always exist functions f ∈ C∞(M) with ¯f < 0 and the property that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) has no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We recall that solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) can be characterised as critical points of the functional Ef : H1(M, ¯g) → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ef(u) := 1 2 � M � |∇¯gu(x)|2 ¯g + 2 ¯Ku(x) − f(x)e2u(x)� dµ¯g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) Under the assumption χ(M) < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=', ¯K < 0, the functional Ef is strictly convex and coercive on H1(M, ¯g) if f ≤ 0 and f does not vanish identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence, as noted in [7], the functional Ef admits a unique critical point uf ∈ H1(M, ¯g) in this case, which is a strict absolute minimiser of Ef and a (weak) solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The situation is more delicate in the case where fλ = f0 + λ, where f0 ≤ 0 is a smooth, nonconstant function on M with maxx∈M f0(x) = 0, and λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the case where λ > 0 sufficiently small (depending on f0), it was shown in [7] and [1] that the corresponding functional Efλ admits a local minimiser uλ and a further critical point uλ ̸= uλ of mountain pass type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' These results motivate our present work, where we suggest a new flow approach to the prescribed Gausss curvature problem in the case χ(M) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' It is important to note here that there is an intrinsic motivation to formulate the static problem in a flow context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Typically, elliptic theories are regarded as the static case of the corresponding parabolic problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' in that sense, many times the better-understood elliptic theory has been a source of intuition to generalise the corresponding results in the parabolic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Examples of this feedback are minimal surfaces/mean curvature flow, harmonic maps/solutions of the heat equation, and the uniformisation theorem/the two-dimensional normalised Ricci flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this spirit, a flow approach to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), the so-called prescribed Gauss curvature flow, was first introduced by Struwe in [18] (and [2]) for the case M = S2 with the standard background metric and a positive function f ∈ C2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' More precisely, he considers a family of metrics (g(t, ·))t≥0 which fulfils the initial value problem ∂tg(t, x) = 2(α(t)f(x) − Kg(t,·)(x))g(t, x) in (0, T) × M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6) g(0, x) = g0(x) on {0} × M, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) with α(t) = � M Kg(t,·)(x)dµg(t,·)(x) � M f(x)dµg(t,·)(x) = 2πχ(M) � M f(x)dµg(t,·)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) This choice of α(t) ensures that the volume of (M, g(t, ·)) remains constant throughout the deformation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=', � M dµg(t,·)(x) = � M e2u(t,x)dµ¯g(x) ≡ volg0 for all t ≥ 0, where g0 denotes the initial metric on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Equivalently one may consider the evolution equation for the associated conformal factor u given by g(t, x) = e2u(t,x)¯g(x): ∂tu(t, x) = α(t)f(x) − Kg(t,·)(x) in (0, T) × M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) u(0, x) = u0(x) on {0} × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10) Here the initial value u0 is given by g0(x) = e2u0(x)¯g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The flow associated to this parabolic equation is usually called the prescribed Gauss curvature flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With the help of this flow, Struwe [18] provided a new proof of a result by Chang and Yang [6] on sufficient criteria for a function f to be the Gauss curvature of a metric g(x) = e2u(x)gS2(x) on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' He also proved the sharpness of these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the case of surfaces with genus greater than one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=', with negative Euler characteristic, the prescribed Gauss curvature flow was used by Ho in [9] to prove that any smooth, strictly negative function on a surface with negative Euler characteristic can be realised as the Gaussian curvature of some metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' More precisely, Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 3 assuming that χ(M) < 0 and that f ∈ C∞(M) is a strictly negative function, he proves that equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) has a solution which is defined for all times and converges to a metric g∞ with Gaussian curvature Kg∞ satisfying Kg∞(x) = α∞f(x) for some constant α∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' While the prescribed Gauss curvature flow is a higly useful tool in the cases where f is of fixed sign, it cannot be used in the case where f is sign-changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Indeed, in this case we may have � M f(x)dµg(t,·)(x) = 0 along the flow and then the normalising factor α(t) is not well-defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' As a consequence, a long-time solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) might not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular, the static existence results of [7] and [1] can not be recovered and reinterpreted with the standard prescribed Gauss curvature flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this paper we develop a new flow approach to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) in the case χ(M) < 0 for general f ∈ C∞(M), which sheds new light on the results in [7], [1] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The main idea is to replace the multiplicative normalisation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) by an additive normalisation, as will be described in details in the next chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' At this point, it should be noted that the normalisation factor α(t) in the prescribed Gauss curvature flow given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) is also not the appropriate choice in the case of the torus, where, as noted before, f has to change sign or be identically zero in order to arise as the Gauss curvature of a conformal metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The case of the torus was considered by Struwe in [19], where, in particular, he used to a flow approach to reprove and partially improve a result by Galimberti [8] on the static problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this approach, the normalisation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) is replaced by α(t) = � M f(x)Kg(t,·)(x)dµg(t,·)(x) � M f 2(x)dµg(t,·)(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) With this choice, Struwe shows that for any smooth u0 ∈ C∗ := � u ∈ H1(M, ¯g) | � M f(x)e2u(x)dµ¯g(x) = 0, � M e2u(x)dµ¯g(x) = 1 � there exists a unique, global smooth solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) satisfying u(t, ·) ∈ C∗ for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, u(t, ·) → u∞(·) in H2(M, ¯g) (and smoothly) as t → ∞ suitably, where u∞ + c∞ is a smooth solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) for some c∞ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In principle, the normalisation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) could also be considered in the case χ(M) < 0, but then the flow is not volume-preserving anymore, which results in a failure of uniform estimates for solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, we were not able to make use of the associated flow in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In Section 2 we set up the framework for the new variant of the prescribed Gauss curvature flow with additive normalisation, and we collect basic properties of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In Section 3, we then present our main result on the long-time existence and convergence of the flow (for suitable times tk → ∞) to solutions of the corresponding static problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular, our results show how sign changing functions of the form fλ = f0 + λ arise depending on various assumptions on the shape of f0 and on the fixed volume A of M with respect to the metric g(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Before proving our results on the time-dependent problem, we first derive, in Section 4, some results on the static problem with volume constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Most of these results will then be used in Section 5, where the parabolic problem is studied in detail and the main results of the paper are proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the appendix, we provide some regularity estimates and a variant of a maximum princple for a class of linear evolution problems with H¨older continuous coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the remainder of the paper, we will use the short form f, g(t), u(t), Kg(t), volg(t) := � M dµg(t) = � M e2u(t)dµ¯g, and so on instead of f(x), g(t, x), u(t, x), Kg(t,·)(x), � M dµg(t,·)(x) = � M e2u(t,x)dµ¯g(x), et cetera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A New Flow Approach and Some of its Properties Let f ∈ C∞(M) be a smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We consider now the additive rescaled prescribed Gauss curvature flow given by ∂tu(t) = f − Kg(t) − α(t) = f − e−2u(t)(∆¯gu(t) − ¯K) − α(t) in (0, T) × M, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) where α(t) is chosen such that the volume volg(t) of M with respect to g(t) = e2u(t)¯g remains constant along the flow, that is, we require the condition 1 2 d dt volg(t) = � M ∂tu(t)dµg(t) = � M (f − Kg(t) − α(t))dµg(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) Solving for α(t) then we find α(t) = 1 volg(t) �� M fdµg(t) − ¯K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 4 Franziska Borer, Peter Elbau, Tobias Weth So, starting with u0 ∈ Cp,A := � v ∈ W 2,p(M, ¯g) | � M e2vdµ¯g = A � , p > 2, for a given A > 0, we have volg(t) = volg(0) = volg0 = A, for all t ≥ 0, hence we can define αA(t) = 1 A �� M fdµg(t) − ¯K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) Therefore in the following we consider the flow ∂tu(t) = f − Kg(t) − αA(t) in (0, T) × M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) u(0) = u0 ∈ Cp,A on {0} × M, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) with αA(t) is chosen like in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We can now state some first properties of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u be a (sufficiently smooth) solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' the volume volg(t) of (M, g(t)) is preserved along the flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=', volg(t) ≡ volg0 = A for all t ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' along this trajectory, we have a uniform bound for α given by α(t) ≥ min x∈M f(x) + | ¯K| A =: α1 > −∞ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6) and α(t) ≤ max x∈M f(x) + | ¯K| A =: α2 < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' the flow is invariant under adding or subtracting a constant C > 0 to the function f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' and the energy Ef, defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5), is decreasing in time along the flow, so Ef(u(t)) ≤ Ef(u0) for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The first statement directly follows by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) and the choice of α in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The second one we get since f is smooth and volg(t) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' To show the invariance of the flow, let C > 0 be a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We then replace f by f ± C in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) and see that f ± C − Kg(t) − 1 A �� M (f ± C)dµg(t) − ¯K � = f − Kg(t) − 1 A �� M fdµg(t) − ¯K � = ∂tu(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, the flow (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) is left unchanged if we replace f by f ± C for a constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' To see that the energy Ef is decreasing along the flow, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) and get d dtEf(u(t)) = � M (−∆¯gu(t) + ¯K − fe2u(t))∂tu(t)dµ¯g = � M ((−∆¯gu(t) + ¯K)e−2u(t) − f)e2u(t)∂tu(t)dµ¯g = � M ((−∆¯gu(t) + ¯K)e−2u(t) − f)∂tu(t)dµg(t) = � M (Kg(t) − f)∂tu(t)dµg(t) = � M (Kg(t) − f + α(t))∂tu(t)dµg(t) = − � M |∂tu(t)|2dµg(t) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) Therefore on an interval [0, T], we have the uniform a-priori bound Ef(u(T)) + � T 0 � M |∂tu(t)|2dµg(t)dt = Ef(u(0)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) for any T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Main Results The following is our first main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M), p > 2, and u0 ∈ Cp,A for a given A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then the initial value problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) admits a unique global solution u ∈ C([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' C(M)) ∩ C([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' H1(M, ¯g)) ∩ C∞((0, ∞) × M) satisfying the energy bound Ef(u(t)) ≤ Ef(u0) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, u is uniformly bounded in the sense that sup t>0 ∥u(t)∥L∞(M,¯g) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore, as t → ∞ suitably, u converges to a function u∞ in H2(M, ¯g) solving the equation − ∆¯gu + ¯K = fλe2u in M, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) where fλ := f + λ with λ = 1 A � ¯K − � M fe2u∞dµ¯g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) In other words, u∞ induces a metric g∞ with Gauss curvature Kg∞ satisfying Kg∞(x) = fλ(x) = f(x) + λ for x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For functions f < 0, the convergence of the flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) is shown in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For the additive rescaled flow (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) with initial data (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) we get convergence for arbitrary functions f ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In general we do not have any information about λ and therefore no information about the sign of fλ in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' On the other hand, more information can be derived for certain functions f ∈ C∞(M) and certain values of A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (i) In the case where A ≤ − ¯ K ∥f∥L∞(M,¯g) , it follows that λ = 1 A � ¯K − � M fe2udµ¯g � ≤ ¯K A + ∥f∥L∞(M,¯g) A � M e2udµ¯g = ¯K A + ∥f∥L∞(M,¯g) ≤ 0 for every solution u ∈ C2,A := � v ∈ H2(M, ¯g) | � M e2vdµ¯g = 0 � of the static problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1), and therefore this also applies to λ in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (ii) The following theorems show that fλ in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 may change sign if A > − ¯ K ∥f∥L∞(M,¯g) , so in this case we get a solution of the static problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) for sign-changing functions f ∈ C∞(M) by using the additive rescaled prescribed Gauss curvature flow (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For every A > 0 and c > − ¯ K A there exists ε = ε(c, A, ¯K) > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' If u0 ≡ 1 2 log(A) ∈ Cp,A and f ∈ C∞(M) with −c ≤ f ≤ 0 and ∥f + c∥L1(M,¯g) < ε is chosen in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, then the value λ defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular, if f has zeros on M, then fλ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) is sign changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Under fairly general assumptions on f, we can prove that λ > 0 if A is sufficiently large and u0 ∈ Cp,A is chosen suitably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M) be nonconstant with maxx∈M f(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then there exists κ > 0 with the property that for every A ≥ κ there exists u0 ∈ Cp,A such that the value λ defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In fact we have even more information on the associated limit u∞ in this case, see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' It remains open how large λ can be depending on A and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The only upper bound we have is λ < − � M fdµ¯g, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) since we must have ¯fλ = 1 vol¯g � M fλdµ¯g = � M fdµ¯g + λ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='< 0, so that fλ fulfills the necessary condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) provided by Kazdan and Warner in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 6 Franziska Borer, Peter Elbau, Tobias Weth 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The static Minimisation Problem with Volume Constraint To obtain additional information on the limiting function u∞ and the value λ ∈ R associated to it by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3), we need to consider the associated static setting for the prescribed Gauss curvature problem with the additional condition of prescribed volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Before going into the details of this static problem, we recall an important and highly useful estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The following lemma (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [5, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7]) is a consequence of the Trudinger’s inequality [20] which was improved by Moser in [15] (for more details see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [19, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2]): Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For a two-dimensional, closed Riemannian manifold (M, ¯g) there are constants η > 0 and CMT > 0 such that � M e(u−¯u)dµ¯g ≤ CMT exp � η∥∇¯gu∥2 L2(M,¯g) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) for all u ∈ H1(M, ¯g) where ¯u := 1 vol¯g � M u dµ¯g = � M u dµ¯g, in view of our assumption that vol¯g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' As a consequence of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, we have � M epudµ¯g = ep¯u � M e(pu− ¯ pu)dµ¯g ≤ ep¯uCMT exp � η∥∇¯g(pu)∥2 L2(M,¯g) � < ∞ for every u ∈ H1(M, ¯g) and p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, for a given A > 0, the set C1,A := � u ∈ H1(M, ¯g) | V (u) := � M e2udµ¯g = A � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) is well defined and coincides with the closure of C2,A with respect to the H1-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We also note that ¯u ≤ 1 2 log(A) for u ∈ C1,A, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) since by Jensen’s inequality and our assumption that vol¯g = 1 we have 2¯u = − � M 2udµ¯g = � M 2udµ¯g ≤ log � − � e2udµ¯g � = log(A) for u ∈ C1,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore we want to recall the Gagliardo–Nirenberg–Ladyˇzhenskaya interpolation, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 (Gagliardo–Nirenberg–Ladyˇzhenskaya inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' There exists a constant CGNL > 0 such that we have for every ζ ∈ H1(M, ¯g) the inequality ∥ζ∥4 L4(M,¯g) ≤ CGNL∥ζ∥2 L2(M,¯g)∥ζ∥2 H1(M,¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Now we enter the details of the static prescribed Gauss curvature problem with volume constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this problem, we wish to find, for given f ∈ C∞(M) and A > 0, critical points of the restriction of the functional Ef defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) to the set C1,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A critical point u ∈ C1,A of this restriction is a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) for some λ ∈ R, where, here and in the following, we put again fλ := f + λ ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In other words, such a critical point induces, similarly as the limit u∞ in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, a metric gu with Gauss curvature Kgu satisfying Kgu(x) = fλ(x) = f(x) + λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The unknown λ ∈ R arises in this context as a Lagrangian multiplier and is a posteriori characterised again by λ = 1 A � ¯K − � M fe2udµ¯g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the study of critical points of the restriction of Ef to C1,A, it is natural to consider the minimisation problem first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For this we set mf,A = inf u∈C1,A Ef(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We have the following estimates for mf,A: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M), A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then we have mf,A ≤ 1 2 � ¯K log(A) − A � M fdµ¯g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) Moreover, if max f ≥ 0, then we have lim sup A→∞ mf,A A ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u0(A) ≡ 1 2 log(A), so that � M e2u0(A)dµ¯g = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence u0(A) is the (unique) constant function in C1,A, and mf,A ≤ Ef(u0(A)) = 1 2 � M (|∇¯gu0(A)|2 ¯g + 2 ¯Ku0(A) − fe2u0(A))dµ¯g = 1 2 � M ( ¯K log(A) − fA)dµ¯g = 1 2 � ¯K log(A) − A � M fdµ¯g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' This shows (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' To show (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5), we let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since f ∈ C∞(M) and max f ≥ 0 by assumption, there exists an open set Ω ⊂ M with f ≥ −ε on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Next, let ψ ∈ C∞(M), ψ ≥ 0, be a function supported in Ω and with ∥ψ∥L∞(M,¯g) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, the set Ω′ := {x ∈ M | ψ > 1} is a nonempty open subset of Ω, and therefore µ¯g(Ω′) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Next we consider the continuous function h : [0, ∞) → [0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' h(τ) = � M e2τψdµ¯g and we note that h(0) = � M dµ¯g = 1, and that h(τ) ≥ � Ω′ e2τψdµ¯g ≥ e2τµ¯g(Ω′) for τ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence for every A ≥ 1 there exists 0 ≤ τA ≤ 1 2 � log(A) − log(µ¯g(Ω′)) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6) with h(τA) = A and therefore τAψ ∈ C1,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, mf,A ≤ Ef(τAψ) = 1 2 � M (|∇¯gτAψ|2 ¯g + 2 ¯KτAψ − fe2τAψ)dµ¯g = τ 2 Ac1 − τAc2 − c3 − 1 2 � Ω fe2τAψdµ¯g with c1 = 1 2 � M |∇¯gψ|2 ¯gdµ¯g, c2 = − ¯K � M ψdµ¯g and c3 = 1 2 � M\\Ω fdµ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since f ≥ −ε on Ω, we thus deduce that mf,A ≤ τ 2 Ac1 − 2τAc2 + c3 + ε 2 � Ω e2τAψdµ¯g ≤ τ 2 Ac1 − 2τAc2 + c3 + εA 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since τA A → 0 as A → ∞ by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6), we conclude that lim sup A→∞ mf,A A ≤ ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since ε > 0 was chosen arbitrarily, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M) nonconstant with maxx∈M f(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For every ε > 0 there exists κ0 > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' If A ≥ κ0 and u ∈ C1,A is a solution of − ∆¯gu + ¯K = (f + λ)e2u (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) for some λ ∈ R with Ef(u) < εA 2 , then we have λ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For given ε > 0, we may choose κ0 > 0 sufficiently large so that | ¯ K| 2 log(A) |A| < ε 2 for A ≥ κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Now, let A ≥ κ0, and let u ∈ C1,A be a solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) satisfying Ef(u) < εA 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) over M with respect to µ¯g and using that vol¯g(M) = 1 and � M e2udµ¯g = A, we obtain λ = 1 A � ¯K − � M fe2udµ¯g � ≤ − 1 A � M fe2udµ¯g = 1 A � Ef(u) − 1 2 � M (|∇¯gu|2 ¯g + 2 ¯Ku)dµ¯g � ≤ 1 A � Ef(u) + | ¯K|¯u � ≤ ε 2 + | ¯K| 2 log(A) A < ε, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Here we used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) to estimate ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 8 Franziska Borer, Peter Elbau, Tobias Weth Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M) be a nonconstant function with maxx∈M f(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, let λn → 0+ for n → ∞, and let (un)n∈N be a sequence of solutions of − ∆¯gun + ¯K = (f + λn)e2un in M (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) which are weakly stable in the sense that � M (|∇¯gh|2 ¯g − 2(f + λn)e2unh2)dµ¯g ≥ 0 for all h ∈ H1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) Then un → u0 in C2(M), where u0 is the unique solution of − ∆¯gu0 + ¯K = fe2u0 in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We only need to show that (un)n∈N is bounded in C2,α(M) for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) Indeed, assuming this for the moment, we may complete the argument as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Suppose by contradiction that there exists ε > 0 and a subsequence, also denoted by (un)n∈N, with the property that ∥un − u0∥C2(M) ≥ ε for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) and the compactness of the embedding C2,α(M) �→ C2(M), we may then pass to a subsequence, still denoted by (un)n∈N, with un → u∗ in C2(M) for some u∗ ∈ C2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Passing to the limit in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8), we then see that u∗ is a solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10), which by uniqueness implies that u∗ = u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' This contradicts (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12), and thus the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) follows by similar arguments as in [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1063 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since the framework is slightly different, we sketch the main steps here for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We first note that, by the same argument as in [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1063 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='], there exists a constant C0 > 0 with un ≥ −C0 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13) Since {f < 0} is a nonempty open subset of M by assumption, we may fix a nonempty open subdomain Ω ⊂⊂ {f < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By [1, Appendix], there exists a constant C1 > 0 with ∥u+ n ∥H1(Ω,¯g) ≤ C1 for all n and therefore � Ω e2undµ¯g ≤ � Ω e2u+ n dµ¯g ≤ C2 for all n (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) for some C2 > 0 by the Moser–Trudinger inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Next, we consider a nontrivial, nonpositive function h ∈ C∞ c (Ω) ⊂ C∞(M) and the unique solution w ∈ C∞(M) of the equation −∆¯gw + ¯K = he2w in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, we let wn := un − w, and we note that wn satisfies −∆¯gwn + he2w = (f + λn)e2un in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Multiplying this equation by e2wn and integrating by parts, we obtain � M (f + λn)e2(un+wn)dµ¯g = � M � −∆¯gwn + he2w� e2wndµ¯g = � M � 2e2wn|∇¯gwn|2 ¯g + he2(w+wn)� dµ¯g = 2 � M |∇¯gewn|2 ¯gdµ¯g + � Ω he2undµ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15) Moreover, applying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) to h = ewn gives � M (f + λn)e2(un+wn)dµ¯g ≤ 1 2 � M |∇¯gewn|2 ¯gdµ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='16) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='16) yields ∥∇¯gewn∥2 L2(M,¯g) ≤ −2 3 � Ω he2undµ¯g ≤ 2 3∥h∥L∞(M,¯g)C2 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='17) Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 9 Next we claim that also ∥ewn∥L2(M,¯g) remains uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Suppose by contradiction that ∥ewn∥L2(M,¯g) → ∞ as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='18) We then set vn := ewn ∥ewn∥L2(M,¯g) , and we note that ∥vn∥L2(M,¯g) = 1 for all n and ∥∇¯gvn∥2 L2(M,¯g) → 0 as n → ∞ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='19) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, we may pass to a subsequence satisfying vn ⇀ v in H1(M, ¯g), where v is a constant function with ∥v∥L2(M,¯g) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='20) However, since ∥ewn∥L2(Ω,¯g) ≤ ∥eun∥L2(Ω,¯g)∥e−w∥L∞(Ω,¯g) ≤ � C2∥e−w∥L∞(Ω,¯g) for all n ∈ N by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) and therefore ∥v∥L2(Ω,¯g) = lim n→∞ ∥vn∥L2(Ω,¯g) = lim n→∞ ∥ewn∥L2(Ω,¯g) ∥ewn∥L2(M,¯g) = 0 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='18), we conclude that the constant function v must vanish identically, contradicting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, ∥ewn∥L2(M,¯g) remains uniformly bounded, which by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='17) implies that ewn remains bounded in H1(M, ¯g) and therefore in Lp(M, ¯g) for any p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since eun ≤ ∥ew∥L∞(M,¯g)ewn on M for all n ∈ N, it thus follows that also eun remains bounded in Lp(M, ¯g) for any p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), the same applies to the sequence un itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Therefore, applying successively elliptic Lp and Schauder estimates to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8), we deduce (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M) be a nonconstant function with maxx∈M f(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then there exists λ♯ and a C1-curve (−∞, λ♯] → C2(M);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' λ �→ uλ with the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (i) If λ ≤ 0, then uλ is the unique solution of − ∆¯gu + ¯K = fλe2u in M (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='21) and a global minimum of Efλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (ii) If λ ∈ (0, λ♯], then uλ is the unique weakly stable solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='21) in the sense of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9), and it is a local minimum of Efλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (iii) The curve of functions λ �→ uλ is pointwisely strictly increasing on M, and so the volume function (−∞, λ♯] → [0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' λ �→ V (λ) := � M e2uλdµ¯g (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='22) is continuous and strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We already know that, for λ ≤ 0, the energy Efλ admits a strict global minimiser uλ which depends smoothly on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, by [1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4], the curve λ �→ uλ can be extended as a C1-curve to an interval (−∞, λ♯] for some λ♯ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We also know from [1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4] that, for λ ∈ (−∞, λ♯], the solution uλ is strongly stable in the sense that Cλ := inf h∈H1(M,¯g) 1 ∥h∥2 H1(M,¯g) � M � |∇¯gh|2 ¯g − 2fλe2uλh2� dµ¯g > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='23) Here we note that the function λ �→ Cλ is continuous since uλ depends continuously on λ with respect to the C2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Next we prove that, after making λ♯ > 0 smaller if necessary, the function uλ is the unique weakly stable solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='21) for λ ∈ (0, λ♯].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Arguing by contradiction, we assume that there exists a sequence λn → 0+ and corresponding weakly stable solutions (un)n∈N of − ∆¯gun + ¯K = (f + λn)e2un in M (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='24) with the property that un ̸= uλn for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5, we know that un → u0 in C2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, vn := un − uλn → 0 in C2(M) as n → ∞, whereas the functions vn solve − ∆¯gvn = (f + λn) � e2un − e2uλn � = (f + λn)e2uλn � e2vn − 1 � in M for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='25) 10 Franziska Borer, Peter Elbau, Tobias Weth Combining this fact with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='23), we deduce that ∥vn∥2 H1(M,¯g) ≤ 1 Cλ � M � |∇¯gvn|2 ¯g − 2(f + λn)e2uλn v2 n � dµ¯g = 1 Cλ � M (f + λn)e2uλn � e2vn − 1 − 2vn � vndµ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since vn → 0 in C2(M), there exists a constant C > 0 with |(e2vn − 1 − 2vn)vn| ≤ C|vn|3 on M for all n ∈ N, which then implies with H¨older’s inequality and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 that ∥vn∥2 H1(M,¯g) ≤ C∥(f + λn)e2uλn ∥L∞(M,¯g)∥vn∥3 L3(M,¯g) ≤ C �� M |vn|3· 4 3 dµ¯g � 3 4 = C∥vn∥3 L4(M,¯g) ≤ C∥vn∥3 H1(M,¯g) with a constant C > 0 independent on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' This contradicts the fact that vn → 0 in H1(M) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The claim thus follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' It remains to prove that the curve of functions λ �→ uλ is pointwisely strictly increasing on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' This is a consequence of the uniqueness of weakly stable solutions stated in (ii) and the fact that, as noted in [7], if uλ0 is a solution for some λ0 ∈ (−∞, λ♯], it is possible to construct, via the method of sub- and supersolutions, for every λ < λ0, a weakly stable solution uλ with uλ < uλ0 everywhere in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f ∈ C∞(M) be nonconstant with maxx∈M f(x) = 0, and let λ♯ > 0 be given as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then there exists κ1 > 0 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' If A ≥ κ1 and u ∈ C1,A is a solution of − ∆¯gu + ¯K = (f + λ)e2u (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26) for some λ ∈ R with Ef(u) < λ♯A 2 , then 0 < λ < λ♯, and u is not a weakly stable solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26), so u ̸= uλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let κ0 > 0 be given as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4 for ε = λ♯ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, let κ1 := max � κ0, V (uλ♯) � with V defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Next, let u ∈ C1,A be a solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26) for some λ ∈ R with Ef(u) < λ♯A 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4, we then deduce that 0 < λ < λ♯, and by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6 (iii) we have u ̸= uλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since uλ is the unique weakly stable solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26), it follows that u is not weakly stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let p > 2, f ∈ C∞(M) be nonconstant with maxx∈M f(x) = 0, and let λ♯ > 0 be given as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then there exists κ > 0 with the property that for every A ≥ κ the set ˜C := � u0 ∈ C1,A ∩ W 2,p(M, ¯g) | Ef(u0) < λ♯A 2 � is nonempty, and for every u0 ∈ ˜C the global solution u ∈ C([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' C(M))∩C([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' H1(M, ¯g))∩C∞((0, ∞)× M) of the initial value problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) converges, as t → ∞ suitably, to a solution u∞ of the static problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26) for some λ ∈ (0, λ♯) which is not weakly stable and hence no local minimiser of Efλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let κ1 > 0 be given by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5), there exists κ ≥ κ1 > 0 with mf,A < λ♯A 4 for fixed A > κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, there exists u0 ∈ C1,A ∩ W 2,p(M, ¯g) with Ef(u0) < λ♯A 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, the global solution u ∈ C([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' C(M)) ∩ C([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' H1(M, ¯g)) ∩ C∞((0, ∞) × M) of the initial value problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) converges, as t → ∞ suitably, to a solution u∞ ∈ C1,A of the static problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26) for some λ ∈ R, whereas Ef(u∞) ≤ Ef(u0) < λ♯A 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, λ ∈ (0, λ♯) by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7, and u∞ is not weakly stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof of the Main Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Notation and Some Regularity Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this chapter we summarise different kind of estimates which will be useful later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the following, for T > 0 we use the notation Lp t Lr x := Lp([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lr(M, ¯g)) and Lp t Hq x := Lp([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hq(M, ¯g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' A first regularity result is therefore given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 11 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We have ∥θ∥4 Lp t L4x ≤ CGNL∥θ∥2 Lp t L2x∥θ∥2 Lp t H1x for θ ∈ Lp t H1 x with p ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 (Sobolev inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' There exists a constant CS > 0 such that for every ρ ∈ L∞ t H1 x, T ≤ 1, we have ∥ρ∥2 L4 t L4x ≤ CS(∥ρ∥2 L∞ t L2x + ∥∇¯gρ∥2 L2 t L2x) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 there exists a constant CGNL > 0 such that we have for all T ≤ 1 ∥ρ∥4 L4 t L4x = � T 0 ∥ρ(t)∥4 L4(M,¯g)dt ≤ CGNL � T 0 ∥ρ(t)∥2 L2(M,¯g)∥ρ(t)∥2 H1(M,¯g)dt ≤ CGNL∥ρ∥2 L∞ t L2x � T 0 (∥ρ(t)∥2 L2(M,¯g) + ∥∇¯gρ(t)∥2 L2(M,¯g))dt ≤ CGNL · T ∥ρ∥4 L∞ t L2x + CGNL∥ρ∥2 L∞ t L2x∥∇¯gρ∥2 L2 t L2x ≤ CGNL � ∥ρ∥4 L∞ t L2x + ∥ρ∥2 L∞ t L2x∥∇¯gρ∥2 L2 t L2x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By using Young’s inequality we have ∥ρ∥L∞ t L2x∥∇¯gρ∥L2 t L2x ≤ 1 2 � ∥ρ∥2 L∞ t L2x + ∥∇¯gρ∥2 L2 t L2x � and therefore ∥ρ∥2 L4 t L4x ≤ C 1 2 GNL � ∥ρ∥4 L∞ t L2x + 1 4(∥ρ∥2 L∞ t L2x + ∥∇¯gρ∥2 L2 t L2x)2 ≤ C 1 2 GNL(∥ρ∥2 L∞ t L2x + 1 2∥ρ∥2 L∞ t L2x + 1 2∥∇¯gρ∥2 L2 t L2x) ≤ 3 2C 1 2 GNL(∥ρ∥2 L∞ t L2x + ∥∇¯gρ∥2 L2 t L2x) =: CS(∥ρ∥2 L∞ t L2x + ∥∇¯gρ∥2 L2 t L2x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since T is finite, ρ ∈ L∞ t H1 x implies that ρ ∈ Lp t H1 x for all p ∈ [1, ∞] which shows that the upper bound is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore, since T < ∞ and vol¯g = 1, with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 we also have for every p, s ∈ [1, ∞] that Lq tLr x ⊂ Ls tLp x for q ≥ s, r ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since we will often use it in the following, we recall that for v ∈ CtCx := C([0, T], C(M)) we have ∥1 − ev∥2 L∞ t L∞ x ≤ e2∥v∥L∞ t L∞ x ∥v∥2 L∞ t L∞ x (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) since for x ∈ R we get with the Taylor expansion |ex − 1| = |1 − ex| ≤ |x|e|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 we get the following statements: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For a (sufficiently smooth) solution u of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) we have ¯u(t) ≥ 1 2 log � A Cup � =: m0(A, Ef(u0), f, CMT, η1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) with Cup = CMT exp(4η1(2Ef(u0) + | ¯K| log(A) + A maxx∈M f(x))) where η1 is a number determined by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, especially for a solution u of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) we have the uniform bound m0 ≤ ¯u(t) ≤ 1 2 log(A), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) where we used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) and the volume preserving property to get the upper bound of ¯u(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For a solution u of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) we have for all p ∈ R that � M e2pu(t)dµ¯g ≤ Cint(A, CMT, Ef(u0), f, ¯K, η1, η2, p), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6) where again, η1, η2 are numbers determined by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 12 Franziska Borer, Peter Elbau, Tobias Weth 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For this part we choose f = f0 where f0 ≤ 0 is a nonconstant, smooth function with maxx∈M f0(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then there exists a constant Clow = Clow(Cint, f0) > 0 such that � M |f0|dµg(t) ≥ Clow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u be a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We then know that u(t) ∈ CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) we have for all t ≥ 0 that ∥∇¯gu(t)∥2 L2(M,¯g) = 2Ef(u(t)) − � M (2 ¯Ku(t) − fe2u(t))dµ¯g = 2Ef(u(t)) + � M (2| ¯K|u(t) + fe2u(t))dµ¯g ≤ 2Ef(u0) + | ¯K| log(A) + A max x∈M f(x), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) where we used the fact that � M 2u(t)dµ¯g ≤ log(A) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) and since � M e2u(t)dµ¯g ≡ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With this and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 we can now estimate A = � M e2u(t)dµ¯g = e2¯u(t) � M e2(u(t)−¯u(t))dµ¯g ≤ e2¯u(t)CMT exp(η1∥∇¯g(2u(t))∥2 L2(M,¯g)) ≤ e2¯u(t)CMT exp(4η1(2Ef(u0) + | ¯K| log(A) + A max x∈M f(x))) =: Cupe2¯u(t), with Cup = Cup(A, CMT, Ef(u0), f, ¯K, η1) > 0 and therefore ¯u(t) ≥ 1 2 log � A Cup � =: m0(A, CMT, Ef(u0), f, ¯K, η1) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, for a solution u(t) ∈ CA of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) we get the uniform bound m0 ≤ ¯u(t) ≤ 1 2 log(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u be a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, u(t) ∈ CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) we directly get for any p ∈ R that � M e2pu(t)dµ¯g = e2p¯u(t) � M e2p(u(t)−¯u(t))dµ¯g ≤ e2p¯u(t)CMT exp(4η2p2∥∇¯gu(t)∥2 L2(M,¯g)) ≤ Cint, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) where Cint = Cint(A, CMT, Ef(u0), f, ¯K, η1, η2, p) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Similar to [19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3] we see by the choice of f0, H¨older’s inequality, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) that 0 < ���� � M � |f0|dµ¯g ���� 2 ≤ � M |f0|e2u(t)dµ¯g � M e−2u(t)dµ¯g ≤ Cint � M |f0|e2u(t)dµ¯g (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10) which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3 is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Now we can turn to the proofs of the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Short-Time Existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let A > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We are looking for a short-time solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) with initial data (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Using the Gauss equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) we can rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) in the following way: ∂tu(t) = f − Kg(t) − αA(t) = e−2u(t)∆¯gu(t) + ¯K � 1 A − e−2u(t) � + f − 1 A � M fe2u(t)dµ¯g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) u(0) = u0 ∈ Cp,A := � u ∈ W 2,p(M, ¯g) | � M e2u = A � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 13 with p > 2, where αA(t) = 1 A �� M fdµg(t) − ¯K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' To find a solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12), we consider the linear equation ∂tu(t) = e−2v(t)∆¯gu(t) + ¯K � 1 A − e−2v(t) � + f − 1 A � M fe2v(t)dµ¯g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13) u(0) = u0 ∈ Cp,A, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) and use a fixed point argument in the space (X, ∥ · ∥X) := (CtCx, ∥ · ∥CtCx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' First we observe that for v ∈ CtCx, equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13) is strongly parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermoren, with p > 2 and the fact that M is compact, we have u0 ∈ Cp,A ⊂ H2(M, ¯g), and therefore u0 ∈ L∞(M, ¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For the fixed point argument we fix R = R(u0) := ∥u0∥L∞(M,¯g) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For fixed T > 0, let X = CtCx = C([0, T], C(M, ¯g)) �→ L∞ t L∞ x with ∥u∥X = max t∈[0,T ], x∈M |u(x, t)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For v ∈ X, by [14, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='32] and the appendix, we get a unique solution uv ∈ W 2,1 p = W 1,p t Lp x ∩Lp t W 2,p x of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) for t ∈ [0, T], x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' On XR = {U ∈ X | ∥U∥X ≤ R}, we now define the function Φ as follows: for v ∈ XR, let Φ(v) =: uv be the unique solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' First, we want to show that Φ : XR → XR if T > 0 is chosen small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' If T > 0 is fixed with T ≤ � | ¯K|e2(∥u0∥L∞(M,¯g)+1) + ∥f∥L∞(M,¯g) � 1 + e2(∥u0∥L∞(M,¯g)+1) A ��−1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15) and v ∈ XR, then Φ(v) ∈ XR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3 (ii) we directly get ∥Φ(v)∥L∞ t L∞ x = ∥uv∥L∞ t L∞ x ≤ ∥u+ 0 ∥L∞(M,¯g) + TdT (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='16) where dT ≤ | ¯K|e2∥v∥L∞ t L∞ x + ∥f∥L∞(M,¯g) + ∥f∥L∞(M,¯g)e2∥v∥L∞ t L∞ x A ≤ | ¯K|e2R + ∥f∥L∞(M,¯g) � 1 + e2R A � , hence ∥Φ(v)∥L∞ t L∞ x ≤ T � | ¯K|e2R + ∥f∥L∞(M,¯g) � 1 + e2R A �� + ∥u+ 0 ∥L∞(M,¯g) ≤ 1 + ∥u0∥L∞(M,¯g) = R, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15) and since R = ∥u0∥L∞(M,¯g) + 1, which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We now use Schauder’s fixed point Theorem [17] to show the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' If u0 ∈ Cp,A ⊂ W 2,p(M, ¯g) and T > 0 is fixed with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15), then there exists a short-time solution u ∈ X ∩ C∞(M × (0, T)) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, any such solution satisfies u ∈ C([0, T), H1(M, ¯g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 1: First we recall Schauder’s Theorem: It asserts that if H is a nonempty, convex, and closed subset of a Banach space B and F is a continuous mapping of H into itself such that F(H) is a relatively compact subset of H, then F has a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In our case, B ˆ=X = C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' C(M, ¯g)), H ˆ=XR = {u ∈ X | ∥u∥X = ∥u∥CtCx ≤ R}, and F ˆ=Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So to show the existence of a fixed point of Φ in XR, it remains to show that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Φ : XR → XR ist continuous and 14 Franziska Borer, Peter Elbau, Tobias Weth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Φ(XR) ⊂ XR is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In a first step we show that Φ : XR → XR ist continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For this, let (vn)n∈N ⊂ XR be a sequence with ∥vn − v∥X → 0 for n → ∞ with v ∈ XR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 we know that for all vn there exists un ∈ W 2,1 p , p > 2, which is the unique solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) such that ∥un∥W 2,1 p ≤ C(∥u0∥W 2,p(M,¯g) + ∥dn∥Lp t Lp x) with dn(t) := ¯K � 1 A − e−2vn(t) � + f − 1 A � M fe2vn(t)dµ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since vn → v in CtCx and therefore vn → v in L∞ t L∞ x , we know that vn → v in Lp t Lp x for all p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore, since the exponential map is continuous, we have e±2vn → e±2v in Lp t Lp x for all p, and therefore dn → d in Lp t Lp x for all p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence, for every ε > 0 there exist NV , Nd ∈ N such that ∥vn − v∥Lp t Lp x < ε for all n ≥ N and ∥dn − d∥Lp t Lp x < ε for all n ≥ N, with N := max{NV , Nd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore we have the estimate ∥e2vn − e2v∥L∞ t L∞ x = ∥(e2vn−2v − 1)e2v∥L∞ t L∞ x ≤ ∥e2vn−2v − 1∥L∞ t L∞ x ∥e2v∥L∞ t L∞ x ≤ ∥2vn − 2v∥e∥2Vn−2V ∥L∞ t L∞ x ∥e2v∥L∞ t L∞ x < 2εe2εe2R, and similarly ∥e−2vn − e−2v∥L∞ t L∞ x < 2εe2εe2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Considering now the difference un − u, where un = Φ(vn) and u = Φ(v), we see that un − u fulfils the equation ∂t(un − u)(t) = e−2vn(t)∆¯gun(t) + dn(t) − e−2v(t)∆¯gu(t) − d(t) = e−2vn(t)∆¯g(un − u)(t) + (e−2vn(t) − e−2v(t))∆¯gu(t) + dn(t) − d(t) with ∥un − u∥W 2,1 p ≤ C∥(e−2vn − e−2v)∆¯gu + dn − d∥Lp t Lp x ≤ C � ∥e−2vn − e−2v∥L∞ t L∞ x ∥∆¯gu∥Lp t Lp x + ∥dn − d∥Lp t Lp x � ≤ C(2εe2εe2R∥∆¯gu∥Lp t Lp x + ε) for n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since ∥∆¯gu∥Lp t Lp x is finite and ε > 0 was arbitrary, we see that ∥Φ(vn) − Φ(v)∥W 2,1 p → 0 for n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, we get ∥Φ(vn) − Φ(v)∥X ≤ C∥Φ(vn) − Φ(v)∥Cα ≤ C∥Φ(vn) − Φ(v)∥W 2,1 p → 0 for n → ∞ which shows the continuity of Φ : XR → XR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In a second step we show that Φ(XR) is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For this let (un)n∈N ⊂ Φ(XR) be an arbitrary sequence in the image of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, again with Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, we see that for every un ∈ Φ(XR) there exists a vn ∈ XR with Φ(vn) = un such that ∥un∥W 2,1 p ≤ C(∥u0∥W 2,p(M,¯g) + ∥dn∥Lp t Lp x) ≤ C � ∥u0∥W 2,p(M,¯g) + T| ¯K| A + ∥ ¯Ke−2vn∥Lp t Lp x + ∥f∥Lp t Lp x + ���� 1 A � M fe2vndµ¯g ���� Lp t Lp x � ≤ C � ∥u0∥W 2,p(M,¯g) + T| ¯K| A + | ¯K|e2R + T∥f∥L∞(M,¯g) + T A∥f∥L∞(M,¯g)e2R � ≤ C(A, f, ¯K, R, T, u0) =: Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, (un)n∈N is uniformly bounded in W 2,1 p ((0, T) × M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Using now that W 2,1 p ((0, T) × M) is continuously embedded in Cα([0, T] × M) for some 0 < α < 1 and this on the other hand is compactly embedded in Cβ([0, T] × M) for some 0 < β < α < 1 we can conclude the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We have thus proved that Φ has a fixed point u in XR, which then is a (strong) solution u ∈ W 2,1 p ((0, T) × M) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 2: We now show that u ∈ C∞(M × (0, T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' To see this, we first note the trivial fact that u ∈ W 2,1 p ((0, T)×M) is a strong solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) with v = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since then v ∈ W 2,1 p ((0, T)×M) ⊂ Cα([0, T]× Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 15 M), [14, Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10] imply the existence of a classical solution ˜u ∈ X ∩ C2+α′,1+α′ loc ((0, T) × M) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) with v = u for some α′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Here C2+α′,1+α′ loc ((0, T) × M) denotes the space of functions f ∈ C2,1((0, T) × M) with the property that ∂tf and all derivatives up to second order of f with respect to x ∈ M are locally α′-H¨older continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular, ˜u ∈ W 2,1 p ((ε, T − ε) × M) for ε ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The function w := u − ˜u ∈ W 2,1 p ((ε, T − ε) × M) is then a strong solution of the initial value problem ∂tw(t) = e−2v(t)∆¯gw(t) for t ∈ (ε, T − ε), w(ε) = u(ε, ·) − ˜u(ε, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3 (ii) we then have |w| ≤ ∥u(ε, ·) − ˜u(ε, ·)∥L∞(M,¯g) on (ε, T − ε) × M, whereas ∥u(ε, ·) − ˜u(ε, ·)∥L∞(M,¯g) → 0 as ε → 0 by the continuity of u and ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' It thus follows that u ≡ ˜u on (0, T) × M), and therefore u ∈ C2+α′,1+α′ loc ((0, T) × M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since u solves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) with v = u ∈ C2+α′,1+α′ loc ((0, T) × M), we can apply [14, Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9] and the above argument again to get u ∈ C4+α′′,2+α′′ loc ((0, T) × M) for some α′′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Repeating this argument inductively, we get u ∈ C k, k 2 loc ((0, T) × M) for every k > 0, and hence u ∈ C∞(M × (0, T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 3: It remains to show that any solution u ∈ X ∩ C∞((0, T) × M) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) also satisfies u ∈ C([0, T), H1(M, ¯g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since u ∈ C∞((0, T) × M), only the continuity in t = 0 needs to be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Setting φ(t) = ∥u(t)∥2 H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) for t ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' we see that 1 2(φ(t2) − φ(t1)) = 1 2 � t2 t1 ∂t∥u(t)∥2 H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) dt = � t2 t1 � M � u(t)∂tu(t) + ∇u(t)∇∂tu(t) � dµ¯gdt = � t2 t1 � M � u(t)∂tu(t) − [∆u(t)]∂tu(t) � dµ¯gdt and therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' by H¨older’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 1 2|φ(t2) − φ(t1)| ≤ � t2 t1 � M � |u||∂tu| + |∆u||∂tu| � dµ¯gdt ≤ C∥∂tu∥Lp((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T )×M) � ∥u∥Lp((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T )×M) + ∥∆u∥Lp((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T )×M) � (t2 − t1)β ≤ C∥u∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 p ((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T )×M)(t2 − t1)β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' for 0 < t1 < t2 < T with some β > 0 depending on p > 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' which implies that the function φ is uniformly continuous and therefore bounded on (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We now assume by contradiction that u is not continuous at t = 0 with respect to the H1(M, ¯g)-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then there exists a sequence (tn)n∈N in (0, T) and ε > 0 with tn → 0+ as n → ∞ and ∥u(tn) − u0∥H1(M,¯g) ≥ ε for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='17) Since ∥u(tn)∥2 H1(M,¯g) = φ(tn) remains bounded as n → ∞, we conclude that, passing to a subsequence, the sequence u(tn) converges weakly in H1(M, ¯g) and therefore strongly in L2(M, ¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since the strong L2-limit of u(tn) must be u0 = u(0) as a consequence of the fact that u ∈ X, we deduce that u(tn) ⇀ u0 weakly in H1(M, ¯g) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Combining this information with Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 from the appendix, we deduce that lim sup n→∞ ∥u(tn)∥2 H1(M,¯g) ≤ ∥u0∥2 H1(M,¯g) ≤ lim inf n→∞ ∥u(tn)∥2 H1(M,¯g) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='18) and therefore ∥u(tn)∥H1(M,¯g) → ∥u0∥H1(M,¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Note here that this part of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 applies since u solves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) with v = u ∈ W 2,1 p ((0, T) × M) ⊂ Cα([0, T] × M) for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='18) and the uniform convexity of the Hilbert space H1(M, ¯g), we conclude that u(tn) → u0 strongly in H1(M, ¯g), contrary to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We now show that the solution from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u0 ∈ W 2,p(M, ¯g), p > 2, and T > 0 be fixed with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then the short-time solution of u ∈ X ∩ C∞(M × (0, T)) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) given by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u1, u2 ∈ X ∩ C∞(M × (0, T)) be two solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The difference u := u1 − u2 ∈ X ∩ C∞(M × (0, T)) then fulfils ∂tu(t) = e−2u1(t)∆¯gu1(t) − e−2u2(t)∆¯gu2(t) − ¯K(e−2u1(t) − e−2u2(t)) − 1 A � M f(e2u1(t) − e2u2(t))dµ¯g = e−2u1(t)∆¯gu(t) + ∆¯gu2(t) � e−2u1(t) − e−2u2(t)� − ¯K(e−2u1(t) − e−2u2(t)) − 1 A � M f(e2u1(t) − e2u2(t))dµ¯g for t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='19) 16 Franziska Borer, Peter Elbau, Tobias Weth In the following, the letter C denotes different positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Multiplying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='19) with 2u and integrating over M gives d dt∥u(t)∥2 L2(M,¯g) = 2 � M u(t)∂tu(t)dµ¯g = 2 � M e−2u1(t)u(t)∆¯gu(t)dµ¯g + 2 � M u(t)∆¯gu2(t) � e−2u1(t) − e−2u2(t)� dµ¯g (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='20) − 2 � M ¯Ku(t)(e−2u1(t) − e−2u2(t))dµ¯g − 2 A � M f(e2u1(t) − e2u2(t))dµ¯g � M u(t)dµ¯g ≤ 2 � M e−2u1(t)u(t)∆¯gu(t) + 2 � M V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' x)u2(t) + 2ρ(t)∥u(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) � M |u(t)|dµ¯g ≤ 2 � − � M e−2u1(t)|∇¯gu(t)|2 ¯g + 2 � M e−2u1(t)u(t)⟨∇¯gu1(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ∇¯gu(t)⟩¯gdµ¯g � + 2∥V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ·)∥Lp(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥u(t)∥2 L2p′(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + C∥u(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ C∥∇¯gu1(t)∥L4(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥u(t)∥L4(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥∇¯gu(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + 2∥V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ·)∥Lp(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥u(t)∥2 L2p′(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + C∥u(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ C � ∥u1(t)∥H2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥u(t)∥2 H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + 2∥V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ·)∥Lp(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥u(t)∥2 H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + ∥u(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) � ≤ C � ∥u1(t)∥H2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + 2∥V (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ·)∥Lp(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + 1 � ∥u∥2 H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='21) with functions V ∈ Lp((0, T) × M) ∩ C∞((0, T) × M) and ρ ∈ L∞(0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Here we used the Sobolev embeddings H1(M, ¯g) �→ Lρ(M) for ρ ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Multiplying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='19) with −2∆u and integrating over M yields d dt∥∇gu(t)∥2 L2(M,¯g) = 2 � M ∇u(t)∇∂tu(t)dµ¯g = −2 � M ∆gu(t)∂tu(t)dµ¯g ≤ −2 � M e−2u1(t)|∆¯gu(t)|2dµ¯g + 2 � M V (x, t)|u(t)||∆u(t)|dµ¯g ≤ −κ∥∆¯gu(t)∥2 L2(M,¯g) + 2∥V (t, ·)∥Lp(M,¯g)∥u∥Lα(M,¯g)∥∆gu∥L2(M,¯g) ≤ −κ∥∆¯gu(t)∥2 L2(M,¯g) + 1 κ∥V (t, ·)∥2 Lp(M,¯g)∥u∥2 Lα(M,¯g) + κ∥∆gu∥2 L2(M,¯g) = 1 κ∥V (t, ·)∥2 Lp(M,¯g)∥u∥2 Lα(M,¯g) ≤ C∥V (t, ·)∥2 Lp(M,¯g)∥u∥2 H1(M,¯g), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='22) where we used first H¨older’s inequality with α = 2p p−2, then Young’s inequality and finally Sobolev embeddings again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Here we note that, by making C > 0 larger if necessary, we may assume that the constants are the same in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='21) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Combining these estimates gives d dt∥u(t)∥2 H1(M,¯g) ≤ g(t)∥u(t)∥2 H1(M,¯g) for t ∈ (0, T) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='23) with the function g ∈ L1(0, T) given by g1(t) = C � ∥u1(t)∥H2(M,¯g) + 3∥V (t, ·)∥Lp(M,¯g) + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Integrating and using the fact that u ∈ C([0, T), H1(M, ¯g)) by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5 with u(0) = u1(0) − u2(0) = 0, we see that ∥u(t)∥2 H1(M,¯g) ≤ � t 0 g(s)∥u(s)∥2 H1(M,¯g) ds for t ∈ [0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' It then follows from Gronwall’s inequality [3] that ∥u(t)∥2 H1(M,¯g) ≡ 0 on [0, T), hence u1 ≡ u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Global Existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' From Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 and Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3 we know that there exists a unique solution u ∈ C([0, T], C(M)) ∩ C([0, 1], H1(M, ¯g)) ∩ C∞((0, T) × M), of the initial value problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular we know that u ∈ L∞ t L∞ x for t ∈ [0, T], where T > 0 is given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this section we want to show that u posses an L∞-a-priori bound on any time interval [0, T], T < ∞, and therefore, u is the unique global solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For this we partially follow the idea of [2, Chapter 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For every T > 0, there exists M(T) > 0 such that we have sup t∈[0,T ] ∥u(t)∥L∞(M,¯g) ≤ M(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let I := � t ≥ 0 ��� u is a solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) on (0, t] × M with initial data u(0) ∈ Cp,A � , Tmax := sup I, and Tk ⊂ I a sequence in I such that Tk → Tmax for k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For any t ∈ [0, Tk] and any xmax(t) ∈ M where u(t, xmax(t)) = max x∈M u(t, x) ≥ 0 we have with ∂tu(t) = ∆g(t)u(t) − e−2u(t) ¯K + f − α(t) and the upper bound for |α| which is given by α0 := max{|α1|, |α2|}, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='24) that d dt [u(t, xmax(t))] = ∂tu(t, xmax(t)) ≤ | ¯K|e−2u(t,xmax(t)) + f(xmax(t)) + α0 ≤ | ¯K| + ∥f∥L∞(M,¯g) + α0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='25) where we used that ∇¯gu(t, xmax(t)) = 0 and therefore d dt [u(t, xmax(t)] = ∂tu(t, xmax(t)) + ∇¯gu(t, xmax(t)) ˙xmax(t) = ∂tu(t, xmax(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Integrating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='25) on both side with respect to t and taking the supremum over t yields (together with the fact that u(0) = u0 ∈ Cp,A) sup t∈[0,Tk] x∈M u(t, x) ≤ Tk(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup x∈M u0(x) → Tmax(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup x∈M u0(x) =: M1(Tmax) < ∞ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='26) for k → ∞ which shows the upper bound for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Analogously, at any point xmin(t) ∈ M where u(t, xmin(t)) = min x∈M u(t, x) ≤ 0 we have with ∂tu(t) = ∆g(t)u(t) − e−2u(t) ¯K + f − α(t), the fact that ¯K < 0, and the upper bound for |α| given by α0 that d dt [u(t, xmin(t))] = ∂tu(t, xmin(t)) ≥ −∥f∥L∞(M,¯g) − α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='27) Integrating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='27) on both side with respect to t and taking the infimum over t yields (together with the fact that u(0) = u0 ∈ Cp,A) inf t∈[0,Tk] x∈M u(t, x) ≥ −Tk(∥f∥L∞(M,¯g) + α0) + inf x∈M u0(x) → −Tmax(∥f∥L∞(M,¯g) + α0) + inf x∈M u0(x) =: M2(Tmax) > −∞ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='28) for k → ∞ which shows the lower bound for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, we get sup t∈[0,T ] x∈M |u(t, x)| ≤ max{|M1(T)|, |M2(T)|} ≤ T(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup x∈M |u0(x)| =: M(T) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='29) which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In fact, with the help of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) we can turn (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='29) into a uniform estimate for all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let u be the global, smooth solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) with u(0) = u0 ∈ Cp,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then we have that supt>0 ∥u(t)∥L∞(M,¯g) ≤ Cuni < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 18 Franziska Borer, Peter Elbau, Tobias Weth Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We follow the proof of [19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By using the fact that u(t) is a volume preserving solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) with u(0) = u0 ∈ Cp,A and therefore � M e2u(t)dµ¯g ≡ A, we get with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) and the fact that ¯K < 0 that Ef(u(t)) = 1 2∥∇¯gu(t)∥2 L2(M,¯g) + � M ¯Ku(t)dµ¯g − 1 2 � M fe2u(t)dµ¯g ≥ ¯K 2 � M 2u(t)dµ¯g − 1 2 � M fe2u(t)dµ¯g ≥ ¯K 2 log(A) − A 2 ∥f∥L∞(M,¯g) > −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='30) Defining F(t) := � M |∂tu(t)|2dµg(t) = � M |∂tu(t)|2e2u(t)dµ¯g and using the uniform lower bound of Ef given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='30), we then get from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9), respectively, the estimate � ∞ 0 F(t)dt = � ∞ 0 � M |∂tu(t)|2dµg(t)dt ≤ Ef(u0) + | ¯K| 2 | log(A)| + A 2 ∥f∥L∞(M,¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='31) Hence, for any T > 0 we find tT ∈ [T, T + 1] such that F(tT ) = inf t∈(T,T +1) F(t) ≤ Ef(u0) + | ¯K| 2 | log(A)| + A 2 ∥f∥L∞(M,¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='32) So, at time tT we get with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1), H¨olders inequality, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='32) that ∥∆¯gu(tT )∥L 3 2 (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ ∥e2u(tT )∂tu(tT )∥L 3 2 (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + ∥ ¯K∥L 3 2 (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + ∥e2u(tT )f∥L 3 2 (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + ∥e2u(tT )α(tT )∥L 3 2 (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ ∥eu(tT )∥L6(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)F(tT ) 1 2 + | ¯K| + �� M e3u(tT )|f| 3 2 dµ¯g � 2 3 + �� M e3u(tT )|α(tT )| 3 2 dµ¯g � 2 3 ≤ C 1 6 int(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ef(u0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ¯K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' η1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' η2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 3) � Ef(u0) + | ¯K| 2 | log(A)| + A 2 ∥f∥L∞(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) � 1 2 + | ¯K| + C 2 3 int � A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ef(u0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ¯K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' η1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' η2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 3 2 � (∥f∥L∞(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + α0) =: C10 � A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Ef(u0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ¯K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' η1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' η2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='33) Furthermore, with Sobolev’s embedding theorem we have W 2, 3 2 ⊂ C0, 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Therefore we get with Poincar´e’s inequality, the Calder´on–Zygmund inequality for closed surfaces, and with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='33) that ∥u(tT ) − ¯u(tT )∥ 3 2 L∞(M,¯g) ≤ C∥u(tT ) − ¯u(tT )∥ 3 2 W 2, 3 2 (M,¯g) ≤ C∥∇2 ¯gu(tT )∥ 3 2 L 3 2 (M,¯g) ≤ C∥∆¯gu(tT )∥ 3 2 L 3 2 (M,¯g) ≤ CC 3 2 10, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='34) and therefore with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) we obtain the uniform bound ∥u(tT )∥L∞(M,¯g) ≤ CC10 + max � |m0|, 1 2| log(A)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='35) Upon shifting time by tT , from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='29) we now get sup s∈[T +1,T +2] ∥u(s)∥L∞(M,¯g) ≤ sup s∈[tT ,T +2] ∥u(s)∥L∞(M,¯g) ≤ 2(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup x∈M |u(tT , x)| ≤ 2(| ¯K| + ∥f∥L∞(M,¯g) + α0) + CC10 + max � |m0|, 1 2| log(A)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='36) Since T > 0 is arbitrary, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Convergence of the Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let f0 ≤ 0 be a smooth, nonconstant function withmaxx∈M f0(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Following here the argumentation of [19], and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='31), we know that for a suitable sequence tl → ∞, l → ∞, with associated metrics gl = g(tl) we obtain convergence � M |∂tu(tl)|2dµg(tl) = � M |f0 − Kgl − α(tl)|2dµg(tl) → 0 for l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='37) Provided that we can also show convergence of the associated sequence of metrics g(tl) to a limit metric g∞ A = e2u∞ A ¯g with Gauss curvature Kg∞ A , it then follows that Kg∞ A = f0 − α∞ A for a constant α∞ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Later we will have a closer look at this constant α∞ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For F(t) = � M |∂tu(t)|2dµg(t) as above, we have F(t) → 0 for t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' First we consider the evolution equation of the curvature Kg(t) and of α(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' From the Gauss equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) we get for the curvature that ∂tKg(t) = ∂t(−e−2u(t)∆¯gu(t) + e−2u(t) ¯K) = −2∂tu(t)Kg(t) − ∆g(t)∂tu(t) = 2Kg(t)(Kg(t) − f0 + α(t)) + ∆g(t)(Kg(t) − f0 + α(t)) = 2(Kg(t) − f0 + α(t))2 + 2(f0 − α(t))(Kg(t) − f0 + α(t)) + ∆g(t)(Kg(t) − f0 + α(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='38) With (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) we get for the evolution equation for α(t): d dtα(t) = 2 A � M f0e2u(t)∂tu(t)dµ¯g = 2 A � M f0(f0 − Kg(t) − α(t))dµg(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='39) So, with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='38) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='39) we arrive at ∂t(Kg(t) − f0 − α(t)) − ∆g(t)(Kg(t) − f0 + α(t)) = 2(Kg(t) − f0 + α(t))2 + 2(f0 − α(t))(Kg(t) − f0 + α(t)) + 2 A � M f0(Kg(t) − f0 + α(t))dµg(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='40) Following the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 in [19] we therefore get 1 2 d dt � M |f0 − Kg(t) − α(t)|2dµg(t) = � M �� ∂tKg(t) + � d dtα(t) �� (Kg(t) − f0 + α(t)) − (Kg(t) − f0 − α(t))3 � dµg(t) = − � M |∇g(t)(Kg(t) − f0 + α(t))|2 g(t)dµg(t) + 2 � M (f0 − α(t))(Kg(t) − f0 + α(t))2dµg(t) + � M (Kg(t) − f0 + α(t))3dµg(t), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='41) where we used in the second step the fact that � d dtα(t) � � M (Kg(t) − f0 + α(t))dµg(t) = 0 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With H¨older’s inequality we can estimate � M (Kg(t) − f0 + α(t))3dµg(t) ≤ ∥∂tu(t)∥3 L3(M,g(t)) ≤ ∥∂tu(t)∥L2(M,g(t))∥∂tu(t)∥2 L4(M,g(t)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='42) 20 Franziska Borer, Peter Elbau, Tobias Weth and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 we further get with the uniform bound for u ∈ CtCx that ∥∂tu(t)∥2 L4(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g(t)) = �� M |∂tu(t)|4e2u(t)dµ¯g � 1 2 ≤ e∥u∥L∞ t L∞ x ∥∂tu(t)∥2 L4(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ e∥u∥L∞ t L∞ x � CGNL∥∂tu(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥∂tu(t)∥H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) = e∥u∥L∞ t L∞ x � CGNL �� M |∂tu(t)|2e2u(t)e−2u(t)dµ¯g � 1 2 �� M |∂tu(t)|2e2u(t)e−2u(t)dµ¯g + � M |∇¯g∂tu(t)|2 ¯gdµ¯g � 1 2 = e∥u∥L∞ t L∞ x � CGNL �� M |∂tu(t)|2e−2u(t)dµg(t) � 1 2 �� M |∂tu(t)|2e−2u(t)dµg(t) + � M |∇g(t)∂tu(t)|2 g(t)dµg(t) � 1 2 ≤ e∥u∥L∞ t L∞ x max{e∥u∥L∞ t L∞ x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' e2∥u∥L∞ t L∞ x } � CGNL∥∂tu(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g(t))∥∂tu(t)∥H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g(t)) =: ˜C2∥∂tu(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g(t))∥∂tu(t)∥H1(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='43) where we used the fact that � M |∇¯g∂tu(t)|2 ¯gdµ¯g = � M |∇g(t)∂tu(t)|2 g(t)dµg(t) =: G(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Plugging in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='43) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='42) we arrive at � M (Kg(t) − f0 + α(t))3dµg(t) ≤ ˜C2∥∂tu(t)∥2 L2(M,g(t))∥∂tu(t)∥H1(M,g(t)) ≤ ˜C2 2 2 ∥∂tu(t)∥4 L2(M,g(t)) + 1 2∥∂tu(t)∥2 H1(M,g(t)) ≤ ˜C2 2F 2(t) + 1 2(F(t) + G(t)), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='44) where we used Young’s inequality in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With α0 = max{|α1|, |α2|} > 0 we furthermore have that 2 � M (f0 − α(t))(Kg(t) − f0 + α(t))2dµg(t) ≤ 2(∥f0∥L∞(M,¯g) + α0)F(t) =: ˜C3(α0, f0)F(t) So, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='41) yields d dtF(t) + G(t) ≤ 2 � ˜C3F(t) + ˜C2 2F 2(t) + 1 2F(t) � = (2 ˜C3 + 1)F(t) + 2 ˜C2 2F 2(t) =: ˜C4F(t) + 2 ˜C2 2F 2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='45) We recall that with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='31) we have lim inft→∞ F(t) = 0 and therefore we know that there exist tl → ∞ with F(tl) → 0 as l → ∞, see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By integrating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='45) over (tl, t) ⊂ (tl, T) and taking the supremum over (tl, T) we get with � T tl G(t)dt ≥ 0 that sup t∈(tl,T ) F(t) ≤ F(tl) + ˜C4 � T tl F(t)dt + 2 ˜C2 2 � T tl F 2(t)dt ≤ F(tl) + ˜C4 � T tl F(t)dt + 2 ˜C2 2 sup t∈(tl,T ) F(t) � T tl F(t)dt ≤ F(tl) + ˜C4 � T tl F(t)dt + 2 ˜C2 2 sup t∈(tl,T ) F(t) � ∞ tl F(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='31) we also have � ∞ tl F(t)dt → 0 for l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, for T > 0 big enough such that for tl < T big enough we have that 2 ˜C2 2 � ∞ tl F(t)dt is small enough to guarantee that 1 − 2 ˜C2 2 � ∞ tl F(t)dt > 0 and therefore the term 2 ˜C2 2 supt∈(tl,T ) F(t) � ∞ tl F(t)dt can be absorbed on the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, we get sup t∈(tl,T ) F(t) ≤ 1 � 1 − 2 ˜C2 2 � ∞ tl F(t)dt � � F(tl) + ˜C4 � T tl F(t)dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 21 Letting T → ∞ yields sup t∈(tl,∞) F(t) ≤ 1 � 1 − ˜C2 2 � ∞ tl F(t)dt � � F(tl) + ˜C4 � ∞ tl F(t)dt � → 0 as l → ∞ which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' To prove now the convergence of the flow, let A > 0 and u0 ∈ Cp,A, p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore let f ∈ C∞(M) be a smooth, nonconstant function, and (f0, λ) ∈ C∞(M) × R the unique pair such that f = f0 + λ with f0 ≤ 0, f0 nonconstant, and maxx∈M f0(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1 the additive rescaled prescribed Gauss curvature flow (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) is invariant under adding or subtracting a constant C > 0 to the function f, for all functions f ∈ {f0 + λ | λ ∈ R} we consider the same flow given by ∂tu(t) = f0 − Kg(t) − αA(t) in (0, T) × M, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='46) which is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) with f replaced by f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' With (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) we know that 1 2 � M (|∇¯gu(T)|2 ¯g + 2 ¯Ku(T) − f0e2u(T ))dµ¯g = Ef0(u(T)) ≤ Ef0(u(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, we get with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) that 1 2 � M |∇¯gu(T)|2 ¯gdµ¯g = Ef0(u(T)) − � M ¯Ku(T)dµ¯g + 1 2 � M f0e2u(T )dµ¯g ≤ Ef0(u(T)) + | ¯K| � M u(T)dµ¯g ≤ Ef0(u(0)) + | ¯K| 2 | log(A)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, u is uniformly (in T) bounded in H1(M, ¯g), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=', ∥u∥L∞ t H1x ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We now consider ul := u(tl) for a suitable sequence tl → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By the Theorem of Banach-Alao˘glu we know that (ul)l is weak∗ relatively compact in H1(M, ¯g) and therefore (since H1 is reflexive) also weak relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' This means that that there exists a subsequence ulk which we again call ul such that ul → u∞ A weakly in H1(M, ¯g) and therefore strongly in L2(M, ¯g) (by a direct consequence of the Rellich–Kondrachov embedding Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) we know that αl := α(tl) → α∞ A as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover we have e±ul → e±u∞ A (as l → ∞) in Lp(M, ¯g) for any 2 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Indeed, with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) we have ∥eul − eu∞ A ∥p Lp(M,¯g) = � M epul|1 − eu∞ A −ul|pdµ¯g ≤ epCuni � M |1 − eu∞ A −ul|pdµ¯g ≤ epCuni � M |u∞ A − ul|pep|u∞ A −ul||dµ¯g ≤ epCunie2pCuni � M |u∞ A − ul|p−2|u∞ A − ul|2dµ¯g ≤ e3pCuni(2Cuni)p−2∥u∞ A − ul∥2 L2(M,¯g) → 0 as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Replacing ul by −ul we get also e−ul → e−u∞ A in Lp(M, ¯g) as l → ∞ for any p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9 we also have e2ul∂tul → 0 in L2(M, ¯g) as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Furthermore we have ∥e2ulαl − e2u∞ A α∞ A ∥L2(M,¯g) ≤ ∥e2ul(αl − α∞ A )∥L2(M,¯g) + ∥α∞ A (e2ul − e2u∞ A )∥L2(M,¯g) ≤ ∥e2ul∥L∞(M,¯g)|αl − α∞ A |A 1 2 + |α∞ A |∥e2ul − e2u∞ A ∥L2(M,¯g) → 0 for l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, considering our evolution equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), we therefore get ∆¯gul = e2ul∂tul + ¯K − e2ulf0 + e2ulαl → ¯K − e2u∞ A f0 + e2u∞ A α∞ A =: (∆¯gu)∞ A 22 Franziska Borer, Peter Elbau, Tobias Weth in L2(M, ¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since the Laplace operator ∆¯g is closed we know that (∆¯gu)∞ A = ∆¯gu∞ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence ∥∆¯g(ul − u∞ A )∥L2(M,¯g) → 0 as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So, we even have strong convergence ul → u∞ A in H2(M, ¯g) and uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Thus, passing to the limit l → ∞ in the equation e2ul∂tul − ∆¯gul = − ¯K + e2ulf0 − e2ulαl we get the identity −∆¯gu∞ A = − ¯K + e2u∞ A f0 − e2u∞ A α∞ A and therefore Kg∞ A = f0 − α∞ A = f0 + 1 A � ¯K + � M |f0|dµg∞ A � which shows the convergence of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The Sign of the Constant α∞ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In this subsection we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4, with other words, under certain assumptions we can now further estimate the expression 1 A � ¯K + � M |f0|dµg∞ A � to show that it is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4 is already covered by the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' So we can turn to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We have seen in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7 that in the case where u0 ≡ 1 2 log(A) ∈ Cp,A, the uniform L∞-bound on the global solution of the initial value problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) only depends on A and an upper bound on ∥f∥L∞(M,¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In other words, if A > 0 and c > 0 are fixed, then there exists τ > 0 with the property that sup t>0 ∥u(t)∥L∞(M,¯g) ≤ τ for every f ∈ C∞(M) with ∥f∥L∞(M,¯g) ≤ c and the corresponding solution u of the initial value problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) with u0 ≡ 1 2 log(A) ∈ Cp,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, we also have ∥u∞∥L∞(M,¯g) ≤ τ under the current assumptions on f, which implies that λ = 1 A � ¯K − � M fe2u∞dµ¯g � = 1 A � ¯K + cA − � M (f + c)e2u∞dµ¯g � ≥ c + ¯K A − ∥f + c∥L1(M,¯g)∥e2u∞∥L∞(M,¯g) ≥ c + ¯K A − ∥f + c∥L1(M,¯g)e2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence, if ∥f + c∥L1(M,¯g) < ε := c+ ¯ K A e2τ , we have λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Appendix As before, let (M, ¯g) be a two-dimensional, smooth, closed, connected, oriented Riemann manifold endowed with a smooth background metric ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For a domain Ω ⊂ M × R and p ≥ 1, we let W 2,1 p (Ω) denote the space of functions u ∈ Lp(Ω) which have weak derivatives Du, D2u and ∂tu in Lp(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the following, we fix p > 2, which implies that W 2,1 p (Ω) is continuously embedded in Cα(Ω) for some α = α(p) > 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [13, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We consider the linear parabolic problem ∂tu(x, t) = a(x, t)∆¯gu(x, t) + c(x, t)u(x, t) + d(x, t), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) with a, c, d ∈ C(Ω) and d ∈ Lp(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We say that a function u ∈ W 2,1 p (Ω) is a (strong) solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) in Ω if (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) holds almost everywhere in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Specifically, we consider (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) on the cylindrical domains ΩT = M × (0, T) and �ΩT = M × (−∞, T) in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In particular, we consider strong solutions of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) together with the initial condition u(0, x) = u0(x) in M (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with u0 ∈ W 2,p(M, ¯g), which is supposed to hold in the (initial) trace sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 23 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let T > 0, a, c ∈ C(ΩT ) with aT := min (x,t)∈ΩT a(x, t) > 0, let d ∈ Lp(ΩT ) for some p > 2, and let u0 ∈ W 2,p(M, ¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then the initial value problem (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) has a unique strong solution u ∈ W 2,1 p (ΩT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, u satisfies the estimate ∥u∥W 2,1 p (ΩT ) ≤ C � ∥u0∥W 2,p(M,¯g) + ∥d∥Lp(ΩT ) � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4) with a constant C > 0 depending only on ∥a∥L∞(ΩT ), ∥c∥L∞(ΩT ) and aT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, C does not increase after making T smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' If, moreover, a, c, d ∈ Cα(ΩT ) for some α > 0, then u ∈ C(ΩT )∩C2,1(ΩT ) is a classical solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3), and we have the inequality ∥u0∥H1(M,¯g) ≥ lim sup t→0+ ∥u(t)∥H1(M,¯g) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the following, the letter C stands for various positive constants depending only on ∥a∥L∞(ΩT ), ∥c∥L∞(ΩT ), and aT , and which do not increase after making T smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 1: We first assume that we are given a strong solution u ∈ W 2,1 p (ΩT ) of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with u0 ≡ 0 ∈ W 2,p(M, ¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We then define v : �ΩT → R by v(x, t) = � u(x, t), for t > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 0, for t ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then v ∈ W 2,1 p (�ΩT ) solves (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2) with a, c, d replaced by suitable extensions ˜a, ˜c, ∈ L∞(�ΩT ), ˜d ∈ Lp(�ΩT ) satisfying ˜a(x, t) = a(x, 0), ˜c(x, t) = c(x, 0) and ˜d(x, t) = 0 for t ≤ 0, x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Therefore, [14, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='22] gives rise to the uniform bound ∥D2v∥Lp(�ΩT ) + ∥∂tv∥Lp(�ΩT ) ≤ C � ∥ ˜d∥Lp(�ΩT ) + ∥v∥Lp(�ΩT ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='6) This translates into the estimate ∥D2u∥Lp(ΩT ) + ∥∂tu∥Lp(ΩT ) ≤ C � ∥d∥Lp(ΩT ) + ∥u∥Lp(ΩT ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) Moreover, setting V (t) := ∥u(t)∥p Lp(M,¯g) for t ∈ R, we have V (0) = 0 and ˙V (t) = p � M |u(t)|p−2u(t)∂tu(t)dµ¯g ≤ pV (t) 1 p′ ∥∂tu(t)∥Lp(M,¯g) ≤ p � V (t) p′ + ∥∂tu(t)∥p Lp(M,¯g) p � = p p′ V (t) + ∥∂tu(t)∥p Lp(M,¯g) for t ∈ (0, T), therefore V (t) = � t 0 ˙V (s) ds ≤ p p′ � t 0 V (s) ds + ∥∂tu∥p Lp(Ωt) ≤ p p′ � t 0 V (s) ds + C � ∥d∥p Lp(Ωt) + ∥u∥p Lp(Ωt) � ≤ C �� t 0 V (s) ds + ∥d∥p Lp(Ωt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By Gronwall’s inequality we get V (t) ≤ C∥d∥p Lp(Ωt) and thus ∥u(t)∥Lp(M,¯g) ≤ C∥d∥Lp(Ωt) for t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8) This already implies the uniqueness of strong solutions of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3), since the difference u of two solutions u1, u2 ∈ W 2,1 p (ΩT ) of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with u0 = 0 and d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, if u ∈ W 2,1 p (ΩT ) is a strong solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3), then the function ˆu ∈ W 2,1 p (ΩT ) given by ˆu(x, t) := u(x, t) − u0(x) safisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with u0 = 0 and d replaced by ˆd given by ˆd(x, t) = d(x, t) + a(x, t)∆¯gu0(x) + c(x, t)u0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='7) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='8), and using an interpolation estimate for Du, we find that ∥u∥W 2,1 p (ΩT ) ≤ ∥ˆu∥W 2,1 p (ΩT ) + ∥u0∥W 2,p(M,¯g) ≤ C � ∥ ˆd∥Lp(ΩT ) + ∥ˆu∥Lp(ΩT ) � + ∥u0∥W 2,p(M,¯g) ≤ C∥ ˆd∥Lp(ΩT ) + ∥u0∥W 2,p(M,¯g) ≤ C � ∥d∥Lp(ΩT ) + ∥u0∥W 2,p(M,¯g) � , 24 Franziska Borer, Peter Elbau, Tobias Weth as claimed in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 2 (Existence): In the case where a, c, d ∈ Cα(ΩT ) and u0 ∈ C2+α(M), the existence of a classical solution u ∈ C(ΩT ) ∩ C2,1(ΩT ) of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) follows as in [14, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' In the general case we consider (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with coefficients an, cn, dn ∈ Cα(ΩT ), u0,n ∈ C2+α(M), in place of a, c, d, u0 with the property that an → a, cn → c in L∞(ΩT ), dn → d ∈ Lp(ΩT ) as well as u0,n → u0 in W 2,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The associated unique solutions un ∈ C(ΩT ) ∩ C2,1(ΩT ) are uniformly bounded in W 2,1 p (ΩT ) by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), and therefore we have un ⇀ u in W 2,1 p (ΩT ) after passing to a subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For every φ ∈ C∞ c (ΩT ), we then have � ΩT � ∂tu(x, t) − a(x, t)∆¯gu(t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='x) − c(x, t)u(x, t) − d(x, t) � φ(x, t)dµ¯g(x)dt = lim n→∞ � ΩT � ∂tun(x, t) − an(x, t)∆¯gun(x, t) − cn(x, t)un(x, t) − dn(x, t) � φ(x, t)dµ¯g(x)dt = 0, and from this we deduce that ∂tu(x, t) − a(x, t)∆¯gu(x, t) − c(x, t)u(x, t) − d(x, t) = 0 almost everywhere in ΩT , so u is a strong solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 3: It remains to show the inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='5) in the case where a, c, d ∈ Cα(ΩT ) for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since u ∈ C(ΩT ) ∩ C2,1(ΩT ) in this case and therefore ∥u0∥L2(M,¯g) = lim t→0+ ∥u(t)∥L2(M,¯g), it suffices to show that ∥∇u0∥L2(M,¯g) ≥ lim sup t→0+ ∥∇u(t)∥L2(M,¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9) If u0 ∈ C2+α(M) for some α > 0, this follows by [14, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14] with lim in place of lim sup, since the function t �→ u(t) is continuous from [0, T) → C2+α(M) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' in this case we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' by H¨older’s and Young’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' d dt∥∇u(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) = − � M ∂tu(t)∆u(t)dµ¯g = − � M � a(t)|∆u(t)|2 + c(t)u(t)∆u(t) + d(t)∆u(t) � dµ¯g ≤ −aT ∥∆¯gu(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + ∥c(t)u(t) + d(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g)∥∆¯gu(t)∥L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ −aT ∥∆¯gu(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + aT ∥∆¯gu(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + 1 4aT ∥c(t)u(t) + d(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) = 1 4aT ∥c(t)u(t) + d(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' and therefore ∥∇u(t)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ≤ ∥∇u(0)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) + 1 4aT � t 0 ∥c(s)u(s) + d(s)∥2 L2(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='¯g) ds for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10) In the general case, we consider (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with a sequence of initial conditions un,0 in place of u0, where un,0 → u0 in H2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' The associated unique solutions un ∈ C(ΩT ) ∩ C2,1(ΩT ) are uniformly bounded in W 2,1 p (ΩT ) by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='4), and they are also uniformly bounded in C2,1([ε, T] × M) by [14, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15] for every ε ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Fix t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Passing to a subsequence, we may assume that un ⇀ u in W 2,1 p (ΩT ), un → u strongly in C0(ΩT ) and un(t) → u(t) strongly in C1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' As in Step 2, we see, by testing with φ ∈ C∞ c (ΩT ), that ∂tu(x, t) − a(x, t)∆¯gu(x, t) − c(x, t)u(x, t) − d(x, t) = 0 almost everywhere in ΩT , so u is the unique strong solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10) we have ∥∇u(t)∥2 L2(M,¯g) = lim n→∞ ∥∇un(t)∥2 L2(M,¯g) ≤ lim n→∞ � ∥∇un(0)∥2 L2(M) + 1 4aT � t 0 ∥c(s)un(s) + d(s)∥2 L2(M,¯g) ds � = ∥∇u(0)∥2 L2(M,¯g) + 1 4aT � t 0 ∥c(s)u(s) + d(s)∥2 L2(M,¯g) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' It thus follows that ∥∇u(t)∥2 L2(M,¯g) − ∥∇u(0)∥2 L2(M,¯g) ≤ 1 4aT � t 0 ∥c(s)u(s) + d(s)∥2 L2(M,¯g) ds Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic 25 and therefore lim sup t→0 � ∥∇u(t)∥2 L2(M,¯g) − ∥∇u(0)∥2 L2(M,¯g) � ≤ 1 4aT lim t→0+ � t 0 ∥c(s)u(s) + d(s)∥2 L2(M,¯g) ds = 0, as claimed in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Next we prove a maximum principle for solutions of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We need the following preliminary lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Let T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (i) For any function u ∈ C2(M) we have � {x∈M|u(x)>0} ∆¯gudµ¯g ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (ii) Let u, ρ ∈ C1([0, T]) be functions with u(0) ≤ 0 and ρ(T) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then � {t∈[0,T ]|u(t)>0} � ρ(t)∂tu(t) + κu(t) � dt ≥ 0 with κ := sup s∈(0,T ) ∂tρ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11) (iii) Let u ∈ C2,1(ΩT ) ∩ C0,1(ΩT ), ρ ∈ C0,1(ΩT ) be functions with u ≤ 0 on M × {0} and ρ ≥ 0 on M × {T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then we have � {(x,t)∈M×[0,T ]|u(x,t)>0} (ρ(x, t)∂tu(x, t) + κu(x, t) − ∆¯gu(x, t))dµ¯g(x)dt ≥ 0 with κ := sup (s,x)∈M×(0,T ) ∂tρ(s, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (i) By Lebesgue’s theorem, it suffices to prove � {x∈M|u(x)>εn} ∆¯gudµ¯g ≤ 0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13) for a sequence εn → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By Sard’s Lemma, we may choose this sequence such that Ωε := {x ∈ M | u(x) > εn} is an open set of class C1, whereas the outer unit vector field of Ωε is given by (x, t) �→ − ∇¯gu(x,t) |∇¯gu(x,t)|¯g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='13) follows from the divergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (ii) The set {t ∈ [0, T] | u(t) > 0} is a union of at most countably many open intervals Ij, j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' For any such interval, partial integration gives � Ij � ρ(t)∂tu(t) + ∂tρ(t)u(t) � dt = � 0, if T ̸∈ Ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' ρ(T)u(T) ≥ 0, if T ∈ Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Consequently, � {t∈[0,T ]|u(t)>0} ρ(t)∂tu(t) dt ≥ − � {t∈[0,T ]|u(t)>0} ∂tρ(t)u(t) dt ≥ − � {t∈[0,T ]|u(t)>0} κu(t) dt with κ given in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' This shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (iii) This is a direct consequence of (i), (ii) and Fubini’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (Maximum principle) Let T > 0, a, c ∈ C(ΩT ) with aT := min (x,t)∈ΩT a(x, t) > 0, let d ∈ Lp(ΩT ) for some p > 2 with dT := sup(x,t)∈ΩT d(x, t) < ∞, and let u0 ∈ W 2,p(M, ¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, let u ∈ W 2,1 p (ΩT ) be the unique solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (i) If u0 ≤ 0 on M and dT ≤ 0, then u ≤ 0 on ΩT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (ii) If c ≡ 0 on ΩT , then u(x, t) ≤ ∥u+ 0 ∥L∞(M,¯g) + tdT for t ∈ [0, T], x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14) 26 Franziska Borer, Peter Elbau, Tobias Weth Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (i) Step 1: We consider the special case a ∈ C0,1(ΩT ), u0 ≤ 0 and dT ≤ −ε for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We put ρ := 1 a ∈ C0,1(ΩT ) and κ := sup (s,x)∈M×(0,T ) ∂tρ(s, x) as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, we consider the function ˘u ∈ W 2,1 p (ΩT ), ˘u(x, t) = e−˘κtu(x, t) with ˘κ = |κ| min(x,t)∈ΩT ρ(x,t) + ∥c∥L∞(ΩT ), noting that ˘u satisfies ρ(x, t)∂t˘u(x, t) − ∆¯g˘u(x, t) + κ˘u(x, t) = e−˘κt� u(x, t)(ρ(x, t)c(x, t) − ρ(x, t)˘κ + κ) + ρ(x, t)d(x, t) � ≤ −ρ(x, t)εe−˘κt almost everywhere in {(x, t) ∈ ΩT | ˘u(x, t) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='15) We now let (un)n∈N be a sequence in C2,1(ΩT ) ∩ C0,1(ΩT ) with un(x, 0) ≤ 0 and un → ˘u in W 2,1 p (ΩT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Since the functions gn := 1{(x,t)∈M×[0,T ]|un(x,t)>0} are bounded in Lp′(ΩT ), we may pass to a subsequence such that gn ⇀ g in Lp′(ΩT ), where g ≥ 0 and g ≡ 1 in {(x, t) ∈ M × [0, T] | ˘u(x, t) > 0}, since un → ˘u uniformly as a consequence of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1) and therefore gn → 1 pointwisely on {(x, t) ∈ M × [0, T] | ˘u(x, t) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Applying Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 (iii) to un,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' we find that 0 ≤ � {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='t)∈M×[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T ]|un(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='t)>0} � ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)∂tun(t) − ∆¯gun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) + κun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � dµ¯g(x)dt = � M×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T ) gn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)∂tun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) − ∆¯gun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) + κun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � dµ¯g(x)dt for all n ∈ N and therefore 0 ≤ lim n→∞ � M×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T ) gn(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)∂tun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) − ∆¯gun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) + κun(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � dµ¯g(x)dt = � M×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T ) g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)∂t˘u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) − ∆¯g˘u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) + κ˘u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t) � dµ¯gdt ≤ − � M×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T ) g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)εe−˘κtdµ¯g(x)dt ≤ − � {(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='t)∈M×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='T )|˘u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='t)>0} ρ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' t)εe−˘κtdµ¯g(x)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' We thus conclude that {(x, t) ∈ M × (0, T) | ˘u(x, t) > 0} = {(x, t) ∈ M × (0, T) | u(x, t) > 0} = ∅ and therefore u ≤ 0 in M × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 2: In the special case where a ∈ C0,1(ΩT ), u0 ≤ 0 and dT ≤ 0, we may apply Step 1 to the functions uε ∈ W 2,1 p (ΩT ) defined by uε(x, t) = u(x, t) − εt, which yields that uε ≤ 0 for every ε > 0 and therefore u ≤ 0 in ΩT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Step 3: In the general case, we consider a sequence an ∈ C0,1(ΩT ) with an → a in C(ΩT ), and we let un denote the associated solutions of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with a replaced by an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' As in the end of the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='1, we then find that, after passing to a subsequence, un ⇀ ˜u in W 2,1 p (ΩT ), where ˜u is a solution of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' By uniqueness, we have u = ˜u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Moreover, since un ≤ 0 for all n by Step 3, we have u = ˜u ≤ 0, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' (ii) We consider the function v ∈ W 2,1 p (ΩT ) given by v(x, t) = u(x, t)−∥u+ 0 ∥L∞(M) −tdT , which, by assumption, satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='3) with c ≡ 0, d − dT in place of d and u0 − ∥u+ 0 ∥L∞(M) in place of u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Then (i) yields v ≤ 0 in ΩT , and therefore u satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Borer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Galimberti, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Struwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' “Large” conformal Metrics of prescribed Gauss Curvature on Surfaces of higher Genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Commentarii Mathematici Helvetici 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='2 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 407–428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' doi: 10.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Struwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' “Bubbling” of the prescribed curvature flow on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' Journal of the European Mathematical Society 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content='10 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 3223–3262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' [20] N.' metadata={'source': 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+page_content=' 17 (1967), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} +page_content=' 473–484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtFLT4oBgHgl3EQfMS8S/content/2301.12015v1.pdf'} diff --git a/MdE3T4oBgHgl3EQfwQuL/vector_store/index.faiss b/MdE3T4oBgHgl3EQfwQuL/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..05383d18ed673d23864c5de92d01edd7343e6594 --- /dev/null +++ b/MdE3T4oBgHgl3EQfwQuL/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9e0b5b13384f5b9099ab9dec26a8cbbe2445fcbe34e5466f1b5aa66451a66da5 +size 3670061 diff --git a/MtE1T4oBgHgl3EQftQWC/vector_store/index.faiss b/MtE1T4oBgHgl3EQftQWC/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..2213d8bee118b955c8279cca5c5ab94829d5ab65 --- /dev/null +++ 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characteristic spin relaxation (T1) time +caused by, e.g., paramagnetic molecules in proximity. However, while only the negatively-charged +NV center is to be probed in these pulsed-laser measurements, an inevitable consequence of the +laser excitation is the conversion to the neutrally-charged NV state, interfering with the result for +the negatively-charged NV centers’ T1 time or even dominating the response signal. In this work, we +perform relaxometry measurements on an NV ensemble in nanodiamond combining a 520 nm excita- +tion laser and microwave excitation while simultaneously recording the fluorescence signals of both +charge states via independent beam paths. Correlating the fluorescence intensity ratios to the fluo- +rescence spectra at each laser power, we monitor the ratios of both charge states during the T1-time +measurement and systematically disclose the excitation-power-dependent charge conversion. Even +at laser intensities below saturation, we observe charge conversion, while at higher intensities, charge +conversion outweighs spin relaxation. These results underline the necessity of low excitation power +and fluorescence normalization before the relaxation time to accurately determine the T1 time and +characterize paramagnetic species close to the sensing diamond. +I. +INTRODUCTION +The negatively-charged nitrogen-vacancy (NV) cen- +ter in diamond constitutes a versatile tool for the detec- +tion of magnetic [1–9] and electric [10] fields with high +sensitivity and spatial resolution. Measurement of the +NV centers’ spin relaxation (T1) time is widely applied +in different fields of science to detect magnetic noise +[11, 12]. +Various so-called relaxometry measurement +schemes employ a reduction of the NV centers’ T1 time +with the host nanodiamond exposed to paramagnetic +molecules fluctuating at the NV centers’ resonance fre- +quency [13–15]. Thus, relaxometry schemes have been +used to detect a superparamagnetic nanoparticle [16], or +paramagnetic Gd3+ ions [15, 17–20]. Further, relaxome- +try with NV− centers has been utilized to trace chemical +reactions involving radicals [21, 22]. Also, the NV cen- +ters’ T1 time as a measure for the presence of paramag- +netic noise gains momentum in biological applications +[7, 12]. Individual ferritin proteins have been detected +[23] and relaxometry has been applied to detect radicals +even inside cells [24–27]. +Especially in the field of biology, T1 measurement +schemes are often conducted only with optical excita- +tion of the NV− centers, while the readout of their spin +states is realized by detection of the ensemble’s fluores- +cence intensity. However, recent results indicate that a +second process impeding the NV− centers’ fluorescence +signal is present in relaxometry measurements [20, 28– +31]. The laser pulse that is fundamental for preparation +∗ Author +to +whom +correspondence +should +be +addressed: +widera@physik.uni-kl.de +of the NV− centers’ spin state can additionally ionize the +NV− center to its neutrally-charged state, NV0. Conver- +sion under illumination and back-conversion in the dark +influence the NV− centers’ fluorescence signal, compli- +cating a seemingly simple measurement. A quantitative +determination of the unwanted contribution of the NV0 +state to the NV− relaxometry data is, however, elusive. +In this work, we compare the results of two relaxometry +schemes well-known in literature for the same nanodi- +amond at varying laser powers. Additionally, we intro- +duce a novel method to extract the ratio of the two NV +charge states from the NV centers’ fluorescence spectra +throughout the entire measurement sequence to give an +insight into the vivid NV charge dynamics we observe +in our data. +A level scheme of the NV center in diamond is de- +picted in Fig. 1, including the negatively-charged NV− +[32–34], the neutrally-charged NV0 and transitions from +NV− to NV0 under green illumination [35, 36]. We in- +clude transitions independent of excitation power from +the NV0’s ground state to NV−, reflecting the observa- +tion of recharging processes in the dark in [28, 29] and +in this work. +Using a 520 nm laser, we non-resonantly excite the +NV− centers from their triplet ground state 3A2 to the +electronically-excited state 3E. Because 3E’s states mS = +±1 are preferentially depopulated via the NV− centers’ +singlet states 1A1 and 1E, illumination with a green laser +will spin polarize the NV− centers into their ground +spin state mS = 0 [34]. +The T1 time describes how +long this spin polarization persists until the spin popu- +lation decays to a thermally mixed state [34]. It can reach +up to 6 ms in bulk diamonds at room temperature [37] +and is influenced by paramagnetic centers within the +arXiv:2301.01063v1 [quant-ph] 3 Jan 2023 + +2 +host diamond or on its surface [38, 39]. In the simplest +T1 measurement scheme, spin polarization is achieved +by a laser pulse, followed by a second readout-laser +pulse after a variable relaxation time τ. Besides differ- +ent durations, the two laser pulses are identical. There- +fore, the readout pulse is capable of spin-polarizing and +ionizing the NV-center ensemble as well as the initial- +ization pulse. Additionally, the spin-polarization pulse +provides information about the charge-conversion pro- +cesses during laser excitation. +To determine the T1 time of NV− centers of a spe- +cific orientation in the diamond crystal, coherent spin +manipulation is introduced in these measurements [38]. +Here, a resonant microwave π pulse transfers the pop- +ulation of these NV− centers from mS = 0 to mS = +1 +or mS = −1 after the spin-polarization pulse. A sec- +ond laser pulse is used for the readout of the spin state. +Repetition of the sequence with the π pulse omitted and +subtracting the readout signals from each other yields a +spin-polarization signal as a function of τ that is robust +against background fluorescence [38, 40]. +In the following, we present our experimental sys- +tem in Section II. Our results are divided into two main +parts. We first analyze fluorescence spectra of NV cen- +ters in a single nanodiamond to assign concentration ra- +tios to count ratios measured with SPCMs in Section III. +This knowledge allows us to quantify the NV0 contribu- +tion during the spin-relaxation dynamics in Section IV. +FIG. 1. Level scheme of the NV center in diamond. Depicted +are levels of the negatively-charged NV− and the neutrally- +charged NV0 and transitions between the two charge states. +Gray arrows show transitions between NV−’s triplet and sin- +glet states, mediated via intersystem crossing (ISC). Green ar- +rows denote transitions driven by a green laser, red and or- +ange arrows mark the fluorescence of the NV charge states. +Light-green dashed arrows between mS states are transitions +driven by microwave radiation at 2.87 GHz at zero magnetic +field. Additionally, the light-green dashed arrows represent +the relaxation of the spin-polarized state to a thermally mixed +state without illumination (T1). The purple dashed arrows de- +note charge transfer processes in the dark. +II. +EXPERIMENTAL SYSTEM +We +perform +our +studies +on +a +single +nanodia- +mond +crystal +of +size +750 nm +commercially +avail- +able +from +Adamas +Nano +as +water +suspension +(NDNV/NVN700nm2mg). +As specified by the man- +ufacturer, +the nanodiamonds’ NV concentration is +[NV] ≈ 0.5 ppm, which is about 2 × 104 NV centers per +diamond. +For sample preparation, the suspension is +treated in an ultrasonic bath to prevent the formation of +crystal agglomerates. We spin-coat the nanodiamonds +to a glass substrate and subsequently remove the sol- +vent by evaporating the residual water on a hot contact +plate. +To probe the NV centers in a single nanodiamond, we +use a microscope consisting of an optical excitation and +detection section and a microwave setup, as shown in +Fig. 2. A CW-laser source of wavelength λ = 520 nm +is used to optically excite the NV centers with a max- +imum laser power of 4.9 mW. +The laser beam is fo- +50:50 NPBS +NF 514 nm +DM 550 nm +laser +520 nm +AOM +objective +sample +permanent magnet +MW antenna +LP 550 nm +ND1 +ND2 +SP 625 nm +LP 665 nm +to SPCM2 +> 665 nm +to SPCM1 + < 600 nm +to spectrometer +spectrometer +camera +tube lens +grating +(a) +(b) +FIG. 2. +Experimental setup for recording NV fluorescence +spectra and relaxometry data. In both setups, the excitation +is the same, but the detection sections are different for the re- +spective application. (a) NV centers in a single crystal nanodi- +amond are excited by a 520 nm-laser in combination with an +acousto-optic modulator (AOM). The light stemming from the +sample is filtered by a dichroic mirror (DM), a longpass filter +(LP) and a notch filter (NF) with given wavelenghts and passes +a non-polarizing beamsplitter (NPBS). The remaining fluores- +cence is spectrally resolved on a camera chip. This setup is +used for the measurement of the NV fluorescence spectra. (b) +The NV fluorescence is split into two arms of a beamsplitter, +additionally filtered with an LP or a shortpass filter (SP) and +detected with fiber-coupled SPCMs. The SP is tilted to only +transmit fluorescence below 600 nm. +To keep the detectors +below saturation, neutral-density (ND) filters are used. Lu- +minescence above 665 nm (NV− fluorescence) is detected in +SPCM2, while light below 600 nm (NV0 fluorescence) is de- +tected in SPCM1. Transitions of the NV− centers’ spin states +mS are driven with a microwave (MW) antenna. This setup is +used for the measurement of the charge-state dependent relax- +ometry. + +3 +cused to a spot-size diameter of 700 nm (1/e2 diame- +ter), reaching a maximum intensity of ∼ 2500 kW cm−2. +Pulses are generated by an AOM with an edge width of +about 120 ns. Laser light is guided through an objective +(NA = 0.5, WD = 2.1 mm) and focused at the position +of the nanodiamond. Fluorescent light stemming from +the sample is guided back through the objective and fil- +tered by a dichroic mirror with a cut-on wavelength of +550 nm. Next, the fluorescence light is filtered by an ad- +ditional 550 nm-longpass filter and a 514 nm-notch filter +to prevent detection of reflected laser light. The filtered +fluorescence light is branched at a 50:50 non-polarizing +beamsplitter, giving the possibility to further filter the +luminescence and collect it in two separate detectors. In +particular, our setup allows for tailoring the transmitted +wavelengths to the spectral regions, where either pho- +ton emission from the neutral or the negative NV charge +state dominates in each beam path individually. Thus, +we can easily discriminate between the emission of both +charge states in our measurements. In this work, we +make use of different detectors. While for spectral anal- +ysis of the NV centers’ fluorescence, we use a spectrom- +eter (Fig. 2 (a)), we employ two single-photon counting +modules (SPCMs) as detectors for our spin-relaxation +measurements (Fig. 2 (b)) in combination with a time- +to-digital converter. +Microwave signals are generated, amplified, and +brought close to the nanodiamond using a microwave +antenna structure written on a glass substrate. +All +experiments are carried out under ambient conditions +and in an external magnetic field in the order of 12 mT +caused by a permanent magnet to split the NV centers’ +ODMR resonances. In our ODMR spectrum, eight reso- +nances appear because of the four existing orientations +of NV centers in the single diamond crystal. We select +one resonance to drive Rabi oscillations, from which we +determine a π-pulse length of 170 ns. +III. +FLUORESCENCE SPECTRA +A. +Setup +To spectrally resolve the NV centers’ fluorescence, we +use a spectrometer. The incoming fluorescence light is +dispersed at a grating (600 grooves/mm), and an achro- +matic tube lens translates the angle dispersion into a +spatial dispersion. Thus, the detection of light of dif- +ferent wavelengths at different positions of a camera’s +chip is facilitated, and spectra are obtained from 500 nm +to 760 nm. With this setup, we achieve a resolution of +∆λ ≈ 0.19 nm/pixel. Each spectrum consists of a mean +of at least 20 spectra recorded at each laser power. We +correct the spectra for the wavelength-dependent prop- +erties of optical elements in the beam path and subtract +a background. +B. +Concentration ratio assignment +Corrected fluorescence spectra of a monocrystalline +nanodiamond for excitation laser powers from 1 % to +100 % are depicted in Fig. 3 (a). Two features, the NV0s’ +ZPL at ∼ 575 nm [41] and the NV−s’ ZPL at ∼ 639 nm +[42] are clearly visible. +The overlapping fluorescence +spectra of both NV charge states show phonon broad- +ening. Conform with the observation in [1], but con- +trary to the results in [31], the NV0s’ ZPL intensity +increases with higher laser power with respect to the +NV−s’ ZPL in our sample. These results indicate a lower +[NV−]/[NV0] ratio at higher laser powers and thus an +increasing charge conversion for higher powers. +We obtain area-normalized extracted spectra for NV− +and for NV0 from our recorded data as shown in +Fig. 3 (b). We conduct the spectra decomposition anal- +ysis of our spectra according to Alsid et al. and follow +the nomenclature given in reference [43]. The fraction of +[NV−] of the total NV concentration [NVtotal] is defined +(a) +(b) +FIG. 3. NV fluorescence spectra. (a) Spectra recorded at laser +powers from 1 % to 100 %. The NV0s’ ZPL at ∼ 575 nm and the +NV−s’ ZPL at ∼ 639 nm are evident and marked in the spec- +trum. For better visibility, spectra were normalized to the sum +of the NV charge states’ ZPL intensities. (b) Area-normalized +decomposed basis functions for NV0 and NV−. + +4 +by +[NV−] +[NVtotal] = +[NV−] +[NV−] + [NV0] = +c− +c− + κ520c0 +. +(1) +Thus, the concentration ratio between NV charge +states [NV−]/[NV0] can be described with +[NV−] +[NV0] = c− +c0 +1 +κ520 +. +(2) +Here, c− and c0 describe the coefficients of the ba- +sis functions of NV− and NV0 used to assemble an +area-normalized composed spectrum at arbitrary laser +power with the condition c− + c0 = 1. +The correc- +tion factor κ520 translates this fluorescence ratio c−/c0 +to the ratio of NV concentrations [NV−]/[NV0], taking +into account the different lifetimes and the absorption +cross sections of the two NV charge states [43]. Note +the different subscript in our work for the excitation +(a) +(b) +FIG. 4. +(a) NV fractions as a function of the laser power we +derived from spectral analysis. (b) NV ratios as a function of +the fluorescence count ratio in the two SPCMs applied as de- +tectors. Using the fit curve, we map the fluorescence count +ratio to an NV ratio during relaxometry measurements. For +fitting the [NV−]/[NV0] concentration ratio with f (x) = axn, +we obtain a = 0.0135 ± 0.0001 and n = 1.334 ± 0.004. The re- +ciprocal ratio [NV0]/[NV−] was not fit separately, displayed is +the function g(x) = a−1x−n. +wavelength of 520 nm compared to κ532 in [43]. +Us- +ing ten spectra recorded at laser powers below the sat- +uration intensity and the deviations from the linearity +of the charge states’ fluorescence intensity with the ap- +plied laser power, we find κ520 = 2.03 ± 0.07. +The +error denotes the statistical error from a weighted fit +we performed on our measurement data. +For a de- +tailed description of the determination of κ520, see Ap- +pendix B. This value is within the reported value for +κ532 = 2.5 ± 0.5 for an excitation wavelength of 532 nm +[43]. +We use our value for κ520 to calculate the frac- +tions of [NV−] and [NV0] and the concentration ratio +[NV−]/[NV0] as a function of the laser power. +As shown in Fig. 4 (a), the fraction of [NV−] is high for +low laser powers and decreases with higher laser pow- +ers. At the lowest laser power of 0.1 %, about 73 % of +the total NV concentration is [NV−], while at the high- +est laser power, only about 21 % [NV−] remain. Already +at laser powers of 2 % (∼ 51 kW cm−2), which is below +saturation intensity (≈ 100 kW cm−2) [44], [NV0] out- +weighs [NV−]. +Therefore, a significant influence due +to charge conversion is to be considered in relaxometry +measurements. +Together with the recorded fluorescence-count-rate +ratio of both SPCMs for each laser power, we assign each +count-rate ratio ρSPCM2/ρSPCM1 a ratio [NV−]/[NV0]. +The results are shown in Fig. 4 (b). +With an increas- +ing ratio of ρSPCM2/ρSPCM1, the ratio [NV−]/[NV0] in- +creases. We fit a power law (inverse-variance-weighted +fit) to the ratio [NV−]/[NV0] to be able to trace the NV- +concentration ratio over a broad range of count-rate ra- +tios during the spin-relaxation measurements. Thereby +we are able to quantitatively trace the contribution of +NV0 during the spin-relaxation dynamics of the NV− +centers in the following. +IV. +SPIN-RELAXATION MEASUREMENTS +A. +Setup and measurement sequences +To separately detect the fluorescence of NV− and NV0 +throughout our measurement, different filters are used +in the optical beam path as depicted in Fig. 2 (b). Af- +ter passing a 50:50 non-polarizing beamsplitter, the sam- +ple’s transmitted fluorescence light is guided through a +665 nm-longpass filter, and mainly NV− fluorescence is +detected. For the luminescence reflected by the beam- +splitter, we use a tilted 625 nm-shortpass filter to collect +NV0 fluorescence below 600 nm. Neutral-density filters +are added in front of the beamsplitter and within its +transmitted beam path to keep the SPCMs below sat- +uration. +For determining the longitudinal spin relaxation time +T1, we conduct and compare two different and fre- +quently used pulsed-measurement schemes, which we + +5 +term P1 and P2 in the following. These two pulse se- +quences are depicted in Fig. 5. +In the pulsed sequence P1, we choose an initialization +pulse of 200 µs duration to spin polarize the NV-center +ensemble to their spin states mS = 0. We apply a 5 µs +normalization pulse 1 µs after the initialization pulse to +probe the fluorescence intensity before a variable relax- +ation time τ in gate Cπ +1 . To assure a depopulation of the +NV− centers’ singlet states, we choose the time between +the two pulses to be longer than the singlet lifetimes of +τmeta ≈ 150 ns at room temperature [45]. Approximately +1.5 µs into τ, a resonant π pulse is applied. After τ, a +readout pulse of duration 5 µs probes the fluorescence of +both NV centers’ charge states in gate Rπ. Subsequently, +the sequence is repeated with the π pulse omitted, ob- +taining fluorescence intensities in gates C1 and R. The +spin polarization as a function of τ for NV− is obtained +by subtracting the fluorescence counts in Rπ from the +counts in gate R. Details on measurement sequence P1 +can be found in [38, 40]. Sequence P1 provides a tech- +nique for determination of the NV− centers’ T1 time ro- +bust against background fluorescence [38, 40] and is be- +lieved to be unaffected by charge-state conversion [29]. +Analysis of the second half of P1 represents an all- +optical T1 measurement scheme as often applied in bi- +ology [7, 24, 27]. Further, using only the second half +of this sequence, we are able to obtain the fluorescence +evolution as a function of τ for NV− and NV0, includ- +ing effects caused by the charge-state conversion. Only +taking into account the signal without the π pulse ap- +laser +laser +MW +MW +detection +detection +(a) Sequence P1 +(b) Sequence P2 +FIG. 5. +Longitudinal spin relaxation time (T1) measurement +schemes applied in this work. The beginning of the second +half of each sequence is indicated by a dashed line. (a) Se- +quence P1. +The NV ensemble is spin polarized by a laser +pulse. Next, the fluorescence is detected by a control pulse +(orange). Within the variable relaxation time τ, a resonant π +pulse is applied (light-green). The spin state is read out by a +third laser pulse (purple). The sequence is repeated with the +π pulse omitted after a pause time tp. (b) Sequence P2. As +opposed to P1, the readout pulse has the same length as the +spin-polarization pulse. The control gates C1 within the ini- +tialization pulse and C2 within the readout pulse will be com- +pared in this work. +plied, we obtain the fluorescence evolution by dividing +the fluorescence counts in gate R by the counts in gate +C1. Charge conversion during the relaxometry measure- +ment has an effect on the NV− fluorescence as well as +on the NV0 fluorescence during the relaxation time τ. +Therefore, by only evaluating P1’s second half, we gain +information about the charge conversion taking place +alongside the spin relaxation. However, to obtain the +NV− centers’ T1 time, the full sequence P1 is evaluated. +As opposed to P1, P2 uses a normalization probe after +the readout of the NV centers’ fluorescence [19, 21, 46]. +We choose the laser readout pulse to have the same du- +ration as the initialization pulse (200 µs) and carry out +the readout gates Rπ and R in the first 5 µs and the nor- +malization probes Cπ +2 and C2 in the last 5 µs of the read- +out pulse. Scheme P2 assumes the NV centers to have +the same fluorescence intensity at the end of the second +pulse as at the end of the first pulse. To test this notion, +we apply second normalization gates, Cπ +1 and C1, within +the last 5 µs of the initialization pulse and compare the +results for both normalized data. +Between readout and the upcoming initialization +pulse, we insert a pause time tp between the sequences +of 1 ms, which is in the order of T1, to minimize build-up +effects from spin polarization during the cycle for both +sequences. Each cycle is repeated 50 000 times, and the +whole measurement is swept multiple times. The se- +quences are repeated for different laser powers, ranging +from 0.1 % to 11 % of the maximum laser power. +B. +Results and discussion +In this section, we present and compare the experi- +mental results for the spin-relaxation measurements for +both sequences, P1 and P2. +Using our experimental +setup as described in Section IV A, we observe laser- +power-dependent dynamics in the NV− and NV0 flu- +orescence throughout our measurement. Fig. 6 depicts +an example for the fluorescence as a function of τ for +the NV0 fluorescence recorded at a laser power of 11 % +with sequence P1. These results show the normalized +fluorescence as a function of τ obtained from the second +part of the measurement sequence without a microwave +π pulse, dividing the fluorescence counts in gate R by +the counts in gate C1. The normalized fluorescence as +a function of τ decays exponentially. Different from the +dynamics of the NV− center, we observe similar behav- +ior for the NV0 fluorescence at all laser powers. We fit a +biexponential function of type +f 0(τ) = A e−τ/TR,1 + B e−τ/TR,2 + d +(3) +to our measurement data and obtain two recharge times +in the order of TR,1 = 100 µs and TR,2 = 2.0 ms for +all laser powers. We assign these time constants TR to +an electron-recapturing process of NV0 during the dark + +6 +time τ, after an ionization from NV− to NV0 has pre- +viously taken place in the initializing laser pulse. Re- +markably, this process occurs even at the lowest laser +power. +Presumably, the presence of two components +of TR is due to the different environments of NV cen- +ters concerning charge transfer sites. Vacancies or elec- +tronegative surface groups on the diamond surface are +known to promote a charge conversion of NV− to NV0 +[47, 48]. We assume that the NV environment similarly +affects the recharging process in the dark. Therefore, we +attribute one component of TR to NV centers closer to +the nanodiamond surface and the other to NV centers +more proximate to the center of the crystal. We empha- +size that both TR,1 and TR,2 we report match previously +reported values for TR of 100 µs [28] and (3 ± 1) ms [20] +and underline that they simultaneously appear as two +components in our sample. We find that neither TR,1 +nor TR,2 changes as a function of the laser power. The +coefficients of the exponential functions A and B do not +change significantly from 1 % to 11 % laser power. How- +ever, for the lowest laser power of 0.1 % A and B are +smaller. We attribute this to little NV0 fluorescence ob- +served at this low laser power due to less charge conver- +sion, resulting in a lower signal-to-noise ratio (SNR) for +the NV0 fluorescence. +Further, we present the results for the normalized +NV− fluorescence as a function of τ in Fig. 7 (a) for +ascending laser powers. We conducted the experiment +with sequence P1, and this data refers to the results with +the π pulse omitted. +The laser-power-dependent dy- +namics of NV− and NV0 result in a drastic change of +shape of the normalized fluorescence as a function of τ. +While we observe an exponential decay in the lowest +laser power, we find an inverted exponential profile of +the NV− fluorescence at 11 % laser power. In-between +laser powers show both an exponential decay and an in- +0 +2 +4 +6 +8 +10 +τ (ms) +0.7 +0.8 +0.9 +normalized fluorescence +R/C1 +biexponential fit +FIG. 6. +NV0 fluorescence as a function of τ as recorded +with sequence P1 (second half) by division of the fluorescence +counts in R by the counts in C1 for 11 % laser power. With +a biexponential fit function, we find TR,1 = (109 ± 7) µs and +TR,2 = (2.1 ± 0.1) ms. +crease, present in the fluorescence. This phenomenon of +inverted exponential components in the recorded nor- +malized fluorescence during a T1 measurement has been +reported by [29] and attributed to a recharging process +of NV0 to NV− during τ. However, a complete flip of +the fluorescence alone by a laser power increase has not +been reported so far. Remarkably, this behavior indi- +cates that NV0 to NV− charge dynamics outweigh the +NV− ensemble’s spin relaxation at high laser powers in +our sample. +To better understand the NV− power-dependent be- +havior, we use the results from the spectral analysis to +map the ratios of [NV−]/[NV0] to our relaxometry mea- +surement data of sequence P1. The ratio [NV−]/[NV0] +as a function of τ for all laser powers is summarized +in Fig. 8 (a) and displayed in more detail in Fig. C1 (a). +For all laser powers, even for the lowest, which lies well +below saturation intensity, we observe an increase of +[NV−]/[NV0] as a function of τ in the readout pulse R. +We conclude that during τ a re-conversion from NV0 to +NV− takes place in the dark, after ionization of NV− +had occurred in the initialization pulse. While for the +lowest power, the ratio [NV−]/[NV0] increases from 4.0 +to 7.6 over the variation of τ, [NV0] outweighs [NV−] at +11 % laser power throughout the entire relaxation mea- +surement, see Fig. C1 (a). As shown in Fig. 8 (a), the ra- +tios increase by a factor of ∼ 2 from shortest to longest +τ for all laser powers. +The ratios [NV−]/[NV0] we find in control pulse C1 +as a function of τ also show a power-dependent be- +havior. While the ratio increases as a function of τ in +the lowest power, it is constant in the control pulse for +the highest power. +These power-dependent recharge +processes in the control pulse we observe appear most +likely due to build-up effects during the measurement +cycle, as we explain in the following. At low powers, +the initializing laser pulse spin polarizes the NV− cen- +ters but does not ionize to a steady state of NV− and +NV0. For short τ, the re-conversion in the dark of NV0 +to NV− has not completed, and the following laser pulse +continues to ionize the NV− centers. However, at the +highest power, each initialization pulse efficiently ion- +izes to a steady state of the two NV charge states, reach- +ing [NV−]/[NV0] ≈ 0.44. These results clearly show +that the normalization in the sequence we perform is +mandatory to only detect the change in the relative flu- +orescence during the relaxation time τ and minimize in- +fluences due to charges passed through cycles. +At the lowest laser power of 0.1 %, we observe the +highest ratio of [NV−]/[NV0] and therefore expect the +most negligible influences of charge conversion on the +NV−s’ spin relaxation. Thus, we fit a monoexponential +function to the relative fluorescence as a function of τ +and obtain T1 = (1.42 ± 0.06) ms for the NV− ensemble +in the nanodiamond. To further underline the necessity +of a normalization of the fluorescence intensity, we fit a + +7 +(a) +(b) +(c) +FIG. 7. NV− fluorescence as a function of τ, recorded with sequence P1 for different laser powers. Insets show the characteristics +of the sequences applied. (a) NV− fluorescence as obtained from the second half of P1, dividing the fluorescence counts in R by +the counts in C1. We observe a transition from an exponential decay of the fluorescence to an inverted exponential profile with +rising laser powers. For 0.1 % laser power, we perform a monoexponential fit and obtain T1 = (1.42 ± 0.06) ms. For laser powers +from 1 % to 11 %, we fit a sum of three exponential functions as explained in the text. (b) Spin polarization of the NV− ensemble +as obtained from the full sequence P1, subtracting the fluorescence counts in Rπ from the counts in R. Unlike (a), we observe an +exponential decay at all laser powers in this measurement data. However, with increasing laser power, we observe a decrease +in the amplitude of the exponential function. Fitting a monoexponential function to the data at 0.1 % laser power, we obtain +T1 = (1.5 ± 0.1) ms, consistent with the T1 time we find in (a) at the same laser power. (c) NV− fluorescence as obtained from +the second half of P2, dividing the fluorescence counts in R by the counts in C1 or C2. Fitting monoexponential functions to the +fluorescence normalized by the counts in C1 or C2 yields T1 = (1.54 ± 0.06) ms or T1 = (1.50 ± 0.07) ms, respectively, consistent +with the results named above. The NV− fluorescence qualitatively behaves as in sequence P1, see (a), and is fitted accordingly. +On the contrary, the shape of the fluorescence depends on the position of the normalization gate, C1 or C2, especially visible at +4 % laser power. +monoexponential function to the non-normalized bare +NV− fluorescence detected in R at 0.1 % laser power. +We obtain a T1 time of (0.94 ± 0.05) ms, see Fig. C2 (a), +which is drastically lower than the T1 time retrieved +with normalization by the fluorescence counts in C1. +For higher laser powers, we fit the normalized data +with a function of type +f −(τ) = −A e−τ/TR,1 − B e−τ/TR,2 + C e−τ/T1 + d +(4) +and restrict the time constants to TR,1 = 100 µs, TR,2 = +2.0 ms and T1 = 1.4 ms. With this, we assume that the +decay of [NV0] causes an increase of [NV−] and, there- +fore, their fluorescence. Thus, the NV− fluorescence is +best described by a sum of an exponential decay due +to the loss of spin polarization and a biexponential in- +verted component due to the recharging process of NV0 +to NV− in the dark. As shown in Fig. 7 (a), our fit func- +tion Eq. (4) describes the measurement data from 1 % to +11 % laser power very well. We emphasize that the mea- +surement data for 1 % laser power does not visibly ap- +pear to show this triexponential behavior. Fitting a mo- +noexponential function to the NV− fluorescence at 1 % +laser power, however, results in T1 = (1.28 ± 0.06) ms, +see Fig. C2 (b), which deviates significantly from the +value obtained at lower laser power. +Measurement +sequence +P1 +is +a +well-established +method to accurately measure the T1 time of the NV− + +8 +centers excited by a resonant π pulse [38]. +Since the +π pulse only acts on the negatively-charged NV cen- +ters, it is said to be independent of charge conversion +processes alongside the spin polarization [29]. We com- +pare the results we obtain in the complete measurement +sequence P1, subtracting fluorescence intensities in Rπ +from the counts in R, to the result we gave for the T1 time +above without the π pulse taken into account. Remark- +ably, although in Fig. 7 (a) we observe vivid dynamics +ranging from exponential decay to an inverted exponen- +tial profile in the NV− fluorescence as a function of τ, +the complete sequence P1 yields a monoexponential de- +crease for all laser powers, see Fig. 7 (b). For the lowest +laser power, we obtain T1 = (1.5 ± 0.1) ms for sequence +P1 comparing the fluorescence intensity with and with- +out the resonant π pulse. This value matches the pre- +viously determined T1 time when only considering the +normalized signal without the π pulse for the lowest +laser power. It does not match the T1 time obtained from +the monoexponential fit we performed on the measure- +ment data for 1 % laser power, stressing the effects of +(a) +(b) +FIG. 8. Changes of the NV-charge-state ratio during the relax- +ometry measurement from lowest to highest τ. (a) Sequence +P1. The change of the ratio [NV−]/[NV0] is constant as a func- +tion of the laser power in the readout pulse R, while it decays +in the control pulse C1. (b) Sequence P2. The change of the +ratio [NV−]/[NV0] in readout and control pulses show qual- +itatively the same behavior as in sequence P1. However, the +changes in the NV-charge-state ratio in C1 and C2 as a func- +tion of the laser power differ. +NV charge conversion within this measurement and the +necessity for consideration of the two components TR,1 +and TR,2 in a triexponential fit function. +However, the measurement sequence P1 is not en- +tirely unaffected by the charge conversion process. Al- +though the resonant π pulse does not directly act on +the NV0 center (we observe no difference in the signals +with and without the π pulse), the fluorescence contrast +in the measurement decreases because of NV0 to NV− +conversion. This lower contrast becomes noticeable in +Fig. 7 (b) due to the decaying amplitude of the mono- +exponential function with increasing laser power. The +effect of NV− spin depolarization due to charge conver- +sion has been previously investigated in [28, 35]. As a +measure for the reliability of our measurement result, +we use the area under the curves showing spin polar- +ization as a function of τ for each laser power as a fluo- +rescence contrast in the respective measurement. We di- +vide this value by the Root mean squared error (RMSE) +value we obtain from the fit result to account for fluc- +tuations in our measurement data and define this value +contrast/RMSE as the SNR. In Fig. 9, we show the SNR +as a function of the laser power. In addition, we dis- +play the value for T1 we obtain in the same graph. With +the SNR decreasing, we observe a decrease in T1, accom- +panied by a larger standard deviation with higher laser +power. We conclude that the T1 time we measured at +the lowest laser power is the most reliable one due to the +highest SNR. In addition, we note that T1 seems to decay +as a function of the laser power, although T1 should be +independent of the excitation power. We attribute this +decay of T1 to the lower SNR in the measurements at +higher laser power due to increased charge conversion. +From the results of sequence P1, we conclude that the +normalization in the T1 measurement is essential to re- +0 +5 +10 +laser power (%) +5 +10 +15 +20 +25 +30 +SNR (arb. units) +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +T1 (ms) +SNR +T1 +FIG. 9. SNR and T1 time as a function of the laser power, ob- +tained from measurements performed in sequence P1. With +higher laser power, the SNR decreases, and so does the T1 +time. +The standard error of T1 that we obtain from mono- +exponential fits increases with higher laser power. The data +is extracted from equal numbers of repetitions of relaxometry +measurements for each laser power. + +9 +flect the charge-state processes alongside the NV− en- +semble’s spin relaxation. Besides P1, sequence P2 is used +in literature to determine a single NV center’s [46] or +an ensemble’s [19] T1 time. While the π pulse is often +omitted in these sequences, we chose to implement it +for low laser powers for better comparison to the results +obtained in P1. For laser powers starting from 3 %, we +repeated the sequence without the π pulse and calcu- +lated the mean values of the control and readout data +taken. The results for sequence P2 with a π pulse in- +cluded for 0.1 % laser power are shown in Fig. C2 (c). +Using the data for 0.1 % laser power and subtracting Rπ +from R, we obtain T1 = (1.45 ± 0.09) ms, which is the +same result as in sequence P1. Since both sequences are +used in the literature to measure an NV− ensemble’s T1 +time, we expect them to produce the same result for our +NV ensemble when neglecting additional effects due +to charge conversion. At this low laser power, charge +conversion is inferior to spin relaxation. Therefore, the +T1 times we obtain from both sequences do not differ. +However, with higher laser power, charge conversion +prevails, and both NV charge states’ fluorescence sig- +nals are greatly affected by recharge in the dark. +In order to evaluate the result of sequence P2 without +the π pulse applied, we normalize the fluorescence in- +tensities. To this end, we divide the counts in R obtained +by the counts measured during the two control gates C1 +or C2, yielding two normalized fluorescence signals for +each NV charge state. This way, we obtain two normal- +ized fluorescence signals as a function of τ. If no charge +conversion effects were present in this measurement, +both signals for the normalized fluorescence should be +equal. However, as pointed out, charge conversion is +prominent in our sample, not only for high laser pow- +ers. We show the NV− fluorescence as a function of τ +we obtain from sequence P2 in Fig. 7 (c). Qualitatively +similar to sequence P1, we see a smooth transition from +an exponential decay at low laser powers to an inverted +exponential profile at high laser powers. Similarly as in +P1, we derive T1 = (1.54 ± 0.06) ms for normalization +with C1 and T1 = (1.50 ± 0.07) ms for normalization +with C2 for the lowest laser power. We emphasize that +all T1 times we derive from the normalized NV− fluo- +rescence in both sequences are equal within their stan- +dard errors. In addition, the values for TR,1 and TR,2 we +obtain from the NV0 fluorescence with sequence P2 are +the same as in sequence P1. We fit the NV− fluorescence +for laser powers from 1 % to 11 % in the same manner as +for P1 using Eq. (4) and the aforementioned values for +T1, TR,1 and TR,2. This triexponential fit function models +our data well, regardless of the normalization we use. +However, the amplitudes of the respective exponen- +tial functions differ depending on the normalization, C1 +or C2, employed. Thus, the shapes of the fluorescence as +a function of τ differ with the gates used for normaliza- +tion, which is especially visible at 4 % laser power. To +understand the difference in the measurement results +that the positions of the normalization gate cause, we +take the ratios [NV−]/[NV0] into account. In Fig. C1 (b) +and (c), [NV−]/[NV0] as a function of τ for sequence P2 +is displayed and summarized as a change from shortest +to longest τ in Fig. 8 (b) for each laser power. The ra- +tios as a function of τ behave similarly to as observed +with sequence P1 discussed above. We note that the ra- +tios we obtain in our measurement for the two control +gates C1 and C2 are different. For τ ≲ 1 ms the ratio +[NV−]/[NV0] is smaller for C2 than for C1, while for val- +ues τ ≳ 1 ms the opposite is the case, see Fig. C1 (c). For +the same reasons discussed in sequence P1, this effect is +prominent in laser powers up to 4 %. In contrast, for the +highest laser power, the ratios in the control gates are +approximately constant with τ and do not differ signifi- +cantly. As pointed out in the discussion of P1, the results +indicate that the first laser pulse does not ionize into a +steady state of [NV−]/[NV0], and the second laser pulse +continues to ionize NV− into NV0. Therefore, especially +for small values of τ, the ratio [NV−]/[NV0] is smaller in +C2 than in C1. For larger values of τ, recharge dynamics +of NV0 to NV− in the dark add to the different ratios of +[NV−]/[NV0] for both control gates. We do not exclude +additional effects due to continued spin polarization of +NV− in the second laser pulse, especially for low laser +powers. +It is for these reasons that in Fig. 8 (b) the changes of +[NV−]/[NV0] as a function of the laser power are higher +for C2 than for C1 for low powers and converge to the +same value for higher laser powers. We therefore at- +tribute the difference in the normalized fluorescences in +Fig. 7 (c) when normalizing to C1 or C2 to the differences +in [NV−]/[NV0] for C1 and C2, respectively. +Both the results from measurement sequences P1 and +P2 and the simultaneous mapping of [NV−]/[NV0] in- +dicate that a charge conversion from NV− to NV0 dur- +ing the spin-polarization pulse of a spin-relaxation mea- +surement is inevitable. We emphasize that a normal- +ization gate is mandatory to correctly display the fluo- +rescence dynamics of NV0 and NV− as a function of τ. +Comparison of the two control gates C1 and C2 shows +that the normalized fluorescence signal depends on the +positions of the gate used for normalization because of +charge conversion processes that take place alongside +the NV− ensemble’s spin relaxation. +V. +CONCLUSIONS +This work examines laser-power-dependent dynam- +ics of NV charge conversion within spin-relaxation mea- +surements of the negatively-charged NV centers in a sin- +gle nanodiamond. We present a new method of trac- +ing the ratio of [NV−] to [NV0] during our sequence, in +which we extract the relative concentrations of NV− to +NV0 from their fluorescence spectra and perform a map- +ping to fluorescence count ratios in two separate detec- + +10 +tors. From the analysis of low-excitation intensity spec- +tra, we find κ520 = 2.03 ± 0.07. This correction factor +κ520 allows us to translate the fluorescence ratio of NV− +to NV0 to a concentration ratio, taking into account dif- +ferent lifetimes and absorption cross sections for the two +charge states. Combining our results, we conclude that +ionization of NV− to NV0 during the optical initializa- +tion and readout is inevitable and occurs even at low +laser powers. A recharge process in the dark of NV0 to +NV− significantly affects the NV− ensemble’s fluores- +cence during the spin-relaxation measurement. We find +the recharging in the dark to be biexponential with com- +ponents TR,1 = 100 µs and TR,2 = 2.0 ms. At high laser +powers, the effect of charge conversion outweighs spin +relaxation, making it impossible to accurately measure a +T1 time, even with a scheme involving a π pulse for two +reasons. Firstly, recharging effects of NV0 to NV− in the +dark dominate the NV− fluorescence signal. Secondly, +the measurement of T1 is crucially impeded by a dimin- +ished fluorescence contrast due to charge conversion. To +determine the NV− centers’ T1 time at low laser powers, +we find it necessary to conduct a pulsed sequence with +a normalization gate included. We prove the normal- +ization mandatory to accurately reflect the charge-state +dynamics as a function of τ and mitigate additional ef- +fects due to charge-state accumulation during the mea- +surement cycle. Additionally, comparing two pulsed se- +quences often used in the literature, we find that the po- +sition of the normalization gates plays an essential role +due to charge conversion during the measurement. We +emphasize that including a normalization gate directly +after the spin polarization before the relaxation time τ is +a simple method to accurately display the fluorescence +dynamics during the relaxation time. This way, com- +paring the fluorescence counts in the readout gate to the +counts in the control gate reliably reflects the spin relax- +ation and the charge dynamics in the relaxometry mea- +surement. +Overall, we emphasize that the results presented in +this work impact relaxometry schemes widely used in +biology, chemistry, and physics. +To further extend +this work, the effects of different duration of the spin- +polarization pulse and the readout pulse can be exam- +ined and give insight into the steady-state dynamics of +the NV centers. Further, the excitation of NV− can be +conducted at longer wavelengths, changing the charge- +state dynamics [49] and impacting the spin relaxation +results. The influence of different NV and nitrogen con- +centrations in diamonds of different sizes on the charge +dynamics can be considered to unravel the mechanisms +of charge conversion in the dark. +ACKNOWLEDGMENTS +We acknowledge support by the nano-structuring +center NSC. This project was funded by the Deutsche +Forschungsgemeinschaft +(DFG, +German +Research +Foundation)—Project-ID No. +454931666. +Further, +I. C. B. thanks the Studienstiftung des deutschen Volkes +for financial support. +We thank Oliver Opaluch and +Elke Neu-Ruffing for providing the microwave antenna +in our experimental setup. Furthermore, we thank Sian +Barbosa, Stefan Dix, and Dennis L¨onard for fruitful +discussions and experimental support. +Appendix A: Methods +To understand the NV centers’ fluorescence evolu- +tion as a function of τ in terms of charge conversion, +we map the fluorescence count ratio detected in both +SPCMs to a ratio of NV− and NV0 throughout the +spin-relaxation measurement. +For this, we combine +the results of recorded NV spectra and spin-relaxation +measurements. We choose a single nanodiamond and +record fluorescence spectra at different laser powers us- +ing the setup in the configuration shown in Fig. 2 (a). +Both charge states, NV− and NV0, contribute to the +recorded spectra between 500 nm and 750 nm because +of the charge states’ overlapping phononic sidebands. +For further analysis, we decompose the obtained spec- +tra into NV− and NV0 basis functions as described by +[43] using the spectra we recorded at the highest and +lowest laser power. Employing our extracted basis func- +tions, we obtain the fluorescence ratio of both NV charge +states for all other laser powers with the help of MAT- +LAB’s function nlinfit. We access the NV-charge-state +ratio from the fluorescence ratio after determining the +necessary correction factor κ520 [43]. A detailed descrip- +tion of κ520’s derivation is given in Appendix B. +Next, we assign the concentration ratio to a count ra- +tio in our SPCM detectors. We alter the setup accord- +ing to Fig. 2 (b). We illuminate the nanodiamond for +1 s with a given laser power and record the fluorescence +counts in both SPCMs. Using the data for each laser +power, we map the NV concentration ratio to a count +ratio in both SPCMs. At this point, we stress that we +do not obtain the NV concentration ratio through fluo- +rescence count ratios in SPCMs, but by analysis of the +NV centers’ fluorescence spectra. This method provides +the advantage that any influence of NV0 fluorescence +in SPCM2 (> 665 nm) can be neglected because only a +count ratio is considered in our analysis and a mapping +to previously-assigned concentration ratios performed. +Appendix B: Determination of κ520 +This section describes how we retrieve the correction +factor κ520 from our measurement data. We derive κ520 +similarly to as described in [43]. +We recorded fluorescence spectra of the single dia- +mond crystal with laser powers well below saturation +intensity with our setup shown in Fig. 2 (a). To achieve + +11 +these laser powers, an additional ND filter was used in +our laser-beam path. We correct the spectra for different +exposure times we set in our camera due to the differ- +ent NV luminescence intensities at different laser pow- +ers. We show the spectra we obtain for different laser +powers in Fig. B1. As can be seen, the overall fluores- +cence counts increase with increasing laser power. We +perform the spectra analysis as described in the main +text to derive the coefficients c− and c0. +Below saturation intensity, the luminescence of NV− +and NV0 should scale linearly with the laser power [43]. +However, due to charge conversion, we observe devia- +tions from this linearity. The coefficients c− and c0 we +obtain directly represent the amount of NV− and NV0 +fluorescence in the given spectra. We scale these factors +with the total integration value of the spectra in Fig. B1 +for each laser power and obtain measured fluorescence +counts for both NV charge states at each laser power. +Further, we take the fluorescence counts for NV− and +NV0 of the lowest-intensity spectrum recorded and scale +it with the laser power. This way, we obtain calculated +fluorescence counts for each NV charge state that strictly +increase linearly with the laser power. +These fluorescence counts for NV− and NV0, mea- +sured and calculated, are shown in Fig. B2 as a func- +tion of the laser power. We note that the measured NV− +fluorescence is lower than the calculated linear integra- +tion value, while the NV0 fluorescence is higher. We per- +form a weighted linear fit (inverse-variance weighting) +for each data set and compare the slopes to one another +for each NV charge state. We divide the two slope ratios +by each other and obtain κ520 = 2.03 ± 0.07, while we +derive the error from the statistical error of the fits we +performed. +550 +600 +650 +700 +750 +wavelength (nm) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +fluorescence counts +×103 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +laser power (%) +FIG. B1. +NV fluorescence spectra recorded at laser pow- +ers from 0.1 % to 1 %. +The laser power is kept well below +saturation intensity with a maximum laser power of ∼ 1 % +(∼ 25 kW cm−2). The spectra were corrected for different cam- +era exposure times used. An overall increase in the fluores- +cence counts is observed with increasing laser power. +(a) +(b) +FIG. B2. +Determination of κ520. (a) Fluorescence counts for +NV− as a function of the laser power. The red curve displays +the fluorescence counts obtained from scaling the counts at +the lowest laser power with the laser power. The blue curve +depicts the fluorescence counts for NV− as a function of the +laser power as we obtain it from the spectra. (b) Calculated +and measured fluorescence counts for NV0 as a function of the +laser power. Error bars are derived from the statistical errors +for c− and c0 and are smaller than the data points shown in +this graph. +Appendix C: Supporting relaxometry data +In Fig. C1, the ratios [NV−]/[NV0] are shown as a +function of τ, recorded with sequences P1 and P2. We +obtained the data as described in the main text. Fig. C2 +shows further supporting relaxometry data recorded +with sequences P1 and P2. + +12 +(a) +(b) +(c) +FIG. C1. +NV-charge-state ratios as a function of τ, obtained from relaxometry measurements. The ratios in R, C1, and C2 are +derived from the count-rate ratios of both SPCMs in the respective gates. (a) Sequence P1. The ratio [NV−]/[NV0] increases in the +readout gate R for all laser powers as a function of τ. For lower laser powers, the ratio [NV−]/[NV0] increases in the control gate +C1, while for the highest laser power, it is constant. (b) Sequence P2. The ratio [NV−]/[NV0] as a function of τ behaves similarly +as in sequence P1. However, the NV-charge-state ratios as a function of τ are different in C1 and C2, indicating charge-conversion +processes during the measurement. (c) Sequence P2. For better visibility, the ratios [NV−]/[NV0] for C1 and C2 as a function of +τ are displayed from panel (b). At laser powers up to 4 %, the ratio [NV−]/[NV0] is smaller for C2 than for C1 for τ ≲ 1 ms. For +values τ ≳ 1 ms, the opposite is the case. For visualization, τ = 1 ms is marked with a dashed line. At 11 % laser power, the ratios +[NV−]/[NV0] are equal in C1 and C2. + +13 +(a) +(b) +(c) +FIG. C2. Supporting results from relaxometry measurements. +(a) NV− fluorescence obtained from sequence P1 at 0.1 % laser +power in gate R. The data shown was not normalized by di- +vision by the fluorescence counts in gate C1. Fitting a mono- +exponential function to the data yields T1 = (0.94 ± 0.05) ms, +which deviates drastically from the T1 time obtained in the full +sequence P1 and in the case of normalization with C1. (b) NV− +fluorescence obtained from sequence P1 at 1 % laser power +by division of the fluorescence counts in R by the counts in +C1. Fitting a monoexponential function instead of the triexpo- +nential function yields T1 = (1.28 ± 0.06) ms, which does not +match the value determined for T1 in the full sequence P1. (c) +NV− spin polarization as obtained from the full sequence P2 +at 0.1 % laser power by subtracting the counts in Rπ from the +counts in R. Fitting a monoexponential function to the data +yields T1 = (1.45 ± 0.09) ms, which matches the previously +determined values for T1 in sequence P1. + +14 +[1] V. M. Acosta, E. Bauch, M. P. Ledbetter, C. Santori, K.- +M. C. Fu, P. E. Barclay, R. G. 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R. Zangara, and C. A. Mer- +iles, Charge Dynamics in near-Surface, Variable-Density +Ensembles of Nitrogen-Vacancy Centers in Diamond, +Nano Letters 18, 4046 (2018). + diff --git a/O9AzT4oBgHgl3EQfIfuL/content/tmp_files/load_file.txt b/O9AzT4oBgHgl3EQfIfuL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b5aa844728ca98f9009b42dc819ad2c0df8f361a --- /dev/null +++ b/O9AzT4oBgHgl3EQfIfuL/content/tmp_files/load_file.txt @@ -0,0 +1,1105 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf,len=1104 +page_content='Impact of Charge Conversion on NV-Center Relaxometry Isabel Cardoso Barbosa, Jonas Gutsche, and Artur Widera∗ Department of Physics and State Research Center OPTIMAS, University of Kaiserslautern-Landau, Erwin-Schroedinger-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 46, 67663 Kaiserslautern, Germany (Dated: January 4, 2023) Relaxometry schemes employing nitrogen-vacancy (NV) centers in diamonds are essential in bi- ology and physics to detect a reduction of the color centers’ characteristic spin relaxation (T1) time caused by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=', paramagnetic molecules in proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, while only the negatively-charged NV center is to be probed in these pulsed-laser measurements, an inevitable consequence of the laser excitation is the conversion to the neutrally-charged NV state, interfering with the result for the negatively-charged NV centers’ T1 time or even dominating the response signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In this work, we perform relaxometry measurements on an NV ensemble in nanodiamond combining a 520 nm excita- tion laser and microwave excitation while simultaneously recording the fluorescence signals of both charge states via independent beam paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Correlating the fluorescence intensity ratios to the fluo- rescence spectra at each laser power, we monitor the ratios of both charge states during the T1-time measurement and systematically disclose the excitation-power-dependent charge conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Even at laser intensities below saturation, we observe charge conversion, while at higher intensities, charge conversion outweighs spin relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' These results underline the necessity of low excitation power and fluorescence normalization before the relaxation time to accurately determine the T1 time and characterize paramagnetic species close to the sensing diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' INTRODUCTION The negatively-charged nitrogen-vacancy (NV) cen- ter in diamond constitutes a versatile tool for the detec- tion of magnetic [1–9] and electric [10] fields with high sensitivity and spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Measurement of the NV centers’ spin relaxation (T1) time is widely applied in different fields of science to detect magnetic noise [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Various so-called relaxometry measurement schemes employ a reduction of the NV centers’ T1 time with the host nanodiamond exposed to paramagnetic molecules fluctuating at the NV centers’ resonance fre- quency [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Thus, relaxometry schemes have been used to detect a superparamagnetic nanoparticle [16], or paramagnetic Gd3+ ions [15, 17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Further, relaxome- try with NV− centers has been utilized to trace chemical reactions involving radicals [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Also, the NV cen- ters’ T1 time as a measure for the presence of paramag- netic noise gains momentum in biological applications [7, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Individual ferritin proteins have been detected [23] and relaxometry has been applied to detect radicals even inside cells [24–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Especially in the field of biology, T1 measurement schemes are often conducted only with optical excita- tion of the NV− centers, while the readout of their spin states is realized by detection of the ensemble’s fluores- cence intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, recent results indicate that a second process impeding the NV− centers’ fluorescence signal is present in relaxometry measurements [20, 28– 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The laser pulse that is fundamental for preparation ∗ Author to whom correspondence should be addressed: widera@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='uni-kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='de of the NV− centers’ spin state can additionally ionize the NV− center to its neutrally-charged state, NV0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Conver- sion under illumination and back-conversion in the dark influence the NV− centers’ fluorescence signal, compli- cating a seemingly simple measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' A quantitative determination of the unwanted contribution of the NV0 state to the NV− relaxometry data is, however, elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In this work, we compare the results of two relaxometry schemes well-known in literature for the same nanodi- amond at varying laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Additionally, we intro- duce a novel method to extract the ratio of the two NV charge states from the NV centers’ fluorescence spectra throughout the entire measurement sequence to give an insight into the vivid NV charge dynamics we observe in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' A level scheme of the NV center in diamond is de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 1, including the negatively-charged NV− [32–34], the neutrally-charged NV0 and transitions from NV− to NV0 under green illumination [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We in- clude transitions independent of excitation power from the NV0’s ground state to NV−, reflecting the observa- tion of recharging processes in the dark in [28, 29] and in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Using a 520 nm laser, we non-resonantly excite the NV− centers from their triplet ground state 3A2 to the electronically-excited state 3E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Because 3E’s states mS = ±1 are preferentially depopulated via the NV− centers’ singlet states 1A1 and 1E, illumination with a green laser will spin polarize the NV− centers into their ground spin state mS = 0 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The T1 time describes how long this spin polarization persists until the spin popu- lation decays to a thermally mixed state [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' It can reach up to 6 ms in bulk diamonds at room temperature [37] and is influenced by paramagnetic centers within the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='01063v1 [quant-ph] 3 Jan 2023 2 host diamond or on its surface [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In the simplest T1 measurement scheme, spin polarization is achieved by a laser pulse, followed by a second readout-laser pulse after a variable relaxation time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Besides differ- ent durations, the two laser pulses are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' There- fore, the readout pulse is capable of spin-polarizing and ionizing the NV-center ensemble as well as the initial- ization pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Additionally, the spin-polarization pulse provides information about the charge-conversion pro- cesses during laser excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' To determine the T1 time of NV− centers of a spe- cific orientation in the diamond crystal, coherent spin manipulation is introduced in these measurements [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Here, a resonant microwave π pulse transfers the pop- ulation of these NV− centers from mS = 0 to mS = +1 or mS = −1 after the spin-polarization pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' A sec- ond laser pulse is used for the readout of the spin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Repetition of the sequence with the π pulse omitted and subtracting the readout signals from each other yields a spin-polarization signal as a function of τ that is robust against background fluorescence [38, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In the following, we present our experimental sys- tem in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Our results are divided into two main parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We first analyze fluorescence spectra of NV cen- ters in a single nanodiamond to assign concentration ra- tios to count ratios measured with SPCMs in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This knowledge allows us to quantify the NV0 contribu- tion during the spin-relaxation dynamics in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Level scheme of the NV center in diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Depicted are levels of the negatively-charged NV− and the neutrally- charged NV0 and transitions between the two charge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Gray arrows show transitions between NV−’s triplet and sin- glet states, mediated via intersystem crossing (ISC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Green ar- rows denote transitions driven by a green laser, red and or- ange arrows mark the fluorescence of the NV charge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Light-green dashed arrows between mS states are transitions driven by microwave radiation at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='87 GHz at zero magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Additionally, the light-green dashed arrows represent the relaxation of the spin-polarized state to a thermally mixed state without illumination (T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The purple dashed arrows de- note charge transfer processes in the dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' EXPERIMENTAL SYSTEM We perform our studies on a single nanodia- mond crystal of size 750 nm commercially avail- able from Adamas Nano as water suspension (NDNV/NVN700nm2mg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' As specified by the man- ufacturer, the nanodiamonds’ NV concentration is [NV] ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='5 ppm, which is about 2 × 104 NV centers per diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For sample preparation, the suspension is treated in an ultrasonic bath to prevent the formation of crystal agglomerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We spin-coat the nanodiamonds to a glass substrate and subsequently remove the sol- vent by evaporating the residual water on a hot contact plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' To probe the NV centers in a single nanodiamond, we use a microscope consisting of an optical excitation and detection section and a microwave setup, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' A CW-laser source of wavelength λ = 520 nm is used to optically excite the NV centers with a max- imum laser power of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='9 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The laser beam is fo- 50:50 NPBS NF 514 nm DM 550 nm laser 520 nm AOM objective sample permanent magnet MW antenna LP 550 nm ND1 ND2 SP 625 nm LP 665 nm to SPCM2 > 665 nm to SPCM1 < 600 nm to spectrometer spectrometer camera tube lens grating (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Experimental setup for recording NV fluorescence spectra and relaxometry data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In both setups, the excitation is the same, but the detection sections are different for the re- spective application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (a) NV centers in a single crystal nanodi- amond are excited by a 520 nm-laser in combination with an acousto-optic modulator (AOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The light stemming from the sample is filtered by a dichroic mirror (DM), a longpass filter (LP) and a notch filter (NF) with given wavelenghts and passes a non-polarizing beamsplitter (NPBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The remaining fluores- cence is spectrally resolved on a camera chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This setup is used for the measurement of the NV fluorescence spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (b) The NV fluorescence is split into two arms of a beamsplitter, additionally filtered with an LP or a shortpass filter (SP) and detected with fiber-coupled SPCMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The SP is tilted to only transmit fluorescence below 600 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' To keep the detectors below saturation, neutral-density (ND) filters are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Lu- minescence above 665 nm (NV− fluorescence) is detected in SPCM2, while light below 600 nm (NV0 fluorescence) is de- tected in SPCM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Transitions of the NV− centers’ spin states mS are driven with a microwave (MW) antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This setup is used for the measurement of the charge-state dependent relax- ometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 3 cused to a spot-size diameter of 700 nm (1/e2 diame- ter), reaching a maximum intensity of ∼ 2500 kW cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Pulses are generated by an AOM with an edge width of about 120 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Laser light is guided through an objective (NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='5, WD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 mm) and focused at the position of the nanodiamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Fluorescent light stemming from the sample is guided back through the objective and fil- tered by a dichroic mirror with a cut-on wavelength of 550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Next, the fluorescence light is filtered by an ad- ditional 550 nm-longpass filter and a 514 nm-notch filter to prevent detection of reflected laser light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The filtered fluorescence light is branched at a 50:50 non-polarizing beamsplitter, giving the possibility to further filter the luminescence and collect it in two separate detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In particular, our setup allows for tailoring the transmitted wavelengths to the spectral regions, where either pho- ton emission from the neutral or the negative NV charge state dominates in each beam path individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Thus, we can easily discriminate between the emission of both charge states in our measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In this work, we make use of different detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' While for spectral anal- ysis of the NV centers’ fluorescence, we use a spectrom- eter (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 2 (a)), we employ two single-photon counting modules (SPCMs) as detectors for our spin-relaxation measurements (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 2 (b)) in combination with a time- to-digital converter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Microwave signals are generated, amplified, and brought close to the nanodiamond using a microwave antenna structure written on a glass substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' All experiments are carried out under ambient conditions and in an external magnetic field in the order of 12 mT caused by a permanent magnet to split the NV centers’ ODMR resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In our ODMR spectrum, eight reso- nances appear because of the four existing orientations of NV centers in the single diamond crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We select one resonance to drive Rabi oscillations, from which we determine a π-pulse length of 170 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' FLUORESCENCE SPECTRA A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Setup To spectrally resolve the NV centers’ fluorescence, we use a spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The incoming fluorescence light is dispersed at a grating (600 grooves/mm), and an achro- matic tube lens translates the angle dispersion into a spatial dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Thus, the detection of light of dif- ferent wavelengths at different positions of a camera’s chip is facilitated, and spectra are obtained from 500 nm to 760 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' With this setup, we achieve a resolution of ∆λ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='19 nm/pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Each spectrum consists of a mean of at least 20 spectra recorded at each laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We correct the spectra for the wavelength-dependent prop- erties of optical elements in the beam path and subtract a background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Concentration ratio assignment Corrected fluorescence spectra of a monocrystalline nanodiamond for excitation laser powers from 1 % to 100 % are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Two features, the NV0s’ ZPL at ∼ 575 nm [41] and the NV−s’ ZPL at ∼ 639 nm [42] are clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The overlapping fluorescence spectra of both NV charge states show phonon broad- ening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Conform with the observation in [1], but con- trary to the results in [31], the NV0s’ ZPL intensity increases with higher laser power with respect to the NV−s’ ZPL in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' These results indicate a lower [NV−]/[NV0] ratio at higher laser powers and thus an increasing charge conversion for higher powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We obtain area-normalized extracted spectra for NV− and for NV0 from our recorded data as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We conduct the spectra decomposition anal- ysis of our spectra according to Alsid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' and follow the nomenclature given in reference [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The fraction of [NV−] of the total NV concentration [NVtotal] is defined (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' NV fluorescence spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (a) Spectra recorded at laser powers from 1 % to 100 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The NV0s’ ZPL at ∼ 575 nm and the NV−s’ ZPL at ∼ 639 nm are evident and marked in the spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For better visibility, spectra were normalized to the sum of the NV charge states’ ZPL intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (b) Area-normalized decomposed basis functions for NV0 and NV−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 4 by [NV−] [NVtotal] = [NV−] [NV−] + [NV0] = c− c− + κ520c0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (1) Thus, the concentration ratio between NV charge states [NV−]/[NV0] can be described with [NV−] [NV0] = c− c0 1 κ520 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (2) Here, c− and c0 describe the coefficients of the ba- sis functions of NV− and NV0 used to assemble an area-normalized composed spectrum at arbitrary laser power with the condition c− + c0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The correc- tion factor κ520 translates this fluorescence ratio c−/c0 to the ratio of NV concentrations [NV−]/[NV0], taking into account the different lifetimes and the absorption cross sections of the two NV charge states [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Note the different subscript in our work for the excitation (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (a) NV fractions as a function of the laser power we derived from spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (b) NV ratios as a function of the fluorescence count ratio in the two SPCMs applied as de- tectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Using the fit curve, we map the fluorescence count ratio to an NV ratio during relaxometry measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For fitting the [NV−]/[NV0] concentration ratio with f (x) = axn, we obtain a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='0135 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='0001 and n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='334 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The re- ciprocal ratio [NV0]/[NV−] was not fit separately, displayed is the function g(x) = a−1x−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' wavelength of 520 nm compared to κ532 in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Us- ing ten spectra recorded at laser powers below the sat- uration intensity and the deviations from the linearity of the charge states’ fluorescence intensity with the ap- plied laser power, we find κ520 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The error denotes the statistical error from a weighted fit we performed on our measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For a de- tailed description of the determination of κ520, see Ap- pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This value is within the reported value for κ532 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='5 for an excitation wavelength of 532 nm [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We use our value for κ520 to calculate the frac- tions of [NV−] and [NV0] and the concentration ratio [NV−]/[NV0] as a function of the laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 4 (a), the fraction of [NV−] is high for low laser powers and decreases with higher laser pow- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' At the lowest laser power of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 %, about 73 % of the total NV concentration is [NV−], while at the high- est laser power, only about 21 % [NV−] remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Already at laser powers of 2 % (∼ 51 kW cm−2), which is below saturation intensity (≈ 100 kW cm−2) [44], [NV0] out- weighs [NV−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Therefore, a significant influence due to charge conversion is to be considered in relaxometry measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Together with the recorded fluorescence-count-rate ratio of both SPCMs for each laser power, we assign each count-rate ratio ρSPCM2/ρSPCM1 a ratio [NV−]/[NV0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' With an increas- ing ratio of ρSPCM2/ρSPCM1, the ratio [NV−]/[NV0] in- creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We fit a power law (inverse-variance-weighted fit) to the ratio [NV−]/[NV0] to be able to trace the NV- concentration ratio over a broad range of count-rate ra- tios during the spin-relaxation measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Thereby we are able to quantitatively trace the contribution of NV0 during the spin-relaxation dynamics of the NV− centers in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' SPIN-RELAXATION MEASUREMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Setup and measurement sequences To separately detect the fluorescence of NV− and NV0 throughout our measurement, different filters are used in the optical beam path as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Af- ter passing a 50:50 non-polarizing beamsplitter, the sam- ple’s transmitted fluorescence light is guided through a 665 nm-longpass filter, and mainly NV− fluorescence is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For the luminescence reflected by the beam- splitter, we use a tilted 625 nm-shortpass filter to collect NV0 fluorescence below 600 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Neutral-density filters are added in front of the beamsplitter and within its transmitted beam path to keep the SPCMs below sat- uration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For determining the longitudinal spin relaxation time T1, we conduct and compare two different and fre- quently used pulsed-measurement schemes, which we 5 term P1 and P2 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' These two pulse se- quences are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In the pulsed sequence P1, we choose an initialization pulse of 200 µs duration to spin polarize the NV-center ensemble to their spin states mS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We apply a 5 µs normalization pulse 1 µs after the initialization pulse to probe the fluorescence intensity before a variable relax- ation time τ in gate Cπ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' To assure a depopulation of the NV− centers’ singlet states, we choose the time between the two pulses to be longer than the singlet lifetimes of τmeta ≈ 150 ns at room temperature [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='5 µs into τ, a resonant π pulse is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' After τ, a readout pulse of duration 5 µs probes the fluorescence of both NV centers’ charge states in gate Rπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Subsequently, the sequence is repeated with the π pulse omitted, ob- taining fluorescence intensities in gates C1 and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The spin polarization as a function of τ for NV− is obtained by subtracting the fluorescence counts in Rπ from the counts in gate R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Details on measurement sequence P1 can be found in [38, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Sequence P1 provides a tech- nique for determination of the NV− centers’ T1 time ro- bust against background fluorescence [38, 40] and is be- lieved to be unaffected by charge-state conversion [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Analysis of the second half of P1 represents an all- optical T1 measurement scheme as often applied in bi- ology [7, 24, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Further, using only the second half of this sequence, we are able to obtain the fluorescence evolution as a function of τ for NV− and NV0, includ- ing effects caused by the charge-state conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Only taking into account the signal without the π pulse ap- laser laser MW MW detection detection (a) Sequence P1 (b) Sequence P2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Longitudinal spin relaxation time (T1) measurement schemes applied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The beginning of the second half of each sequence is indicated by a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (a) Se- quence P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The NV ensemble is spin polarized by a laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Next, the fluorescence is detected by a control pulse (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Within the variable relaxation time τ, a resonant π pulse is applied (light-green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The spin state is read out by a third laser pulse (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The sequence is repeated with the π pulse omitted after a pause time tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (b) Sequence P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' As opposed to P1, the readout pulse has the same length as the spin-polarization pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The control gates C1 within the ini- tialization pulse and C2 within the readout pulse will be com- pared in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' plied, we obtain the fluorescence evolution by dividing the fluorescence counts in gate R by the counts in gate C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Charge conversion during the relaxometry measure- ment has an effect on the NV− fluorescence as well as on the NV0 fluorescence during the relaxation time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Therefore, by only evaluating P1’s second half, we gain information about the charge conversion taking place alongside the spin relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, to obtain the NV− centers’ T1 time, the full sequence P1 is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' As opposed to P1, P2 uses a normalization probe after the readout of the NV centers’ fluorescence [19, 21, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We choose the laser readout pulse to have the same du- ration as the initialization pulse (200 µs) and carry out the readout gates Rπ and R in the first 5 µs and the nor- malization probes Cπ 2 and C2 in the last 5 µs of the read- out pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Scheme P2 assumes the NV centers to have the same fluorescence intensity at the end of the second pulse as at the end of the first pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' To test this notion, we apply second normalization gates, Cπ 1 and C1, within the last 5 µs of the initialization pulse and compare the results for both normalized data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Between readout and the upcoming initialization pulse, we insert a pause time tp between the sequences of 1 ms, which is in the order of T1, to minimize build-up effects from spin polarization during the cycle for both sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Each cycle is repeated 50 000 times, and the whole measurement is swept multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The se- quences are repeated for different laser powers, ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 % to 11 % of the maximum laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Results and discussion In this section, we present and compare the experi- mental results for the spin-relaxation measurements for both sequences, P1 and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Using our experimental setup as described in Section IV A, we observe laser- power-dependent dynamics in the NV− and NV0 flu- orescence throughout our measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 6 depicts an example for the fluorescence as a function of τ for the NV0 fluorescence recorded at a laser power of 11 % with sequence P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' These results show the normalized fluorescence as a function of τ obtained from the second part of the measurement sequence without a microwave π pulse, dividing the fluorescence counts in gate R by the counts in gate C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The normalized fluorescence as a function of τ decays exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Different from the dynamics of the NV− center, we observe similar behav- ior for the NV0 fluorescence at all laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We fit a biexponential function of type f 0(τ) = A e−τ/TR,1 + B e−τ/TR,2 + d (3) to our measurement data and obtain two recharge times in the order of TR,1 = 100 µs and TR,2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='0 ms for all laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We assign these time constants TR to an electron-recapturing process of NV0 during the dark 6 time τ, after an ionization from NV− to NV0 has pre- viously taken place in the initializing laser pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Re- markably, this process occurs even at the lowest laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Presumably, the presence of two components of TR is due to the different environments of NV cen- ters concerning charge transfer sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Vacancies or elec- tronegative surface groups on the diamond surface are known to promote a charge conversion of NV− to NV0 [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We assume that the NV environment similarly affects the recharging process in the dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Therefore, we attribute one component of TR to NV centers closer to the nanodiamond surface and the other to NV centers more proximate to the center of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We empha- size that both TR,1 and TR,2 we report match previously reported values for TR of 100 µs [28] and (3 ± 1) ms [20] and underline that they simultaneously appear as two components in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We find that neither TR,1 nor TR,2 changes as a function of the laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The coefficients of the exponential functions A and B do not change significantly from 1 % to 11 % laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' How- ever, for the lowest laser power of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 % A and B are smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We attribute this to little NV0 fluorescence ob- served at this low laser power due to less charge conver- sion, resulting in a lower signal-to-noise ratio (SNR) for the NV0 fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Further, we present the results for the normalized NV− fluorescence as a function of τ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 7 (a) for ascending laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We conducted the experiment with sequence P1, and this data refers to the results with the π pulse omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The laser-power-dependent dy- namics of NV− and NV0 result in a drastic change of shape of the normalized fluorescence as a function of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' While we observe an exponential decay in the lowest laser power, we find an inverted exponential profile of the NV− fluorescence at 11 % laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In-between laser powers show both an exponential decay and an in- 0 2 4 6 8 10 τ (ms) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='9 normalized fluorescence R/C1 biexponential fit FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' NV0 fluorescence as a function of τ as recorded with sequence P1 (second half) by division of the fluorescence counts in R by the counts in C1 for 11 % laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' With a biexponential fit function, we find TR,1 = (109 ± 7) µs and TR,2 = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1) ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' crease, present in the fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This phenomenon of inverted exponential components in the recorded nor- malized fluorescence during a T1 measurement has been reported by [29] and attributed to a recharging process of NV0 to NV− during τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, a complete flip of the fluorescence alone by a laser power increase has not been reported so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Remarkably, this behavior indi- cates that NV0 to NV− charge dynamics outweigh the NV− ensemble’s spin relaxation at high laser powers in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' To better understand the NV− power-dependent be- havior, we use the results from the spectral analysis to map the ratios of [NV−]/[NV0] to our relaxometry mea- surement data of sequence P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The ratio [NV−]/[NV0] as a function of τ for all laser powers is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 8 (a) and displayed in more detail in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' C1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For all laser powers, even for the lowest, which lies well below saturation intensity, we observe an increase of [NV−]/[NV0] as a function of τ in the readout pulse R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We conclude that during τ a re-conversion from NV0 to NV− takes place in the dark, after ionization of NV− had occurred in the initialization pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' While for the lowest power, the ratio [NV−]/[NV0] increases from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='0 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='6 over the variation of τ, [NV0] outweighs [NV−] at 11 % laser power throughout the entire relaxation mea- surement, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' C1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 8 (a), the ra- tios increase by a factor of ∼ 2 from shortest to longest τ for all laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The ratios [NV−]/[NV0] we find in control pulse C1 as a function of τ also show a power-dependent be- havior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' While the ratio increases as a function of τ in the lowest power, it is constant in the control pulse for the highest power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' These power-dependent recharge processes in the control pulse we observe appear most likely due to build-up effects during the measurement cycle, as we explain in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' At low powers, the initializing laser pulse spin polarizes the NV− cen- ters but does not ionize to a steady state of NV− and NV0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For short τ, the re-conversion in the dark of NV0 to NV− has not completed, and the following laser pulse continues to ionize the NV− centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, at the highest power, each initialization pulse efficiently ion- izes to a steady state of the two NV charge states, reach- ing [NV−]/[NV0] ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' These results clearly show that the normalization in the sequence we perform is mandatory to only detect the change in the relative flu- orescence during the relaxation time τ and minimize in- fluences due to charges passed through cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' At the lowest laser power of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 %, we observe the highest ratio of [NV−]/[NV0] and therefore expect the most negligible influences of charge conversion on the NV−s’ spin relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Thus, we fit a monoexponential function to the relative fluorescence as a function of τ and obtain T1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='06) ms for the NV− ensemble in the nanodiamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' To further underline the necessity of a normalization of the fluorescence intensity, we fit a 7 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' NV− fluorescence as a function of τ, recorded with sequence P1 for different laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Insets show the characteristics of the sequences applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (a) NV− fluorescence as obtained from the second half of P1, dividing the fluorescence counts in R by the counts in C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We observe a transition from an exponential decay of the fluorescence to an inverted exponential profile with rising laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 % laser power, we perform a monoexponential fit and obtain T1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='06) ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For laser powers from 1 % to 11 %, we fit a sum of three exponential functions as explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (b) Spin polarization of the NV− ensemble as obtained from the full sequence P1, subtracting the fluorescence counts in Rπ from the counts in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Unlike (a), we observe an exponential decay at all laser powers in this measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, with increasing laser power, we observe a decrease in the amplitude of the exponential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Fitting a monoexponential function to the data at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 % laser power, we obtain T1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1) ms, consistent with the T1 time we find in (a) at the same laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (c) NV− fluorescence as obtained from the second half of P2, dividing the fluorescence counts in R by the counts in C1 or C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Fitting monoexponential functions to the fluorescence normalized by the counts in C1 or C2 yields T1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='06) ms or T1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='07) ms, respectively, consistent with the results named above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The NV− fluorescence qualitatively behaves as in sequence P1, see (a), and is fitted accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' On the contrary, the shape of the fluorescence depends on the position of the normalization gate, C1 or C2, especially visible at 4 % laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' monoexponential function to the non-normalized bare NV− fluorescence detected in R at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 % laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We obtain a T1 time of (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='05) ms, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' C2 (a), which is drastically lower than the T1 time retrieved with normalization by the fluorescence counts in C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For higher laser powers, we fit the normalized data with a function of type f −(τ) = −A e−τ/TR,1 − B e−τ/TR,2 + C e−τ/T1 + d (4) and restrict the time constants to TR,1 = 100 µs, TR,2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='0 ms and T1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='4 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' With this, we assume that the decay of [NV0] causes an increase of [NV−] and, there- fore, their fluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Thus, the NV− fluorescence is best described by a sum of an exponential decay due to the loss of spin polarization and a biexponential in- verted component due to the recharging process of NV0 to NV− in the dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 7 (a), our fit func- tion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (4) describes the measurement data from 1 % to 11 % laser power very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We emphasize that the mea- surement data for 1 % laser power does not visibly ap- pear to show this triexponential behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Fitting a mo- noexponential function to the NV− fluorescence at 1 % laser power, however, results in T1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='06) ms, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' C2 (b), which deviates significantly from the value obtained at lower laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Measurement sequence P1 is a well-established method to accurately measure the T1 time of the NV− 8 centers excited by a resonant π pulse [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Since the π pulse only acts on the negatively-charged NV cen- ters, it is said to be independent of charge conversion processes alongside the spin polarization [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We com- pare the results we obtain in the complete measurement sequence P1, subtracting fluorescence intensities in Rπ from the counts in R, to the result we gave for the T1 time above without the π pulse taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Remark- ably, although in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 7 (a) we observe vivid dynamics ranging from exponential decay to an inverted exponen- tial profile in the NV− fluorescence as a function of τ, the complete sequence P1 yields a monoexponential de- crease for all laser powers, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 7 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For the lowest laser power, we obtain T1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1) ms for sequence P1 comparing the fluorescence intensity with and with- out the resonant π pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This value matches the pre- viously determined T1 time when only considering the normalized signal without the π pulse for the lowest laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' It does not match the T1 time obtained from the monoexponential fit we performed on the measure- ment data for 1 % laser power, stressing the effects of (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Changes of the NV-charge-state ratio during the relax- ometry measurement from lowest to highest τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (a) Sequence P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The change of the ratio [NV−]/[NV0] is constant as a func- tion of the laser power in the readout pulse R, while it decays in the control pulse C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (b) Sequence P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The change of the ratio [NV−]/[NV0] in readout and control pulses show qual- itatively the same behavior as in sequence P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, the changes in the NV-charge-state ratio in C1 and C2 as a func- tion of the laser power differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' NV charge conversion within this measurement and the necessity for consideration of the two components TR,1 and TR,2 in a triexponential fit function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, the measurement sequence P1 is not en- tirely unaffected by the charge conversion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Al- though the resonant π pulse does not directly act on the NV0 center (we observe no difference in the signals with and without the π pulse), the fluorescence contrast in the measurement decreases because of NV0 to NV− conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This lower contrast becomes noticeable in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 7 (b) due to the decaying amplitude of the mono- exponential function with increasing laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The effect of NV− spin depolarization due to charge conver- sion has been previously investigated in [28, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' As a measure for the reliability of our measurement result, we use the area under the curves showing spin polar- ization as a function of τ for each laser power as a fluo- rescence contrast in the respective measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We di- vide this value by the Root mean squared error (RMSE) value we obtain from the fit result to account for fluc- tuations in our measurement data and define this value contrast/RMSE as the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 9, we show the SNR as a function of the laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In addition, we dis- play the value for T1 we obtain in the same graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' With the SNR decreasing, we observe a decrease in T1, accom- panied by a larger standard deviation with higher laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We conclude that the T1 time we measured at the lowest laser power is the most reliable one due to the highest SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In addition, we note that T1 seems to decay as a function of the laser power, although T1 should be independent of the excitation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We attribute this decay of T1 to the lower SNR in the measurements at higher laser power due to increased charge conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' From the results of sequence P1, we conclude that the normalization in the T1 measurement is essential to re- 0 5 10 laser power (%) 5 10 15 20 25 30 SNR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='6 T1 (ms) SNR T1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' SNR and T1 time as a function of the laser power, ob- tained from measurements performed in sequence P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' With higher laser power, the SNR decreases, and so does the T1 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The standard error of T1 that we obtain from mono- exponential fits increases with higher laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The data is extracted from equal numbers of repetitions of relaxometry measurements for each laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 9 flect the charge-state processes alongside the NV− en- semble’s spin relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Besides P1, sequence P2 is used in literature to determine a single NV center’s [46] or an ensemble’s [19] T1 time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' While the π pulse is often omitted in these sequences, we chose to implement it for low laser powers for better comparison to the results obtained in P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For laser powers starting from 3 %, we repeated the sequence without the π pulse and calcu- lated the mean values of the control and readout data taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The results for sequence P2 with a π pulse in- cluded for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 % laser power are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' C2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Using the data for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 % laser power and subtracting Rπ from R, we obtain T1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='09) ms, which is the same result as in sequence P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Since both sequences are used in the literature to measure an NV− ensemble’s T1 time, we expect them to produce the same result for our NV ensemble when neglecting additional effects due to charge conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' At this low laser power, charge conversion is inferior to spin relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Therefore, the T1 times we obtain from both sequences do not differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, with higher laser power, charge conversion prevails, and both NV charge states’ fluorescence sig- nals are greatly affected by recharge in the dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In order to evaluate the result of sequence P2 without the π pulse applied, we normalize the fluorescence in- tensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' To this end, we divide the counts in R obtained by the counts measured during the two control gates C1 or C2, yielding two normalized fluorescence signals for each NV charge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This way, we obtain two normal- ized fluorescence signals as a function of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' If no charge conversion effects were present in this measurement, both signals for the normalized fluorescence should be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, as pointed out, charge conversion is prominent in our sample, not only for high laser pow- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We show the NV− fluorescence as a function of τ we obtain from sequence P2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 7 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Qualitatively similar to sequence P1, we see a smooth transition from an exponential decay at low laser powers to an inverted exponential profile at high laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Similarly as in P1, we derive T1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='06) ms for normalization with C1 and T1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='07) ms for normalization with C2 for the lowest laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We emphasize that all T1 times we derive from the normalized NV− fluo- rescence in both sequences are equal within their stan- dard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In addition, the values for TR,1 and TR,2 we obtain from the NV0 fluorescence with sequence P2 are the same as in sequence P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We fit the NV− fluorescence for laser powers from 1 % to 11 % in the same manner as for P1 using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (4) and the aforementioned values for T1, TR,1 and TR,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This triexponential fit function models our data well, regardless of the normalization we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, the amplitudes of the respective exponen- tial functions differ depending on the normalization, C1 or C2, employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Thus, the shapes of the fluorescence as a function of τ differ with the gates used for normaliza- tion, which is especially visible at 4 % laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' To understand the difference in the measurement results that the positions of the normalization gate cause, we take the ratios [NV−]/[NV0] into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' C1 (b) and (c), [NV−]/[NV0] as a function of τ for sequence P2 is displayed and summarized as a change from shortest to longest τ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 8 (b) for each laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The ra- tios as a function of τ behave similarly to as observed with sequence P1 discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We note that the ra- tios we obtain in our measurement for the two control gates C1 and C2 are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For τ ≲ 1 ms the ratio [NV−]/[NV0] is smaller for C2 than for C1, while for val- ues τ ≳ 1 ms the opposite is the case, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' C1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For the same reasons discussed in sequence P1, this effect is prominent in laser powers up to 4 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' In contrast, for the highest laser power, the ratios in the control gates are approximately constant with τ and do not differ signifi- cantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' As pointed out in the discussion of P1, the results indicate that the first laser pulse does not ionize into a steady state of [NV−]/[NV0], and the second laser pulse continues to ionize NV− into NV0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Therefore, especially for small values of τ, the ratio [NV−]/[NV0] is smaller in C2 than in C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For larger values of τ, recharge dynamics of NV0 to NV− in the dark add to the different ratios of [NV−]/[NV0] for both control gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We do not exclude additional effects due to continued spin polarization of NV− in the second laser pulse, especially for low laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' It is for these reasons that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 8 (b) the changes of [NV−]/[NV0] as a function of the laser power are higher for C2 than for C1 for low powers and converge to the same value for higher laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We therefore at- tribute the difference in the normalized fluorescences in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 7 (c) when normalizing to C1 or C2 to the differences in [NV−]/[NV0] for C1 and C2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Both the results from measurement sequences P1 and P2 and the simultaneous mapping of [NV−]/[NV0] in- dicate that a charge conversion from NV− to NV0 dur- ing the spin-polarization pulse of a spin-relaxation mea- surement is inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We emphasize that a normal- ization gate is mandatory to correctly display the fluo- rescence dynamics of NV0 and NV− as a function of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Comparison of the two control gates C1 and C2 shows that the normalized fluorescence signal depends on the positions of the gate used for normalization because of charge conversion processes that take place alongside the NV− ensemble’s spin relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' CONCLUSIONS This work examines laser-power-dependent dynam- ics of NV charge conversion within spin-relaxation mea- surements of the negatively-charged NV centers in a sin- gle nanodiamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We present a new method of trac- ing the ratio of [NV−] to [NV0] during our sequence, in which we extract the relative concentrations of NV− to NV0 from their fluorescence spectra and perform a map- ping to fluorescence count ratios in two separate detec- 10 tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' From the analysis of low-excitation intensity spec- tra, we find κ520 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This correction factor κ520 allows us to translate the fluorescence ratio of NV− to NV0 to a concentration ratio, taking into account dif- ferent lifetimes and absorption cross sections for the two charge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Combining our results, we conclude that ionization of NV− to NV0 during the optical initializa- tion and readout is inevitable and occurs even at low laser powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' A recharge process in the dark of NV0 to NV− significantly affects the NV− ensemble’s fluores- cence during the spin-relaxation measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We find the recharging in the dark to be biexponential with com- ponents TR,1 = 100 µs and TR,2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='0 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' At high laser powers, the effect of charge conversion outweighs spin relaxation, making it impossible to accurately measure a T1 time, even with a scheme involving a π pulse for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Firstly, recharging effects of NV0 to NV− in the dark dominate the NV− fluorescence signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Secondly, the measurement of T1 is crucially impeded by a dimin- ished fluorescence contrast due to charge conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' To determine the NV− centers’ T1 time at low laser powers, we find it necessary to conduct a pulsed sequence with a normalization gate included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We prove the normal- ization mandatory to accurately reflect the charge-state dynamics as a function of τ and mitigate additional ef- fects due to charge-state accumulation during the mea- surement cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Additionally, comparing two pulsed se- quences often used in the literature, we find that the po- sition of the normalization gates plays an essential role due to charge conversion during the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We emphasize that including a normalization gate directly after the spin polarization before the relaxation time τ is a simple method to accurately display the fluorescence dynamics during the relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This way, com- paring the fluorescence counts in the readout gate to the counts in the control gate reliably reflects the spin relax- ation and the charge dynamics in the relaxometry mea- surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Overall, we emphasize that the results presented in this work impact relaxometry schemes widely used in biology, chemistry, and physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' To further extend this work, the effects of different duration of the spin- polarization pulse and the readout pulse can be exam- ined and give insight into the steady-state dynamics of the NV centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Further, the excitation of NV− can be conducted at longer wavelengths, changing the charge- state dynamics [49] and impacting the spin relaxation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The influence of different NV and nitrogen con- centrations in diamonds of different sizes on the charge dynamics can be considered to unravel the mechanisms of charge conversion in the dark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge support by the nano-structuring center NSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This project was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 454931666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Further, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' thanks the Studienstiftung des deutschen Volkes for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We thank Oliver Opaluch and Elke Neu-Ruffing for providing the microwave antenna in our experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Furthermore, we thank Sian Barbosa, Stefan Dix, and Dennis L¨onard for fruitful discussions and experimental support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Appendix A: Methods To understand the NV centers’ fluorescence evolu- tion as a function of τ in terms of charge conversion, we map the fluorescence count ratio detected in both SPCMs to a ratio of NV− and NV0 throughout the spin-relaxation measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For this, we combine the results of recorded NV spectra and spin-relaxation measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We choose a single nanodiamond and record fluorescence spectra at different laser powers us- ing the setup in the configuration shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Both charge states, NV− and NV0, contribute to the recorded spectra between 500 nm and 750 nm because of the charge states’ overlapping phononic sidebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For further analysis, we decompose the obtained spec- tra into NV− and NV0 basis functions as described by [43] using the spectra we recorded at the highest and lowest laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Employing our extracted basis func- tions, we obtain the fluorescence ratio of both NV charge states for all other laser powers with the help of MAT- LAB’s function nlinfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We access the NV-charge-state ratio from the fluorescence ratio after determining the necessary correction factor κ520 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' A detailed descrip- tion of κ520’s derivation is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Next, we assign the concentration ratio to a count ra- tio in our SPCM detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We alter the setup accord- ing to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We illuminate the nanodiamond for 1 s with a given laser power and record the fluorescence counts in both SPCMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Using the data for each laser power, we map the NV concentration ratio to a count ratio in both SPCMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' At this point, we stress that we do not obtain the NV concentration ratio through fluo- rescence count ratios in SPCMs, but by analysis of the NV centers’ fluorescence spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This method provides the advantage that any influence of NV0 fluorescence in SPCM2 (> 665 nm) can be neglected because only a count ratio is considered in our analysis and a mapping to previously-assigned concentration ratios performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Appendix B: Determination of κ520 This section describes how we retrieve the correction factor κ520 from our measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We derive κ520 similarly to as described in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We recorded fluorescence spectra of the single dia- mond crystal with laser powers well below saturation intensity with our setup shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' To achieve 11 these laser powers, an additional ND filter was used in our laser-beam path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We correct the spectra for different exposure times we set in our camera due to the differ- ent NV luminescence intensities at different laser pow- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We show the spectra we obtain for different laser powers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' As can be seen, the overall fluores- cence counts increase with increasing laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We perform the spectra analysis as described in the main text to derive the coefficients c− and c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Below saturation intensity, the luminescence of NV− and NV0 should scale linearly with the laser power [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, due to charge conversion, we observe devia- tions from this linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The coefficients c− and c0 we obtain directly represent the amount of NV− and NV0 fluorescence in the given spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We scale these factors with the total integration value of the spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' B1 for each laser power and obtain measured fluorescence counts for both NV charge states at each laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Further, we take the fluorescence counts for NV− and NV0 of the lowest-intensity spectrum recorded and scale it with the laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' This way, we obtain calculated fluorescence counts for each NV charge state that strictly increase linearly with the laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' These fluorescence counts for NV− and NV0, mea- sured and calculated, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' B2 as a func- tion of the laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We note that the measured NV− fluorescence is lower than the calculated linear integra- tion value, while the NV0 fluorescence is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We per- form a weighted linear fit (inverse-variance weighting) for each data set and compare the slopes to one another for each NV charge state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We divide the two slope ratios by each other and obtain κ520 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='07, while we derive the error from the statistical error of the fits we performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 550 600 650 700 750 wavelength (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='0 fluorescence counts ×103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='0 laser power (%) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' NV fluorescence spectra recorded at laser pow- ers from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 % to 1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The laser power is kept well below saturation intensity with a maximum laser power of ∼ 1 % (∼ 25 kW cm−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The spectra were corrected for different cam- era exposure times used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' An overall increase in the fluores- cence counts is observed with increasing laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Determination of κ520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (a) Fluorescence counts for NV− as a function of the laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The red curve displays the fluorescence counts obtained from scaling the counts at the lowest laser power with the laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The blue curve depicts the fluorescence counts for NV− as a function of the laser power as we obtain it from the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (b) Calculated and measured fluorescence counts for NV0 as a function of the laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Error bars are derived from the statistical errors for c− and c0 and are smaller than the data points shown in this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Appendix C: Supporting relaxometry data In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' C1, the ratios [NV−]/[NV0] are shown as a function of τ, recorded with sequences P1 and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' We obtained the data as described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' C2 shows further supporting relaxometry data recorded with sequences P1 and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 12 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' NV-charge-state ratios as a function of τ, obtained from relaxometry measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The ratios in R, C1, and C2 are derived from the count-rate ratios of both SPCMs in the respective gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (a) Sequence P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The ratio [NV−]/[NV0] increases in the readout gate R for all laser powers as a function of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For lower laser powers, the ratio [NV−]/[NV0] increases in the control gate C1, while for the highest laser power, it is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (b) Sequence P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The ratio [NV−]/[NV0] as a function of τ behaves similarly as in sequence P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' However, the NV-charge-state ratios as a function of τ are different in C1 and C2, indicating charge-conversion processes during the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (c) Sequence P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For better visibility, the ratios [NV−]/[NV0] for C1 and C2 as a function of τ are displayed from panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' At laser powers up to 4 %, the ratio [NV−]/[NV0] is smaller for C2 than for C1 for τ ≲ 1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For values τ ≳ 1 ms, the opposite is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' For visualization, τ = 1 ms is marked with a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' At 11 % laser power, the ratios [NV−]/[NV0] are equal in C1 and C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 13 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Supporting results from relaxometry measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (a) NV− fluorescence obtained from sequence P1 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 % laser power in gate R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' The data shown was not normalized by di- vision by the fluorescence counts in gate C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Fitting a mono- exponential function to the data yields T1 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='05) ms, which deviates drastically from the T1 time obtained in the full sequence P1 and in the case of normalization with C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (b) NV− fluorescence obtained from sequence P1 at 1 % laser power by division of the fluorescence counts in R by the counts in C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Fitting a monoexponential function instead of the triexpo- nential function yields T1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='06) ms, which does not match the value determined for T1 in the full sequence P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' (c) NV− spin polarization as obtained from the full sequence P2 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='1 % laser power by subtracting the counts in Rπ from the counts in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' Fitting a monoexponential function to the data yields T1 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content='09) ms, which matches the previously determined values for T1 in sequence P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' 14 [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AzT4oBgHgl3EQfIfuL/content/2301.01063v1.pdf'} +page_content=' M.' metadata={'source': 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b/ONAzT4oBgHgl3EQfWfxA/content/tmp_files/2301.01301v1.pdf.txt @@ -0,0 +1,1791 @@ +Strong correlation effects observed by an ANN-MFT encoder trained on α-RuCl3 high +magnetic field data +Michael J. Lawler,1, 2, 3, ∗ Kimberly A. Modic,4 and B. J. Ramshaw2 +1Dept. of Physics, Applied Physics, and Astronomy, Binghamton University, Binghamton, NY 13902 +2Dept. +of Physics, Cornell University, Ithaca, NY 14853 +3Dept. +of Physics, Harvard University, Cambridge, MA 02138 +4Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria +(Dated: January 5, 2023) +α-RuCl3 is a magnetic insulator exhibiting quantum spin liquid phases possibly found in the +Kitaev honeycomb model. +Much of the effort towards determining Hamiltonian parameters has +focused on low magnetic field ordered phases. We study this problem in the high magnetic field limit +where mean-field theory is better justified. We do so by machine-learning model parameters from +over 200,000 low dimensional data points that include magnetization, torque, and torsion data. Our +machine, an artificial neural network-mean-field theory (ANN-MFT) encoder, maps thermodynamic +conditions (temperature and field vector) to model parameters via a fully connected time-reversal +covariant (equivariant) neural network and then predicts observable values using mean-field theory. +To train the machine, we use PyTorch to enable backpropagation through mean-field theory with +a pure PyTorch implementation of the Newton-Raphson method. The results at 20 K and 34.5 +T are consistent with other parameter inference studies in the literature at low magnetic field but +strikingly have magnitudes that scale with temperature from 1.3 K up to 80 K in the 34.5 − 60 T +range . We conclude that the data presents physics beyond the scope of the mean-field theory and +that strong interactions dominate the physics of α-RuCl3 up to field strengths of at least 60 Tesla. +I. +INTRODUCTION +α-RuCl3 is a frustrated magnet with unexplained +properties. +Early experiments observed its insulating +character[1] and that quasi-one dimensional β-RuCl3 or- +ders antiferromagnetically at 600K while the stacked +honeycomb structure of α-RuCl3 antiferromagnetically +orders at 13 K[2]. +Experiments in 2015 then recog- +nized that the comparatively low antiferromagnetic tran- +sition temperature in α-RuCl3 results from frustration, +despite the large Curie-Weiss temperature of order 150 +K[3, 4] (although the precise size of this number is now +in debate[5]). In 2017, Neutron scattering then found a +continuum of spin excitations at high energy [6] with a +star-like pattern near the gamma point. Further stud- +ies reveal for Hab > 6T and/or Hc > 30T, the an- +tiferromagnetic order melts[7, 8], a continuum of exci- +tations emerges[9], a thermal Hall conductivity appears +quantized[10] between 6T ≲ Hab ≲ 8T, scaling emerges +in thermodynamic observables[11] over a wide range of +magnetic fields below 80 Kelvin, and an in-plane field +generates quantum oscillations[12]. How is it that one +material can behave in this way? +The simultaneous explanation of these properties is +difficult, despite the simplicity of the setting: a van der +Waals-coupled honeycomb lattice of spins[13] that can be +studied both in bulk—a kind of “magnetic graphite”— +and as isolated layers—a “magnetic graphene”. One rea- +son it is hard, is because frustration arises unexpectedly. +Since the honeycomb lattice is bipartite, a nearest neigh- +∗ mlawler@binghamton.edu +bor Heisenberg model predicts N´eel ordering and no frus- +tration is expected. +It is therefore the spin-orbit cou- +pling that generates frustration. But this frustration is +somehow not only classical, as is understood in Heisen- +berg magnets in the large-S limit , but also quantum +mechanical—computational studies find small terms nor- +mally neglected in spin models dramatically change the +phase diagram[14] and are necessary to stabilize the ob- +served antiferromagnetic order. Another reason an ex- +planation is hard is the initial indications that α-RuCl3 +is simply described by the Kitaev model has not sur- +vived closer scrutiny. Symmetry allows for other terms +in the Hamiltonian placing the problem into a more gen- +eral spin-orbit coupled model space[15, 16]. Finally, the +experiments themselves do not seem to agree with each +other. +Quantum oscillations from an in-plane field[12] +clearly evident below 3 Kelvin suggest a quantum spin +liquid phase with complex-Fermionic excitations form- +ing a three-dimensional Fermi surface while a half-integer +quantized thermal hall effect observed between 3 and 5 +Kelvin suggest Majorana fermion edge modes forming a +topological phase with gapped fermionic excitations in +the bulk[10]. +One strategy to resolve these issues is to infer the +Hamiltonian model parameters to enable computational +methods to aid in the interpretation of the data. Begin- +ning with the Kitaev model to capture high energy fea- +tures in neutron scattering, it was argued that “modest +amounts of additional neighbor correlation or simple per- +turbations based on mean-field approaches” reproduce +the star shaped signal near the Gamma point[6]. These +fits, either with an antiferromagnetic Kitaev term[6] or +with ferromagnetic Kitaev coupling, via a parton the- +ory, not not quantitatively reproduce the star pattern[17]. +arXiv:2301.01301v1 [cond-mat.str-el] 3 Jan 2023 + +2 +But the sign of the Kitaev term is intimately connected to +the direction of the ordering moments[18]. Though hard +to determine with neutron scattering[19], resonant elastic +x-ray scattering (REXS) unambiguously show the Kitaev +term is ferromagnetic, a conclusion consistent with fits to +many other experiments[4]. Taken together, these results +show the parameter inference problem itself is hard, per- +haps only resolved in this region of the phase diagram +by a quantum dynamics simulation capable of uniting +inelastic and elastic neutron scattering. +This confusion motivates a second strategy, inferring +parameters at large magnetic field. Such a strategy was +successful in a quantum spin ice material[20]]. Further- +more, there is published high field magnetization[21], +torque[22], and torsion[11] data, and even magnetization +data in pulsed fields up to 100 T[23] data on α-RuCl3. +Qualitative fits to these datasets[23–25] are consistent +with ferromagnetic Kitaev term. But these fits did not +explore the entire parameter space, instead choosing pa- +rameters similar to other studies and focused on employ- +ing powerful computational methods to compare theory +to experiments. So they did not quantify the uncertainty +in their fits or take advantage of the simplicity of the high +field limit. +In this paper, we combine mean-field theory and arti- +ficial neural networks (MFT-ANN) to solve the param- +eter inference problem at high field. +By using an en- +coding scheme , we combine magnetization, torque, and +torsion data sets into one data set with over 200,000 +low-dimensional data points . +The ANN constructs a +smooth map from the thermodynamic state (T, ⃗H) and +data category C of a data point to the Hamiltonian model +parameters from which a MFT estimates the experi- +mental observable corresponding to C. A trained ANN +then shows a variation of these parameters over the data +set, a variation that may have physical implications as +well as providing a degree of uncertainty quantification. +There is no guarantee that training the ANN multiple +times will produce the same state-to-parameter map. In- +stead, the interpretation of the result is reliable when +the MFT is able to fit the data well, even if two qualita- +tively different maps produce equally good fits. Namely, +like MFT can have multiple saddle point solutions, data +fitting via an MFT-ANN also has this property. +Our +trained MFT-ANN shows the scaling behavior of ther- +modynamic observables—previously observed in the in- +termediate field regime[11]—is reflected in the ANN, with +the Kitaev and Gamma couplings scaling with tempera- +ture. Hence, this scaling is a strong correlation effect not +captured by MFT and the mapping shows it persists up +to 60 T. +II. +AN ANN-MFT ENCODER FOR LEARNING +MODEL PARAMETERS +A theory/model of condensed matter physics can be +thought of as a decoder. 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Combining a physics model with a neural network +for parameter inference. a a flow chart describing the ANN- +MFT encoder. Here the thermodynamic state of a condensed +matter system, taken to be magnetic field and temperature +in this paper, together with an observable category label C is +sent through an ANN that encodes this data into parameters +of a physics model. +A backpropagatable mean-field theory +then computes the observable OC for that model. +b The +honeycomb-Kitaev model geometry that forms the foundation +for a study of α-RuCl3. These models live on the honeycomb +lattice involving two types of sites, A sites denoted by an +open circle, and B sites denoted by a filled circle, and three +types of bonds denoted by orange (x-bond), purple (y-bond), +and turquoise (z-bond). At each site lives a spin S that is +anisotropically coupled via g-factors g to a magnetic field H, +and on each bond lives an exchange coupling matrix J that is +different on each of the three types of bonds. +parameters to a physical system. There are many ma- +chine learning models that could be useful for science if +we can think of a physics theory as a layer in a machine +learning model that allows us to backpropagate through +it. Specifically, we need to view the MFT as computing +an observable O that is a function of the model parame- +ters O(θ) and be able to take derivatives ∂O/∂θn. This +will allow the training of a neural network that predicts +the model parameters θ(W, b) given its weights W and +biasesb . Hence, a backpropagatable MFT layer would +solve the problem of adding domain knowledge to a data +science algorithm. +In this paper, we will show how to achieve backpropa- +gation through a mean-field theory(MFT) layer and use + +3 +it together with an artificial neural network (ANN) en- +coding layer forming an ANN-MFT encoder as shown in +Fig. 1, to learn the parameters of RuCl3. +In a sufficiently large magnetic field, spins collectively +precess about the magnetic field direction and a mean- +field Si → m develops. Lowering the magnetic field, an- +tiferromagnetic spin exchange interactions begin to fight +this tendancy, and a staggered Ne´el component to the +mean-field m → mi could develop, producing a differ- +ent mean field on the A and B sublattices. +Lowering +further, the ordering could become even more complex +and/or novel states not captured by mean field theory +could arise. +With this in mind, we begin with the K-Γ model, be- +lieved to capture the physics of RuCl3, defined as +HK−Γ = 1 +2 +� +⟨ij⟩ +Jiα,jβSα +i Sβ +j −µB ⃗H·g· +� +i +Si +(1) += +� +⟨ij⟩ +� +KSγ +i Sγ +j +Γ(Sα +i Sβ +j +Sβ +i Sα +j ) +� +(2) +−µBga ⃗H⊥ · +� +i +S⊥ +i − µBgcHz +� +i +Sz +i +where spin operators are dimensionless with S += +1 +2(σx, σy, σz), σα the Pauli matrices, and in the second +line, we choose the convenient “theory basis” to express +the couplings K and Γ where α, β, γ take on the values +(x, y, z) if ⟨ij⟩ is a “z”–bond, (z, x, y) if ⟨ij⟩ is a “y”– +bond, and, (y, z, x) if ⟨ij⟩ is a “x”–bond, as described +in Fig. +1, and we choose the convenient “experimen- +tal basis” (a, b, c) to express the g-factors that relate the +coupling of the spins to the external magneteic field ⃗H +in terms of the unit cell geometry, setting ga = gb due to +assumed rotational invariance in the ab-plane . Since we +are interested in thermodynamic data, we then construct +the partition function that is the generating function of +all thermodynamic observables: +Z = Tr exp +� +− 1 +2 +� +ijαβ +jiα,jβSα +i Sβ +j + h · +� +i +Si +� +(3) +where jiα,jβ = Jiα,jβ/kBT and h = µBH · g/kBT. We +write the model parameters in this unusual way because +now they are easier to work with in a machine learning +context as they are dimensionless parameters of order 1 . +In this way, we have set up a model that given K/kBT, +Γ/kBT, ga, and gc, we can predict any thermodynamic +observable provided sufficient computational resources. +The exact evaluation of experimental observables, such +as the magnetization Mα = −kBT +∂ +∂Hα ln Z, is hard for +systems with more than 16 spins. So we next make the +physically-reasonable mean-field approximation, reason- +able as discussed above when there is a large magnetic +field. It makes the replacement +Sα +i Sβ +j = mα +i Sβ +j + Sα +i mβ +j − mα +i mβ +j +(4) +where mi are the mean fields, the mean value of the spin +operators Si on a given site. We have found stopping +the complexity of the mean-field theory at the staggered +component, i.e. defining just two vectors mA and mB +placed in a periodic-in-the-unit-cell arrangement, with +sites i ∈ A belonging to the A sublattice and i ∈ B +belonging to the B sublattice, as shown in Fig. 1, is suf- +ficient to produce stable solutions to the mean-field equa- +tions for typical values of the model parameters. These +equations at a finite temperature T are +mA = +1 +|A| +� +i∈A +⟨Si⟩ = 1 +2 +heff +A +|heff +A | +Tanh(|heff +A |/2) +(5) +and similarly for mB, where heff +A = h − 3¯ȷ · mB is the +dimensionless effective field felt by spins on the A sub- +lattice due to a combination of the external magnetic +field and the couplings to the three surrounding spins on +the B sublattice. Here ¯ȷ = diag(jxx, jxx, jzz) is the av- +erage coupling over the three bonds in the unit cell with +jxx = (K − Γ)/3kBT and jzz = (K + 2Γ)/3kBT. Solv- +ing these mean-field equations then fixes the values of +mA and mB and enables us to compute any observable +within the mean-field approximation. +To solve the parameter inference problem using an +ANN-MFT encoder, we need to solve the mean field +equations in a way that allows us to backpropagate from +a calculated observable like the magnetization, through +the model parameters to the weights and biases of the +ANN. Defining the ANN as a trainable map f(x; W, b) +from a point x ∈ X in a thermodynamic data set X with +x = (H, T) to the model parameters θ = (ga, gc, K, Γ), +with weights and biases W, b, and the magnetization +computed within the mean-field theory M = M(θ, x), +we see by the chain rule +∂M +∂Wa += ∂M +∂θm +∂θm +∂Wa += ∂M +∂θm +∂fm +∂Wa +, +(6) +where we used θm = fm(x; W, b), that the key challenge +is to be able to compute quantities like ∂M +∂K and ∂M +∂ga . +The solution we take to accomplish this is numeric +differentiation, which is used via automatic differentia- +tion in modern machine learning packages to compute +derivatives like +∂fm +∂Wa when training the neural network +in stochastic gradient descent. As such, we implemented +this solution in pytorch with a native implementation of +the Newton-Raphston algorithm as shown in Listing 1. +The key to implementing such a method is to use na- +tive pytorch tensors with requires_grad enabled. For +this to work consistently, it is helpful to use the deco- +rator @torch.enable_grad() above functions that call +newtons method. Hence, using a native pytorch implemen- +tation of the numerical solution to mean-field equations, +one can directly use physics models in this approximation +as layers within a larger machine learning model connect- +ing experimental data to theoretical predictions. +In addition to backpropagation, a key benefit of using +pytorch to implement the solution to the mean field equa- +tions is the ability to solve them for different model pa- +rameters in parallel on a gpu. We solve them in batches + +4 +1def newtons method ( function , +i n i t i a l , +2 +i t e r a t i o n s=100 , +t o l=torch . f i n f o () . eps ) : +3 +i f +not( i n i t i a l . r e q u i r e s g r a d) : +4 +i n i t i a l . r e q u i r e s g r a d = True +5 +for +i +in range( i t e r a t i o n s ) : +6 +previous = i n i t i a l . clone () +7 +value = function ( i n i t i a l ) +8 +value . backward ( torch . o n e s l i k e ( value ) , +9 +retain graph=True) +10 +with torch . no grad () : +11 +i n i t i a l −= ( value / +i n i t i a l . grad ) +12 +i n i t i a l . grad . z e r o +() +13 +14 +i f +( i n i t i a l − previous ) . abs () .max( ) < \ +15 +torch . tensor ( t o l ) : +16 +return +i n i t i a l +17 +return +i n i t i a l +Listing +1. +Native +pytorch +code +for +Newton-Raphson +method (also known as Newton’s method) adapted from a +stackoverflow.com question[26]. +1 +A = Id + 3.0∗ torch .bmm( chi0 , j ) +2 +u n s t a b l e c h i = torch . zeros (A. shape [ 0 ] ) +3 +chi = torch . z e r o s l i k e( chi0 ) +4 +for +i +in range( len (A) ) : +5 +try : +6 +chi [ i ]=torch . l i n a l g . s o l v e (A[ i ] , chi0 [ i ] ) +7 +except RuntimeError : +8 +print ( ” Singular A matrix +found ! ” ) +9 +chi [ i ] = 1.0E6∗ Id [ i ] +10 +try : +11 +t e s t = torch . l i n a l g . cholesky ( chi [ i ] ) +12 +except RuntimeError : +13 +u n s t a b l e c h i [ i , 0 ] = 1.0 +Listing 2. Code that calculates the susceptibility efficiently. +of 1000, a number we found to be efficient while pre- +serving the stochastic property of the gradient descent +used to train the ANN. We have been able to solve them +in parallel with 100,000 model parameters on a desktop +with an Nvidia Titan X GPU, a features that might be +valuable in other applications. +In the traditional setting, mean-field equations are +solved for a particular choice of model parameters and so +the mean-fields themselves are chosen to be those which +produce stable solutions to these equations. In the set- +ting of this paper, however, we need stable mean-field +solutions for all model parameter possibilities, which is +not easily achieved. We overcome this challenge by com- +puting the susceptiblity χ at the mean-field solution and +using this calculation to check if a stable solution was +obtained, flagging those that are found to be unstable. +Naively, this is done by computing (see Appendix A 3 for +derivation) +χ = (I + 3χ0 · j)−1χ0 +(7) +There are two ways this calculation can fail. The first is if +the matrix A = I+3χ0 ·j is itself singular. This happens +at a phase transition where χ → ∞ and is not a sign +that an unstable solution was obtained. The second is if +Torque +Magnetization +ACHXicbZC7TsMwFIadcivhF +mBksaiomKoEVcBYqUvHIvUmNVXkuE5r6lxkO6ht1Bdh4VYGECI +gQXxNjhpBmg5kn0+/ec2ed3I0aFNM1vrbCxubW9U9zV9/YPDo ++M45OCGOSRuHLOQ9FwnCaEDakpGehEnyHcZ6bqTelrvPhAua +Bi05CwiAx+NAupRjKSHKNaHjnItnWVcJrunWQ6XeQ0n2eU2K4H +Gxm20qvuGCWzYmYB18HKoQTyaDrGpz0MceyTQGKGhOhbZiQHCeK +SYkYWuh0LEiE8QSPSVxgn4hBkm23gBdKGUIv5OoEmbq74kE+U +LMfFd1+kiOxWotFf+r9WPp3Q4SGkSxJAFePuTFDMoQplbBIeUES +zZTgDCn6q8QjxFHWCpDdWCtbryOnSuKtZ1pXpXLdXKuR1FcAb +OwSWwA2ogQZogjbA4BE8g1fwpj1pL9q79rFsLWj5zCn4E9rXD5 +fQnwc=ga +gc +jxx +jzz +H +T +C +MFT +Mean +Field +Susceptibility +Torsion +AB+nicbVBNS8NAEJ3Ur1q/Uj16CRaLp5JIU9S6MWbFewHNCFstpt26WYTdjdKif0pXjwo4tVf4s1/47bNQVsfDzem2FmXpAwK +pVtfxuFtfWNza3idmlnd2/wCwfdmScCkzaOGax6AVIEkY5aSuqGOklgqAoYKQbjJszv/tAhKQxv1eThHgRGnIaUoyUlnyzXL31m65 +bqrp4RP3s2p76ZsWu2XNYq8TJSQVytHzyx3EOI0IV5ghKfuOnSgvQ0JRzMi05KaSJAiP0ZD0NeUoItL5qdPrVOtDKwFrq4subq +74kMRVJOokB3RkiN5LI3E/z+qkKr7yM8iRVhOPFojBloqtWQ7WgAqCFZtogrCg+lYLj5BAWOm0SjoEZ/nlVdI5rzkXtfpdvdKo5n +EU4RhO4AwcuIQG3EAL2oDhEZ7hFd6MJ+PFeDc+Fq0FI585gj8wPn8ALTaSlg=OC +�>0 +Merge by Category +FIG. 2. Mean field theory implemented for machine learn- +ing purposes in two layers. +The first solves the mean field +equations to determine the values of the mean fields following +Listing 1. The second computes the observables for the re- +quested category, either magnetization, torsion, or torque in +this study, and a flag χ>0 that is 0 if the mean field solution +is unstable and 1 if it is stable. +the matrix χ is not positive. Such a situation implies the +mean-field solution is thermodynamically unstable. We +have found the code presented in Listing 2 was able to +perform these checks efficiently. It runs in series on each +solution of the mean-field equations previously computed +in parallel and achieves efficient checking by a) using torch +. linalg . solve to solve the system of equations A · χ = χ0 +for χ and use torch. linalg .cholesky to check via failure of +this algorithm for positive-definitness of χ. +Combining the above ideas we can construct the MFT +layer as a combination of two layers as shown in Fig. 2. +One layer solves the mean field equations using Listing 1, +and the other computes the observables for the category +of the data and the flag χ>0 denoting whether or not the +MFT equations were stable. +With the MFT upgraded to perform as a layer in a ma- +chine learning algorithm, it remains to specify the ANN. +In the context of high magnetic field data, it turns out +we can preserve the rotational symmetry about the c axis +and the time-reversal symmetry of the data in the archi- +tecture of the ANN , a symmetry covariant/equivariant +neural network. +To build in the rotational invariance +about the c-axis we choose the magnetic field H to always +lie in the ac-plane. To build in time-reversal symmetry +so that sending in (H, T, C) to the ANN will produce the +same model parameters (ga, gc, jxx, jzz) in the output as +sending in (−H, T, C), we need to design the neurons +more carefully. +We achieve time-reversal symmetry by separating the +neurons into two groups, those that act on time-reversal +odd variables like H and those that act on time-reversal +even variables like T. Then it is important to preserve +this property throughout the network, including when +standardizing the data. In general, a neuron acts on data +via σ(w · x + b) where w is a weight matrix, x a data + +5 +Concatenate +Concatenate +A +B+HicbVBNS8N +AEJ3Ur1o/GvX +oZbEonkoiRT0 +WeumxQr+gCW +z3bRLN5uwuxFq +6C/x4kERr/4U +b/4bt20O2vpg +4PHeDPzgoQz +pR3n2ypsbe/s +7hX3SweHR8dl ++S0q+JUEtoh +MY9lP8CKciZo +RzPNaT+RFEcB +p71g2lj4vUcqF +YtFW8S6kd4L +FjICNZGtrlz +AtC1Jx7Xqltq +jG0K07VWQJtE +jcnFcjRGtpf3 +igmaUSFJhwrN +XCdRPsZlpoRT +uclL1U0wWSKx +3RgqMARVX62PH +yOLo0yQmEsTQ +mNlurviQxHSs +2iwHRGWE/Uur +cQ/MGqQ7v/Y +yJNVUkNWiMO +VIx2iRAhoxSY +nmM0Mwkczcis +gES0y0yapkQn +DX94k3Zuqe1u +tPdQq9as8jiK +cwVcgwt3UIc +mtKADBFJ4hld +4s56sF+vd+li +1Fqx85gz+wPr +8Ac1Vkc0=H +T +C +ANN +ACHXicbZC7TsMwFIadcivhFmBksaiomKoEVcBYqUvHIvUmNVXkuE5r6lxkO6ht1Bdh4VYGECIgQXxNjhpBmg5kn0+/e +c2ed3I0aFNM1vrbCxubW9U9zV9/YPDo+M45OCGOSRuHLOQ9FwnCaEDakpGehEnyHcZ6bqTelrvPhAuaBi05CwiAx+NAup +RjKSHKNaHjnItnWVcJrunWQ6XeQ0n2eU2K4HGxm20qvuGCWzYmYB18HKoQTyaDrGpz0MceyTQGKGhOhbZiQHCeKSYkYWuh0L +EiE8QSPSVxgn4hBkm23gBdKGUIv5OoEmbq74kE+ULMfFd1+kiOxWotFf+r9WPp3Q4SGkSxJAFePuTFDMoQplbBIeUESzZT +gDCn6q8QjxFHWCpDdWCtbryOnSuKtZ1pXpXLdXKuR1FcAbOwSWwA2ogQZogjbA4BE8g1fwpj1pL9q79rFsLWj5zCn4E9rXD +5fQnwc=ga +gc +jxx +jzz +H +T +C +Standardization +Seagull(Linear(2,N)) +LeakyReLU(Linear(2,N)) +Softplus(Linear(N,2)) +Linear(N,2) +LeakyReLU(Linear(2N,2N)) +LeakyReLU(Linear(2N,2N)) +LeakyReLU(Linear(2N,2N)) +AB7Hicb +VBNS8NAEJ3Ur1q/qh69LBbFU +0mkqMeClx4rmLbQhrLZbtqlm +03YnQil9Dd48aCIV3+QN/+N2z +YHbX0w8Hhvhpl5YSqFQdf9dg +obm1vbO8Xd0t7+weFR+fikZ +JM+6zRCa6E1LDpVDcR4GSd1 +LNaRxK3g7H93O/cS1EYl6xEn +Kg5gOlYgEo2glv5eKfqNfrh +VdwGyTrycVCBHs1/+6g0SlsV +cIZPUmK7nphMqUbBJ+Vepn +hKWVjOuRdSxWNuQmi2Nn5MI +qAxIl2pZCslB/T0xpbMwkDm1n +THFkVr25+J/XzTC6C6ZCpRly +xZaLokwSTMj8czIQmjOUE0so +08LeStiIasrQ5lOyIXirL6+T +1nXVu6nWHmqV+mUeRxHO4Byuw +INbqEMDmuADAwHP8ApvjnJen +HfnY9lacPKZU/gD5/MHlAWOd +Q=⇡H +AB73icb +VDLSgNBEOyNrxhfUY9eBoPiK +exKUI+BXDxGyAuSJcxOJsmQ2 +dl1plcIS37CiwdFvPo73vwbJ8 +keNLGgoajqprsriKUw6LrfTm +5jc2t7J79b2Ns/ODwqHp+0TJ +RoxpskpHuBNRwKRvokDJO7 +HmNAwkbweT2txvP3FtRKQaOI2 +5H9KREkPBKFqp04tFP23UZv1 +iyS27C5B14mWkBnq/eJXbxC +xJOQKmaTGdD03Rj+lGgWTfFb +oJYbHlE3oiHctVTkxk8X987 +IhVUGZBhpWwrJQv09kdLQmGkY +2M6Q4tisenPxP6+b4PDOT4WK +E+SKLRcNE0kwIvPnyUBozlBO +LaFMC3srYWOqKUMbUcG4K2+ +vE5a12Xvplx5qJSql1kceTiDc +7gCD26hCvdQhyYwkPAMr/DmP +DovzrvzsWzNOdnMKfyB8/kD+ +aGP2g=⇡T C +FIG. 3. The archecture of the ANN. We choose to build the fully connected neural network that is time-reversal symmetric. +We do so by setting the bias to zero and using the symmetric seagull activation function in the first layer for magnetic field +inputs. Otherwise, the rest of the network is an ordinary fully connected neural network, here using the common LeakyReLU +function with a slope 0.1 for negative values of its input. We also pass the output for parameters ga and gc to the softplus +function log(1 + ex), a smooth version of ReLU which guarantees they are positive. +vector, b a bias vector, and σ(...) is a non-linear function +that separately acts on each of the components of the +vector in its argument. For a time reversal odd variable, +we need to demand σ(w · (−x) + b) = −σ(w · x + b). +This requires b = 0, and σ(−x) = −σ(x). For the latter, +one choice is σ = tanh, the hyperbolc tangent function, +but we won’t need this in our network. For time-reversal +even variables, there are no restrictions on σ, w, and b. +But since our network outputs only time-reversal even +variables, the parameters of the model, we need a neuron +that accepts a time-reversal odd variable but outputs a +time-reversal even variable. To achieve this, we use the +seagull function log(1 + x2), previously used in Ref. 27. +Putting these ideas together, our ANN is presented in 3. +It immediately converts the time-reversal odd magnetic +field variables into time-reversal even ones via the seagull +function and then successive layers are normal with a +leakyReLU activation function. We present our ANN in +Fig. 3. A central benefit of this network is not efficiency, +but interpretation for due to the preservation of time- +reversal symmetry, angular sweeps of the magnetic field +will be strictly periodic in θ → θ +π where θ is the angle +of the magnetic field. +III. +RESULTS: SCALING AND A STRONG +INTERACTION REGIME +The MFTANN encoder presented in the previous sec- +tion was applied to a dataset of 213803 thermodynamic +data points taken on α-RuCl3 built from a combination +of publically available magnetization data at high mag- +netic fields taken from Ref. 21, high magnetic field torque +data taken from Ref. 22, and high magnetic field torsion +data taken from Ref. 11. We have combined all three +thermodynamic data sets into one set by attaching a cat- +egory label to each measurement observable. This label +is converted to a vector via one-hot encoding. Specifi- +cally, the in-plane magnetization category was mapped +to the vector (1, 0, 0, 0, 0), the c-axis magnetization to +(0, 1, 0, 0, 0), the torque category to (0, 0, 1, 0, 0), the tor- +sion field sweep data to (0, 0, 0, 1, 0) and the torsion an- +gular sweep data to (0, 0, 0, 0, 1). In this way, the ANN +can think about these different data types differently due +to different weights and biases associated with each in +the initial layers. We have made our data set available +at zenodo[28]. +In addition, we design the MFT layers +following 2 to compute the experimental observables as +shown in appendix A and combine multiple data types +and thereby insert domain knowledge into the machine +learning algorithm. +To train the ANN-MFT, we need a loss function that + +6 +0 +50 +100 +150 +200 +Epoch +1 +0.1 +0.01 +Log(Mean Squared Error) +Training Loss +Validation Loss +FIG. 4. Loss during training. Training loss (blue line) falls +slightly below validation loss (orange line) over 200 epochs +with a dropout fraction of 0.25 used to control validation loss. +captures the problem. A common choice would be to use +the mean-squared error +1 +|X| +� +(x,y)∈(X,Y )(ˆy(x)−y)2. But +it is arbitrary in the sense that it weights all data points +equally. It turns out, the data sets are not all equally im- +portant for an interpolation scheme occurs when a data +set is taken rapidly rendering not all data points as inde- +pendent measurements. In this case, we can weight each +data set separately in the loss function via +L = +� +n +pn +Nn +� +(x,y)∈(Xn,Yn) +(y(x) − y)2 +(8) +where Nn is the number of data points in data set n +and pn is a normalized weight factor satisfying pn > 0, +� +n pn = 1. +Such a loss function could balance the +ANN-MFT so that it weighs each independent data point +equally. +In this paper, we kept the standard mean- +squared-error loss, and its drawbacks, but it would be +interesting to explore the alternative loss function in the +future. +An example of the behaviour of the loss during train- +ing is shown in Fig. +4. +It shows a rapid decay down +to e−5 = 0.0067 over 200 epochs. Each epoch, the time +needed for the machine to see each data point once, took +about 10 minutes on our desktop using an Nvidia Ti- +tan X GPU. We find that later stages of the training +can still have a sizable improvement in the predictions +for sparsely populated regions of the data space, regions +which are overwhelmed by more densely populated re- +gions at earlier stages of the training. +In what follows, we will present the predictions of a +single trained ANN-MFT model on all the data sets com- +bined as discussed above but then feed this machine in- +dividual data sweeps to see what it predicts for those +portions of the full data set. +The simplest data to interpret are the torsion angle +sweeps shown in Figure 5. By covering all angles up to a +time-reversal symmetry transformation, the ANN-MFT +can learn all four parameters ga, gc, K, and Γ. All other +data sets fix the angle θ of the magnetic field to the c- +0.2 +0.0 +0.2 +Torsion [kJ rad +2 mol +1] +Angle Sweep T=20 K, H = 34.5 T +ANN-MFT +Exp. Data +MFT +0 +/2 +Angle [radians] +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +g-factor values +50 +40 +30 +20 +10 +Coupling strength [K] +ga +gc +K +FIG. 5. Predictions for the angular sweep torsion data and +the couplings used to produce these predictions. (top) Fit to +the data showing a good fit throughout much of the angular +sweep from θ = 0 to θ = π, including a decent fit for the MFT +predictions with a fixed value of the couplings. (bottom) Cou- +plings used to make the predictions in (top) with the average +values of ga = 1.85,gc = 0.54,K = −35.26K,Γ = −11.80 cho- +sen for the MFT prediction. Note the discrepancies near the +c-axis with θ = 0. +axis so, for example, if they are in the ab-plane, they +would be sensitive to ga and jxx ∝ K − Γ while if they +were along the c-axis, they would be sensitive to gc and +jzz. However, the angle sweeps are at a fixed magnetic +field strength of 34.5 T, so it is unclear if the parameters +have saturated to their high-field values. Nevertheless, +we find they are roughly constant at all angles and equal +to the values ga = 1.85, gc = 0.54, K = −35.26 K,and +Γ = −11.80 K. These values are consistent with those +observed in previous studies, especially note the ferro- +magnetic nature of the K coupling, that |K| > |Γ|, and +that ga > gc. +A second observation about the ANN-MFT predic- +tions for the torsion angle sweep data is the behavior +near θ = 0. +Here we see the data has an unexpected +peak which is not present in the mean-field results if we +fix the values of the parameters. It is also hard for the +ANN-MFT to fit this peak, with its attempt introducing +a deviations for angles between θ = π/2 to θ = π, which + +7 +0.0 +0.5 +1.0 +Mx [ +B per Ru3 + ] +Magnetization field Sweep T=18K +ANN-MFT +Experimental Data +MFT Fixed Params +0 +20 +40 +60 +Magnetic Field Strength [T] +1.8 +2.0 +2.2 +2.4 +g-Factor [Dimensionless] +40 +30 +20 +Coupling Strength [K] +ga +K +FIG. 6. Predictions for magnetization data and the couplings +used to produce these predictions. a predictions for the 18 +kelvin, high-field, in-plane magnetization sweep and the pre- +diction of the mean-field theory alone with a fixed choice for +the couplings. b The variation in the couplings used to make +the predictions in a during the magnetic field sweep. +The +fixed choice of couplings for the MFT predictions were taken +from the ANN-MFT predictions at 60 Tesla. +was unnecessary between angles θ = 0 and θ = π/2. We +believe this struggle of the ANN-MFT is due to an in- +trinsically interacting feature in the data not captured +by the MFT. +The predictions for the magnetization data are shown +in Fig. 6. We see that the ga g-factor remains relatively +stable, ranging between 1.9 and 2.4. The machine cannot +accurately predict the gc value from this in-plane data +so we do not present it. +The coupling K − Γ reaches +about -40 Kelvin at 60 Tesla and -30 Kelvin at H = 35T, +consistent with the torsion angular sweep predictions. +The torque data shown in Fig. 7 is harder to interpret +than others. +Here the ANN-MFT struggled to fit the +data so we can trust its predictions less. However, we see +the large field predictions at θ = −10.2o are ga = 0.42 +K − Γ = −1.2K, while at θ = 93.3o are gc = 0.71, +K + 2Γ = −.11. Here the machine is predicting ga < gc, +though both are less than 1. solving for K and Γ from +the two data sweeps, we see it predicts K = −0.83 and +Γ = 0.36 and are vastly scaled down in size compared +to the predictions of the ANN-MFT on the torsion angle +sweep data. +The torsion field sweep data presents a further surprise +but explains the substantial discrepancy between the tor- +sion angle sweeps and magnetization and the couplings +predicted by from the torque data. The couplings pre- +0 +10 +20 +30 + [J mol +1] +Torque Field Sweep T=1.3K +ANN-MFT at += +10.2o +Exp. Data at += +10.2o +MFT at += +10.2o +ANN-MFT at += 93.3o +Exp. Data at += 93.3o +MFT at += 93.3o +0 +20 +40 +60 +Magnetic Field Strength [T] +0.4 +0.6 +g-Factor [Dimensionless] +1.5 +1.0 +0.5 +0.0 +0.5 +Coupling Strength [K] +ga at += +10.2o +gc at += 93.3o +K + at += +10.2o +K + 2 at += 93.3o +FIG. 7. Predictions for the Torque data and the couplings +used to produce these predictions along nearly the a-axis +(θ = −10.2o) and c-axis (θ = 93.3o). (top) Fit to the data +showing a relatively poor fit at larger magnetic field strength. +We also plot the predictions of the MFT with a fixed set +of couplings chosen to match the ANN-MFT predictions at +60 Tesla. (bottom) Couplings used to make the predictions. +Torque along the a-axis is sensitive to ga and torque along +the c-axis is sensitive to gc so only these are plotted. The +Couplings K and Γ are much smaller that those found in Fig. +6 but consistent with other data in the data set at low tem- +peratures (here 1.3 Kelvin). +dicted from these field sweeps with a a-axis magnetic field +approximately scale with temperature, doubling in value +when the temperature changes from 20K to 40K and +again doubling in value when the temperature changes +from 40K to 80K as shown in Fig. 5. At temperatures +above 80K, the doubling stops (not shown). This sur- +prising behavior is consistent with the observed scaling +properties of this data set[11]. As a result, the torque +data now fits into the general picture, it presents scaled +down couplings due to the low temperature at which it +was taken. If we multiply by the ratio of the tempera- +tures, say for the 40 K torsion field magnitude sweeps, we +get ((K − Γ)|torque ∗ 40/1.3 = −36K, not so far from the +K − Γ ≈ −100K observed in the torsion at 40K. Hence, +the results are roughly consistent across all data sets if +one takes into account that the parameters K and Γ scale +with temperature. + +8 +0 +100 +200 +300 + [muB per Ru3 + ] +Torsion In-Plane Field Sweeps +ANN-MFT 20K +ANN-MFT 40K +ANN-MFT 80K +Exp. Data 20K +Exp. Data 40K +Exp. Data 80K +0 +20 +40 +60 +Magnetic Field Strength [Tesla] +0.4 +0.6 +0.8 +1.0 +1.2 +g-factor values +350 +300 +250 +200 +150 +100 +50 +Couple Strength [K] +ga coupling 20K +ga coupling 40K +ga coupling 80K +K + 20K +K + 40K +K + 80K +FIG. 8. The couplings used to make torsion field sweep pre- +dictions at a 20K, b 40K, and c 80K. (top) The ANN-MFT +fit to the torsion field sweep data at the three temperatures. +(bottom) the K − Γ couplings predicted by the ANN-MFT +for the three temperatures showing an increase in magnitude +for this combination of couplings as a function of temperature +approximately doubling between T = 20K and T = 40K and +again doubling (or more) between T = 40K and T = 80K. +IV. +DISCUSSION +We have introduced an ANN-MFT to study parame- +ter inference with domain knowledge inserted into a ma- +chine learning algorithm via a backpropagatable mean- +field theory layer. +There are several weaknesses to this new approach one +should be careful of before interpreting the results. +1. Different training can give rise to distinctly dif- +ferent results due to different saddle points of the +MFT. +2. The ANN-MFT may not fit all data points or sub- +sets well. +3. The ANN-MFT is only as good as the MFT. Is a +more general MFT warranted? +4. What does it mean when we do not discover +roughly consistent model parameter values across +all data sets? +Let us first address point 1. +Through several trials +at training the ANN-MFT, it is possible to get wildly +different results, perhaps at a cost of a higher loss and +poorer fit to the data. These results still fit some data +well, but in a different regime of parameters. We inter- +pret this different regime as a saddle point of the MFT. +Unlike neural networks that have the magical property +that different solutions for the weights behave similarly, +the MFT has no such magic. It can have distinct saddle +points that correspond to metastable states. It would be +interesting in the future to add the energy for each data +point to the loss function and either guarantee the results +are the lowest energy saddle point solution or use such a +modified loss function to isolate metastable states. What +we present here is the most common and best trained +machines but we do not know if the solution we find is a +metastable or stable state. +For point 2, we emphasize that while the ANN-MFT +may not fit all data points well, whenever it does, we can +trust the results. If even just a local fit to a portion of +the data is good, it implies there is a mapping between +thermodynamic state and model parameters that agrees +with experiment. +For points 3 and 4, one possibility is that one should +study a more general model, to address point 3, and +hope the more general model provides a consistent set +of parameters across all data points, to address point 4. +Presumably one can generalize the MFT of this paper +well beyond the four parameters we study. Introducing +the complete spin exchange matrix Jiα,jβ between not +only nearest neighbors but also next nearest neighbors +and even third neighbors seems quite possible given the +efficiency of the algorithm we have presented. Perhaps +generalizing the MFT in this way will yield consistent +parameter values across all data sets. But this doesn’t +seem likely in this case since a scaling with temperature +does not seem to be readily captured by further neighbor +couplings. +In addition to the weaknesses, there are many ways +to improve the simple ANN parameter inference method +used in this manuscript. +One is to upgrade the ANN +encoder to behave like a variational autoencoder that +encodes not directly to the parameters but instead to +a probability distribution from which a parameter sam- +ple is drawn. While such a probabilistic approach would +seem to introduce more noise into the machine, varia- +tional autoencoders are actually more efficient than or- +dinary autoencoders. Such an upgrade, in addition to +efficientcy, would then provide error bars on the predic- +tion of the parameter values. +Despite the weaknesses and simplicity of the ANN- +MFT machine we have used to study α-RuCL3 high +magnetic field data, we find it striking that a scal- +ing with temperature is the predominant observa- +tion across the entire data set from three differ- +ent experiments. +A naive explanation for scaling +is that at these large magnetic fields, +the system + +9 +becomes approximately non-interacting. +By scaling, +κ/T = f(µBH/kBT, K/kBT, Γ/kBT). +If we can set +(K, γ)/kBT ≈ 0, then we obtain κ/T = f(µBH/kBT), +a scaling function. But our observations are that K and +Γ are proportional to temperature. Hence, κ/T scales +but we are never in the non-interacting regime. We fur- +ther notice near fields pointing along the c-axis, that +anomalies appear in the data that the ANN-MFT can- +not fit accurately (see torsion angle sweep results). This +is strong evidence that the temperature regime 1.3K to +80K is captured by an interacting theory beyond the +Curie-Weiss mean-field theory approximation. +Unlike the data set as a whole, data at temperatures +between 20 and 80 K saturated at large magnetic field. +It would be nice to understand this from a perturba- +tion theory perspective. +In the limit |h| ≫ |¯ȷ|, owing +to a gap in the spectrum, the system is adiabatically +connected to a non-interacting paramagnet. +However, +the limit we find our selves in this manuscript, as it ap- +pears within MFT say from the torsion angle sweeps, is +|¯ȷ| = max(|K − Γ|/3kBT, |K + 2Γ|/3kBT) = 0.98 when +0.6 ≲ |h| ≲ 1.7 so that both |h| and |¯ȷ| are of the same +order or magnitude. These values do not to appear to +appreciably change with temperature from 1.3 Kelvin to +80 Kelvin and only above this temperature do we begin +to see a departure. This suggests this entire regime is not +easily captured by a perturbation theory and so there is +no controlled approximation to study it. +Spin liquids like α-RuCl3 are hard to study experi- +mentally, yet an ANN-MFT or similar machine could +prove to be a powerful tool. It is striking that from bulk +data we obtain parameters that are consistent with in- +formation rich techniques like neutron scattering and x- +ray scattering. We believe ANN-MFTs and related ma- +chine learning techniques will enable a powerful relation- +ship between experimental data and model parameters, +especially in the common situation of many spin liquid +candidate materials where neutron scattering and X-ray +scattering data are unavailable. +ACKNOWLEDGMENTS +This material is based upon work supported by the +National Science Foundation under Grant No. +OAC- +1940260. +Appendix A: Derivation of mean-field observables +To derive observables within the mean field theory, we +begin with the mean field partition function with mean +fields mA and mB satisfying Eq. 5 +ZMF =Tr exp +� +−heff +A · +� +i∈A +Si−heff +B · +� +i∈B +Si+3βNBmA¯JmB +� +(A1) +where NB is the number of bonds, ¯J = +1 +NB +� +⟨ij⟩ Jij is +the average exchange matrix, heff +A = h − 3j · mB is the +effective magnetic fields felt by spins on the A sublattice +due to their interactions with spins on the B sublattice +and similarly for the heff +B +with A and B swapped. We +can evaluate this partition function directly and obtain +ZMF = 2|A|+|B|eNBmA¯JmB +× cosh|A| ����heff +A /2 +��� +� +cosh|B| ����heff +B /2 +��� +� +(A2) +From this partition function we can obtain all the observ- +ables we need to compare with the experimental data. +Specifically, we can compute the magnetization, mag- +netic susceptibility, magnetic torque, and magnetic tor- +sion observables, as shown in the next few sections. +1. +Magnetization +To compute magnetization, we use the definition +M α = −∂FMF +∂Hα +(A3) +where FMF = −kBT ln ZMF and Hα is a component of +the external magnetic field H. Evaluating this expression +gives +M α = kBT |A| +2 +∂|heff +A | +∂Hα +tanh +����heff +A /2 +��� +� ++ A → B+ +∂ +∂Hα +� +NBmA¯JmB +� +(A4) +where +∂|heff +A | +∂Hα += ˆheff +A +· ∂heff +A +∂Hα . +(A5) +Recognizing that mA = ˆheff +A +tanh +����heff +A /2 +��� +� +we see this +simplifies to +M α = kBT|A|mA·∂heff +A +∂Hα +A → B+ +∂ +∂Hα +� +NBmA¯JmB +� +. +(A6) +Using +∂heff +A +∂Hα = µB +kBT eα · g − 3j∂mB +∂Hα +(A7) +with h = µBH · g/kBT then obtain +M α = µB(|A|mA + |B|mB) · g · eα +−3|A|mA¯J∂mB +∂Hα −3|B|mB¯J∂mA +∂Hα + +∂ +∂Hα +� +NBmA¯JmB +� +(A8) +where we used j = ¯J/kBT. For periodic boundary con- +ditions, |A| = |B| = Nu and NB = 3Nu and we see that +the quadratic inn mA, mB terms cancel leaving us with +M α = NµB ¯m · g · eα +(A9) + +10 +We see then this result could have been obtained another +way. +The cancellation is precisely what is required to +allow us to first compute M α exactly and then perform +the mean-field approximation. +Numerically, for an Avogadro’s number of atoms, we +can express this as M a,b = (µB/kB)gaR(ma,b +A + ma,b +B )/2 +and M c = (µB/kB)gcR(mc +A + mc +B)/2 where µB/kB = +0.6714[K/T] and R = 8.314[J/K]. Or for a data set ex- +pressed in units of per Bohm magneton per spin we would +use M a,b = ga(ma,b +A +ma,b +B )/2 and M c = gc(mc +A+mc +B)/2, +as is the case of the data studied in this manuscript. +2. +Magnetic Torque +To compute the torque τ, we can use the definition +τ = M × H +(A10) +and take the component in the direction of increasing +azimuthal angle φ to obtain: +τφ = ˆφ · (M × H) = M · (H × ˆφ) = −HM · ˆθ +(A11) +We recognize we can compute this directly from the free +energy +τφ ≡ ∂F +∂θ = ∂Hα +∂θ +∂F +∂Hα = −H ˆθ · M +(A12) +Here +and +throughout +this +paper +we +choose +the +spherical +polar +coordinate +system +H += +H(cos φ sin θ, sin φ sin θ, cos θ) with +ˆφ and +ˆθ the di- +rections of increasing φ and θ respectively. If we were to +follow Ref. [22] and parameterize the magnetic field from +the ab plane as H = H′(cos φ′ cos θ′, sin φ′ cos θ′, sin θ′) +then we would find ˆθ′ = −ˆθ and ∂F/∂θ′ = −∂F/∂θ. As +a result, the torque data presented in this paper differs +in the sign convention for the observable τφ. +A convenient numerical expression for τφ, with an Avo- +gadro’s number of atoms and a magnetic field in the ac- +plane, is obtainable by inserting Eq. A9 in to our expres- +sion for τ and writing it as +τ = −RT dh +dθ · ¯m +(A13) +where R = 8.314[J/K] and +dh +dθ = µB +kBT (gaHc, 0, −gcHa). +(A14) +We are free to place H in the ac-plane, i.e. set φ = 0, +because the rotation symmetry about the c-axis allows us +to always choose this unknown parameter. Fixing φ = 0 +then always produces mean field solutions with mA and +mB also in the ac-plane so we choose to ignore the y +components in our calculation of observables. +3. +Magnetic Susceptibility and Thermodynamic +Stability +The magnetic susceptibility is defined as +χαβ = − +∂2F +∂Hα∂Hβ +(A15) +Technically speaking it is defined in the limit |H| → 0. +However, at any finite magnetic field H, thermodynamic +stability demands +δHδM + δTδS + δµδN ≥ 0 +(A16) +so that using δM α = ∂M α/∂HβδHβ = χαβδHβ we see +that χ ≥ 0 both at finite and the limit of zero magnetic +field. Hence, the observable χ is a useful quantity at any +value of the magnetic field. +It is worthwhile simplifying the computation of the +magnetic susceptibility. It is related to the spin suscep- +tibility, via the chain rule +χαβ = − ∂hγ +∂Hα +∂2F +∂hγ∂hδ +∂hδ +∂Hα = µ2 +B +kBT gαγχγδ +s gδβ +(A17) +where we used ∂hα/∂Hβ = µBgαβ/kBT and defined the +spin susceptibility as χαβ +s += +∂2 +∂hα∂hβ (− log Z). Since it +is straightforward to convert from χ to χs, via χ = +(µ2 +B/kBT)gχsg, we will proceed by focusing on the sim- +pler χs. +We can further simplify the problem of computing +the magnetic susceptibility. Within the MFT, the spin +susceptibility is the derivative χαβ +s += ∂ ¯mα/∂Hβ with +¯m = 1 +2 +� +µ mµ, where µ = A, B denotes sublattice (see +Eq. A9). Hence we need compute ∂mα +µ/∂Hα. We can +do so by computing the sublattice dependent spin sus- +ceptibility χαβ +sµν = ∂mα +µ/∂Hβ +ν where we have introduced +a hypothetical magnetic field HA and HB that acts sepa- +rately on the A and B sublattices and this new suscepti- +bility captures the response to a change in the magnetic +field in just one of the sublattices. It is related to χαβ +s +by +the chain rule +χαβ +s += +� +µ +∂mα +µ +∂hβ = +� +µν +∂mα +µ +∂hβ +ν +���� +hA=hB=h +(A18) +where we thought of mµ as a function of two independent +fields hA and hB, i.e. mµ(hA, hB) and then took the +derivative of mµ(h, h) with respect to h. As a result, we +can focus on the sublattice dependent susceptiblity χαβ +sµν, +a quantity we can compute efficiently as a 6x6 matrix. +An additional benefit is this quantity also must satisfy +χ6 ≥ 0 by thermodynamic stability and so fully expresses +whether the mean field theory is stable. +To compute χ6, it is helpful to work with a 6- +component vector notation, m6 = mA ⊕ mB, combining +the three components of magnetization on the A sublat- +tice and B sublattice. In this language, the mean-field +equations define a non-linear map +m6 = V(h6 − 3¯ȷ6 · m6) +(A19) + +11 +where h6 = h ⊕ h, ¯ȷ6 = ¯J6/kBT with J6 = ¯J ⊗ σx, and +V is the map V(x6) = 1 +2x6 ⊙ W(x6) where +W(x6) = +� +� +� +� +� +� +� +|xA|−1 tanh(|xA|/2) +|xA|−1 tanh(|xA|/2) +|xA|−1 tanh(|xA|/2) +|xB|−1 tanh(|xB|/2) +|xB|−1 tanh(|xB|/2) +|xB|−1 tanh(|xB|/2) +� +� +� +� +� +� +� +(A20) +with ⊙ denoting the broadcast matrix operation such as +A ⊙ B = (A1B1, A2B2, A3B3). Hence by the chain rule, +we have +∇h6m6 = ∂V · (I6 − 3¯ȷ∇h6m6) +(A21) +and so the sublattice dependent spin susceptibility χ6 ≡ +∇h6m6 is given by +χ6 = (I6 + 3∂V ¯ȷ6)−1 ∂V +(A22) +where we recognize ∂V = χ0 +s is the “bare” sublattice +spin susceptibility evaluated at the effective magnetic +field h6 − 3¯ȷm6. +Evaluating the derivative, we see it +is given by +∂V (x6) = 1 +2diag(W(x6))+ +1 +2 [(xA ⊗ xA)T(|xA|)] ⊕ [(xB ⊗ xB)T(|xB|)] +(A23) +where T(x) = +1 +2x2 (1 − tanh(x/2)2) − tanh(x/2)/x3. +Hence, we have reduced the calculation of the magnetic +susceptibility χ to the determination of χ6 via Eq. A22. +The implementation of Eq. A22 was discussed in the +main manuscript and was done robustly via Listing 2. +A highlight of this calculation was the determination of +wether the mean-field theory was thermodynamically sta- +ble. This was achieved by checking whether the Cholesky +decomposition failed with a try-catch statement. In this +way, unstable solutions to the mean-field theory could +be caught. We chose to deal with such cases by drop- +ping those data points from the loss function. So long as +the number of data points that map to unstable MFTs +were small, say 1% or 2%, we found the overall training +of the ANN-MFT worked successfully in that it defines +a mapping onto the parameters of the Hamiltonian that +accurately fits the data for most data points. +4. +Torsion +The last observable we need to calculation is torsion κ. +It is related to magnetizataion via +κ ≡ ∂2F +∂θ2 = ∂τφ +∂θ +(A24) +By chain rule, we can write this as +κ = ∂ +∂θ +�∂Hα +∂θ +∂F +∂Hα +� += −∂2H +∂θ2 · M − ∂H +∂θ · ∂M +∂θ +(A25) +Recognizing that ∂2H/∂θ2 = −H and using ∂H/∂θ = +H ˆθ we obtain +κ = H · M − H ˆθ · ∂M +∂θ +(A26) +It remains to place κ in a numerically convenient form. +By using Eq. 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B., High magnetic +field thermodynamic data set for α-rucl3 (2022). + diff --git a/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf b/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fb51f159ea5f14105000b9392eb944cd83c830fa --- /dev/null +++ b/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ea8a13c00bf317f29ad693dbb4e53c939468027c1ccd81c678529fbe07c58c63 +size 3508014 diff --git a/PNAyT4oBgHgl3EQftfl6/vector_store/index.faiss b/PNAyT4oBgHgl3EQftfl6/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..3617621ece3fd4dee990b6e40a57b7d3e283e698 --- /dev/null +++ b/PNAyT4oBgHgl3EQftfl6/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e62f8fead53019a6996a90cdb0e0a6e83f323c943da299589012e21ddd5c0281 +size 2031661 diff --git a/PNAyT4oBgHgl3EQftfl6/vector_store/index.pkl b/PNAyT4oBgHgl3EQftfl6/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..d4b2a0deac727a13df77f2966a5cdbb475d8d8cf --- /dev/null +++ b/PNAyT4oBgHgl3EQftfl6/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1efcdfa226c2c80fa348b68f6b0de32e58abdd31f5e2e50951dc018b36d28868 +size 73034 diff --git a/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf b/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5888104521077ade48bb59aa1d6f5535602ce4b1 --- /dev/null +++ b/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c30afdcb5e5ca706074a211a4488b51f0123f139a385a6a2ff557c206e66f03b +size 349105 diff --git a/PNAzT4oBgHgl3EQfIfvC/vector_store/index.pkl b/PNAzT4oBgHgl3EQfIfvC/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..5aef8794130b09ea26fb25faa838afad00a71239 --- /dev/null +++ b/PNAzT4oBgHgl3EQfIfvC/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e8e590b6bcea8b758dface575e0c081562a2beb0ec25f46c160e1a33c01c9705 +size 101922 diff --git a/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf b/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..77b6cb83fd76b40b1a4a17daa9f5618ad6f5ee16 --- /dev/null +++ b/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:91d9083156d6d533778ca44d1cc102c1481490c663b69860d0340c6028c91c56 +size 850427 diff --git a/PdFPT4oBgHgl3EQfnjXp/vector_store/index.faiss b/PdFPT4oBgHgl3EQfnjXp/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..498101e7cd8296f607d13e0f3364f8fb25979d7a --- /dev/null +++ b/PdFPT4oBgHgl3EQfnjXp/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:850f4915fe1703686a308c30d7e6ebf75dc483bf37c067d1f89f01e8389d154f +size 2293805 diff --git a/PtFAT4oBgHgl3EQfzh7F/content/2301.08699v1.pdf b/PtFAT4oBgHgl3EQfzh7F/content/2301.08699v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..705dc2aea491f7c7628d080bf02f96c8d3f6a9c3 --- /dev/null +++ b/PtFAT4oBgHgl3EQfzh7F/content/2301.08699v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c63ab016e57697a27f1d420233322c07c0eba1195c64239e0dd0e394276b523d +size 24641600 diff --git a/QNAzT4oBgHgl3EQfW_wb/content/tmp_files/2301.01309v1.pdf.txt b/QNAzT4oBgHgl3EQfW_wb/content/tmp_files/2301.01309v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f37947bb65e88a0c6e6ae08e6e1755febda57857 --- /dev/null +++ b/QNAzT4oBgHgl3EQfW_wb/content/tmp_files/2301.01309v1.pdf.txt @@ -0,0 +1,2463 @@ +Regular black holes from Loop Quantum Gravity +Abhay Ashtekar1,* Javier Olmedo2,† and Parampreet Singh3‡ +1 Physics Department, and, Institute for Gravitation & the Cosmos, +Penn State, University Park, PA 16802, USA +2 Departamento de F´ısica Te´orica y del Cosmos, +Universidad de Granada, Granada-18071, Spain +3 Department of Physics and Astronomy, +Louisiana State University, Baton Rouge, LA 70803, USA +There is rich literature on regular black holes from loop quantum gravity (LQG), where +quantum geometry effects resolve the singularity, leading to a quantum extension of the +classical space-time. As we will see, the mechanism that resolves the singularity can also +trigger conceptually undesirable features that can be subtle and are often uncovered only +after a detailed examination. Therefore, the quantization scheme has to be chosen rather +astutely. We illustrate the new physics that emerges first in the context of the eternal black +hole represented by the Kruskal space-time in classical general relativity, then in dynami- +cal situations involving gravitational collapse, and finally, during the Hawking evaporation +process. The emphasis is on novel conceptual features associated with the causal structure, +trapping and anti-trapping horizons and boundedness of invariants associated with curvature +and matter. This Chapter is not intended to be an exhaustive account of all LQG results on +non-singular black holes. Rather, we have selected a few main-stream thrusts to anchor the +discussion, and provided references where further details as well as discussions of related +developments can be found. In the spirit of this Volume, the goal is to present a bird’s eye +view that is accessible to a broad audience.a +I +Introduction +There is general agreement in the gravity community that black hole singularities of +classical general relativity (GR) offer excellent opportunities to probe physics beyond +Einstein. However, as of now, there is no consensus on the fate of black hole singularities +in full quantum gravity. Indeed, there is an ongoing debate even on a central question +in the subject: Will singularities of classical GR be naturally resolved in full quantum +gravity, or will they persist? As the very name of this Volume suggests, in many circles +an affirmative answer is taken to be a necessary condition for the viability of a proposed +quantum gravity theory. But this is not an universally accepted viewpoint. For example, +it has been argued that taming of black hole singularities in asymptotically anti-deSitter +aInvited Chapter for the book Regular Black Holes: Towards a New Paradigm of Gravitational Col- +lapse, Ed. C. Bambi, Springer Singapore (2023) +* ashtekar.gravity@gmail.com +† javolmedo@ugr.es +‡ psingh@lsu.edu +arXiv:2301.01309v1 [gr-qc] 3 Jan 2023 + +2 +space-times would violate a “No Transmission Principle” motivated by the AdS/CFT cor- +respondence [1]. More generally, discussions of the black hole evaporation process are +often based on the assumption that there is a singularity also in quantum gravity. These +expectations are based on the Penrose diagram of an evaporating black hole that Hawking +drew over 40 years ago [2], where the singularity persists as part of the future boundary +of space-time even after the black hole has completely disappeared (see Fig. 4). How- +ever, this feature of the diagram was not arrived at from a calculation, and indeed such a +calculation is not available even today. Furthermore, some forty years later Hawking him- +self changed his mind: A new Penrose diagram was proposed to represent an evaporating +black hole in which there is no singularity (see Fig. 2 of [3]). Nonetheless, interestingly, +Hawking’s first paradigm continues to feature prominently in discussions on the issue of +information loss: see, e.g., Ref. [4] where the persistence of this singularity leads to a +non-unitary evolution from I − to I +, and Refs. [5–7] where proposals are made on +how unitarity could be rescued in spite of this singularity, thereby preventing information +loss. +Loop quantum gravity (LQG) provides a systematic avenue to investigate the fate of +singularities of classical GR because it is based on quantum Riemannian geometry. Con- +sequently, new physics arises in the Planck regime where the continuum space-time of +classical GR becomes inadequate (see, e.g., [8]). Implications of this new physics have +been analyzed in detail in the commonly used cosmological models. Non-perturbative +quantum corrections to Einstein’s equations imply that, once a curvature invariant ap- +proaches the Planck scale, quantum geometry modifications of Einstein dynamics intro- +duce strong ‘repulsive corrections’ that dilute that invariant, preventing a blow-up (see, +e.g., [9–11]). Thus, the big-bang/big-crunch singularity is replaced by a quantum bounce +in loop quantum cosmology (LQC). Once the curvature drops to about ∼ 10−4 Planck +scale, quantum corrections can be neglected and classical GR becomes a good approxi- +mation. +A natural question then is whether the same phenomenon occurs at the black hole +singularities. Results to date provide considerable evidence that it does. However, tech- +nically, the situation is more complicated than that in cosmological models for two rea- +sons. First, even in the Schwarzschild solution, although space-time is homogeneous in +the vicinity of the singularity, it is not isotropic. Second, the nature of the blow up of +curvature is different from that in the commonly used cosmological models: As Pen- +rose has emphasized, while the Weyl curvature vanishes identically at the big-bang in +homogeneous isotropic cosmologies, it diverges at the Schwarzschild singularity. As a +result, although the singularity is resolved in all LQG investigations, as of now, results +in the black hole sector are not as strong as they are in LQC. Nonetheless, a large num- +ber of investigations, carried out since 2004, have provided conceptual insights as well +as detailed technical results on the nature of the resolution of the Schwarzschild singu- +larity. Our goal is to convey an overall picture at a technical level that is accessible to +beginning researchers, emphasizing conceptual issues, novel elements, and problems that +remain. We also provide references where details can be found. Also for convenience of +non-experts, throughout the Chapter, we pause to summarize the main points after each + +3 +technical discussion and also at the end of subsections. +In Sections II and III we focus on the quantum extension of the Kruskal space-time. +Because the static Killing field is space-like in the ‘interior’ region –bounded by the sin- +gularity in the future and the horizon in the past– the space-time metric is spatially ho- +mogeneous (but not isotropic). As is well-known, this portion of Kruskal space-time is +isometric with the vacuum Kantowski-Sachs cosmological model. Therefore techniques +from LQC have been used to analyze the fate of the Schwarzschild singularity in a number +of investigations within LQG (See, e.g.,[12–38]). While some of these analyses present +us with the equations that dictate the evolution of the quantum state of the system, the +detailed results are based on the so-called ‘effective equations’ whose goal is to incor- +porate the leading order quantum corrections to the classical geometry in sharply peaked +quantum states.1 At a conceptual level, all these investigations follow the same strategy. +However, the technical implementation of this procedure differs, leading to different ef- +fective geometries in the interior region. Nonetheless, in all these cases, the singularity +is resolved due to quantum corrections. We will discuss the strategy and compare and +contrast various results in Section II. Singularity resolution in the Kruskal space-time +provides several sharp results on the causal structure of its quantum extension. In particu- +lar, the singularity is replaced by a ‘transition surface’ to the immediate past of which we +have a trapped region and to the immediate future, an anti-trapped region. This geometry +is sometimes referred to as depicting ‘a black hole to white hole transition’. We will avoid +this terminology because it has other connotations that are not realized. In particular, the +terms ‘black hole’ and ‘white hole’ normally go hand in hand with singularities and event +horizons. In LQG, singularities are absent and, in dynamical situations, there are also no +event horizons either. +In Section III we consider the Schwarzschild exterior, i.e. the region bounded by the +horizon and I ±. Space-time is again foliated by homogeneous 3-dimensional surfaces +but they are now time-like rather than space-like. We discuss a possible extension of +the ‘interior’ geometry to this exterior region, following [28, 29, 39]. This extension +has several attractive properties [30], but it also has some puzzling features: while the +quantum corrected metric is again asymptotically flat in a precise sense (that suffices to +define the ADM mass, for example), the approach to the flat metric is weaker than the one +generally used in the physics literature. There are alternate proposals to arrive at effective +metrics with the standard asymptotic behavior (see, e.g., [33, 38]) but a definitive picture +is yet to emerge. +Now, the Kruskal space-time itself is an idealization since it represents an ‘eternal +black hole’; black holes encountered in nature are formed dynamically, e.g., via a gravita- +tional collapse, or compact binary mergers. Nonetheless, one would expect the qualitative +features of the causal structure that arises from taming of the singularity due to quan- +tum effects would be robust. In Section IV we discuss models of dynamical situations +1For the conceptual framework underlying effective equations see, e.g., Section V of [9]. Note that the +term ‘effective equations’ has a very different connotation here than in standard quantum field theory. This +has caused occasional confusion in the literature. In LQG one does not integrate out ‘high energy modes’; +Planck scale effects are retained. In LQC, for example, there are states that remain sharply peaked even +in the Planck regime and the effective equations capture the evolution of the peak of the quantum wave +function in these states, ignoring the fluctuations. + +4 +that have been analyzed within LQG and summarize the current status, focusing on the +Lemaˆıtre-Tolman-Bondi type models of collapse and critical phenomena discovered by +Choptuik. In Section V we turn to the issue of black hole evaporation and ‘information +loss’. The LQG discussion of these issues is characterized by two key features [40]. First, +as discussed above, in contrast to the Penrose diagram in Hawking’s seminal paper [2], +there is no singularity in the space-time interior which can serve as a ‘sink of informa- +tion’. Second, as the LQG Penrose diagram of Fig. 6 shows, there is no event horizon: +what forms and evaporates is a dynamical horizon [41–43]. Much of the discussion in the +literature assumes that there is an event horizon which serves as a boundary of an ‘inte- +rior’ region from which no causal signal can ever be sent to the asymptotic region. One +is then led one to either conclude that information is lost, or, to introduce ‘exotic’ ideas +such as quantum Xerox machines, firewalls and fast scramblers to restore unitarity. As we +discuss, there is a more direct pathway to unitarity once it is realized that there is no event +horizon. However, as in every other approach, important issues remain: the precise nature +quantum radiation at the final stages of the evaporation process require full LQG and this +analysis has only begun. We summarize the current status in Section V. In Section VI we +collect the key features of regular black holes in LQG compare and contrast the regular +LQG black holes with this in other approaches. +Our conventions are the following. Space-time metric gab has signature -,+,+,+ and +the curvature tensors are defined by Rabcdkd = 2∇[a∇b]kc; Rac = Rabcb; and R = gabRab. +By macroscopic black holes we mean those for which GM =: m ≫ ℓPl. +II +The Schwarzschild interior +Denote by (M,gab) the Kruskal extension of the Schwarzschild metric (see Fig. 1) and +by (MII, gab) the quadrant of this space-time that represents the (open) ‘interior region’ II, +bounded by the black hole singularity and future horizons. This region is foliated by the +rsch = const space-like manifolds, with topology S2 × R2. Each leaf admits 3 rotational +Killing fields tangential to its 2-dimensional spherical cross sections that are mapped to +one another by the translational Killing field. +Consequently, (MII, gab) is spatially ho- +mogeneous, but not isotropic; it is isometric to the (vacuum) Kantowski-Sachs cosmolog- +ical model. Therefore LQG approaches use the procedure from homogeneous cosmolo- +gies. Now, while the big-bang and big-crunch singularities persist in the Wheeler-DeWitt +(WDW) theory based on metric variables, they are naturally resolved in LQC because of +the quantum geometry resulting from the use of connection variables (see, e.g., [9]). For +the Schwarzschild interior, then, LQG investigations also begin with a 3+1 decomposition +of Einstein’s equations using connection variables. In the classical theory, components of +the curvature tensor that features in these equations can be obtained by first evaluating +holonomies of the gravitational connections around suitable closed loops (called plaque- +ttes) and then taking the limit as the area enclosed by these plackets tends to zero. In +LQG, the corresponding quantum operator is obtained by shrinking these plaquettes till +the area they enclose reaches the smallest non-zero eigenvalue of the area operator. This +eigenvalue is called the area gap and denoted by ∆. As a consequence, information about + +5 +quantum geometry gets encoded in the dynamical equations. Observables such as curva- +ture scalars can acquire finite upper bounds on entire dynamical trajectories, whence the +singularity is resolved. ∆ appears in the denominator of the expressions of these upper +bounds; classical singularities emerge as ∆ → 0. +For black holes, while operator equations have been written down [12–15, 34, 37], +detailed investigations of the singularity resolution and ensuing quantum corrected geom- +etry have been obtained using ‘effective equations’ discussed in Section I. Solutions to +effective equations show that the central singularity is resolved due to quantum correc- +tions. However, different investigations within LQG have made different choices to arrive +at the quantum corrected curvature operators. Intuitively these choices represent quanti- +zation ambiguities that then affect detailed predictions. For brevity, in Sections II A and +II B we will present the general framework and results following a recent approach that is +free of limitations of the earlier investigations and in Section II C we will briefly compare +and contrast other approaches. Due to space limitation, by and large we will only include +motivations behind various constructions and summarize the final results. For detailed +derivations and other details, see in particular [12, 13, 19, 20, 29]. +A +The framework +In connection-dynamics, the initial data for space-time geometry consists of an SU(2)- +valued connection Ai +a and its conjugate ‘electric field’ Ea +i as in Yang-Mills theory. In +I +II +III +IV +J +J +J +J +i +i +o +0 ++ ++ + + +FIG. 1: The Penrose diagram of the Kruskal space-time. In this section we discuss the quantum +extension of part II, bounded to the past by future horizons and the future by the singularity. The quantum +corrected effective geometry of region I is discussed in Section II B. + +6 +the final solutions to Einstein’s equations, Ai +a has the interpretation of the gravitational +connection that parallel transports SU(2) spinors, and Ea +i , represent the ortho-normal spa- +tial triads (with density weight 1). Because of spatial homogeneity of the model, various +spatial integrals in the Hamiltonian framework have a trivial divergence. Therefore, one +introduces an ‘infrared cut-off’. Thus one truncates the homogeneous slices to be finite +(rather than infinite) cylinders, with coordinates (θ,φ,x) with x ∈ (0, L◦) (rather than +x ∈ (0, ∞)). One has to make sure, of course, that none of the final results depend on +L◦. One can solve the ‘kinematical’ constraint equations and use gauge-fixing to cast the +basic variables in the form +Ai +a τi dxa = c/L◦ τ3 dx+b(τ2dθ −τ1 sinθ dφ)+τ3 cosθ dφ, +Ea +i τi∂a = pc τ3 sinθ ∂x +(pb/L◦)τ2 sinθ ∂θ −(pb/L◦) τ1 ∂φ +(2.1) +where τi are SU(2) generators related to Pauli spin matrices σi via τi = −iσi/2. Real +valued connection components b,c and the triad components pb, pc are functions only of +time and serve as conjugate coordinates on the 4-dimensional phase space. It is conve- +nient to choose an orientation of the triads so that b, c, pc are positive and pb is negative. +It follows from (2.1) that physical quantities can only depend on b, (pb/L◦), (c/L◦), pc. +Given a time coordinate τ that labels the spatially homogeneous surfaces and the corre- +sponding lapse Nτ, in region II the space-time metric has the form +gabdxadxb ≡ ds2 = −N2 +τ dτ2 + p2 +b +pcL2◦ +dx2 + pc(dθ 2 +sin2 θdφ2). +(2.2) +At the horizon, b, pb vanish and the translation Killing field X = ∂/∂x becomes null. +When pc vanishes, the radius of the metric 2-spheres shrinks to zero, making the curvature +scalars diverge there. This is Schwarzschild singularity. +It turns out that Einstein’s equations that govern the dynamics of the basic variables +simplify significantly if one uses the lapse Ncl = (γ √pc)/b (which is different from the +standard lapse in the Schwarzschild coordinates.) The γ in this expression is the dimen- +sionless Barbero-Immirzi parameter of LQG. It is analogous to the θ-parameter of QCD +in that it represents a quantization ambiguity: classical physics is insensitive to the precise +value of γ; we only need γ > 0. In terms of the corresponding time-coordinate Tcl, the +dynamical trajectories are given by: +b(Tcl) = γ +� +e−Tcl −1 +�1/2 +and +pb(Tcl) = p(◦) +b eTcl � +e−Tcl −1 +�1/2, +(2.3) +and +c(Tcl) = c(◦) e−2Tcl +and +pc(Tcl) = p(◦) +c e2Tcl . +(2.4) +Here c(◦), p(◦) +b , p(◦) +c +are integration constants. Comparison with the standard form of the +Schwarzschild solution yields p(◦) +c += 4m2, p(◦) +b /L◦ = −2m, and c(◦)/L◦ = γ/4m, where +m is related to the mass of the Schwarzschild solution via m = GM. At the horizon Tcl = 0 +and at the singularity Tcl = −∞. + +7 +The dynamical variables are subject to the Hamiltonian constraint +Hcl[Ncl] ≡ − 1 +2Gγ +�� +b+ γ2 +b +� +pb + 2c pc +� += 0. +(2.5) +It is easy to verify that the terms in the b and c sectors on the right side of (2.5) are +separately conserved in time, and equal −m and m respectively on solutions. Therefore, +if the constraint (2.5) is satisfied at one instant Tcl, then it holds for all T ∈ (−∞,0). +As explained above, in the passage to quantum theory the spatial curvature is expressed +using the holonomoly of the gravitational connection Ai +a around appropriately chosen pla- +quettes that enclose the minimum non-zero area, ∆ = 4 +√ +3πγℓ2 +Pl. (Thus, while classical +physics is insensitive of the value of the Barbero-Immirzi parameter γ, quantum physics +is not. Its value is generally taken to be γ = 0.2375 via black hole entropy calculation.) +As a consequence, the effective equations that capture the leading quantum corrections +inherit new ‘quantum parameters’, denoted by δb and δc, that refer to edge lengths of +these plackets, and go to zero in the classical limit, ℓPl → 0 (or, ∆ → 0, keeping γ fixed). +Different choices of these quantum parameters represent quantization ambiguities men- +tioned above. In this section we will use a strategy [28–30] that is free of the physically +undesirable features encountered in other approaches (discussed in Section II C). +A key idea behind this strategy is to use δb and δc that are ‘Dirac observables’ i.e. +phase space functions that are constant along dynamical trajectories.2 Let us restrict +ourselves to such δb, δc from now on. Then, again, the evolution equations simplify if +we include the appropriate quantum corrections in the choice of the lapse, defining it as +N := (γ √pc)δb/sin(δbb). (Note that as the area gap ∆ goes to zero, so does δb and N +reduces to Ncl.) Denote by T the corresponding time parameter and by ‘dot’ the derivative +with respect to T. Then, as in the classical theory, the effective evolution equations b and +the c sectors separate: +˙b = −1 +2 +�sin(δbb) +δb ++ +γ2δb +sin(δbb) +� +, +˙pb = pb +2 cos(δbb) +� +1− +γ2δ 2 +b +sin2(δbb) +� +, +(2.6) +and +˙c = −2 sin(δcc) +δc +, +˙pc = 2 pc cos(δcc). +(2.7) +But, again as in the classical theory, the two sectors are linked by the (now, effective) +Hamiltonian constraint: +Heff[N] ≡ − 1 +2Gγ +��sin(δbb) +δb ++ +γ2δb +sin(δbb) +� +pb +2sin(δcc) +δc +pc +� += 0. +(2.8) +2Because the spatial curvature features on the right side of Einstein’s evolution equations, the quantum +corrected version of the classical dynamical trajectories (2.3) and (2.4) along which δb and δc are to remain +constant themselves feature δb and δc (see (2.9), (2.10) and (2.11)). Therefore the issue of finding δb and +δc that are Dirac observables is rather subtle conceptually and quite intricate technically. These subtleties +has led to some concerns [31]. This issue is analyzed in detail [35–37, 44]. Consistency of the final results +directly follows from the effective equations (2.6) - (2.8). + +8 +A direct calculation shows that the constraint (2.8) is preserved in time. +To summarize, conditions ˙δb = 0, ˙δc = 0, the evolution equations (2.6), (2.7) and the +constraint equation (2.8) constitute a set of consistent equations that generalize the clas- +sical constraint and evolution equations. A notable difference from the classical theory +arises because in LQG there is a well-defined operator in the quantum theory correspond- +ing only to the holonomy defined by the gravitational connection Ai +a, rather than Ai +a itself. +As a consequence, only trigonometric functions of δb b and δc c appear. Hence the do- +main of these variables is compactified (just as in LQC [45]): they take values in the open +interval (0, π). The momenta pb, pc, by contrast, continue to assume values pb < 0 and +pc > 0 as in the classical theory. +To solve the evolution equations, it is convenient to first obtain solutions c(T), pc(T) +and b(T). Now, in the c sector, equations of motion (2.7) immediately imply that mc = +(sin(δcc) pc)/(γL◦δc) is a constant of motion. This fact simplifies the form of the solu- +tions. One obtains +tan +�δc c(T) +2 +� += γLoδc +8mc +e−2T, +pc(T) = 4m2 +c +� +e2T + γ2L2 +oδ 2 +c +64m2c +e−2T� +, +(2.9) +cos +� +δb b(T) +� += bo tanh +�1 +2 +� +boT +2tanh−1 � 1 +bo +��� +, +(2.10) +where there constant bo is given by bo = (1+γ2δb +2)1/2. One then uses the Hamiltonian +constraint to determine pb(T): +pb(T) = −2sin(δc c(T)) +δc +sin(δb b(T)) +δb +pc(T) +sin2(δb b(T)) +δ 2 +b ++γ2. +(2.11) +Eqs (2.9) - (2.10) provide the dynamical trajectories of the effective theory. It is easy to +verify that in the limit δb → 0, δc → 0, one recovers the classical trajectories. To sum- +marize, the quantum corrected, effective trajectories are given by (2.9) - (2.11) for any +choice of constants of motion δb, δc. Since these equations only involve the combina- +tions b, (pb/L◦), δb; (c/L◦), pc, and L◦δc, the metric (2.2) and all physical results are +insensitive to the choice of the infrared cut-off L◦. +So far δb, δc could be any quantum parameters satisfying ˙δb = 0 and ˙δc = 0. The +following considerations provide a natural avenue to determine them. Recall that on +classical solutions, the c part of Hcl[Ncl] equals m, and the b part equals −m. Therefore +in the effective theory, one is led to set +1 +2γ +�sin(δbb) +δb ++ +γ2δb +sin(δbb) +� pb +L◦ += −mb +and +sin(δcc) +γL◦δc += mc . +(2.12) +Equations of motion (2.6) and (2.7) imply that both mb and mc are constants of motion +and the effective Hamiltonian constraint reads mb = mc. On solutions, we will drop the +suffix and set mb = mc = m. The fact that mb and mc are constants of motion suggests a + +9 +natural strategy to restrict the form of δb,δc: Require that δb be a function only of mb, and +δc be a function only of mc. To constrain the functional form requires additional input, +summarized in Section II B. Here we only note that the final answer has a rather simple +form for large black holes (i.e. for solutions for which m ≫ ℓPl): δb and δc are extremely +well-approximated by +δb = +� +√ +∆ +√ +2πγ2mb +�1/3 +, +and +Loδc = 1 +2 +� γ∆2 +4π2mc +�1/3 +. +(2.13) +(Recall that physical results can only depend on the combination Loδc.) +To summarize, the effective metric in the interior region is given by (2.14), where c, pc +are given by (2.9), b, pb by (2.10), (2.11), and δb, δc by (2.13). By inspection we see +that as the area gap ∆ goes to zero, δb and δc both go to zero and the effective theory +reduces to the classical GR. +B +Singularity Resolution, Causal Structure and Curvature Bounds +Let us explore properties of the space-time metric +gabdxadxb ≡ ds2 = −N2dT 2 + p2 +b +pcL2◦ +dx2 + pc(dθ 2 +sin2 θdφ2). +(2.14) +of the effective theory. The past boundary of the open region under consideration is +again given by b = 0, pb = 0 which occurs at T = 0 on every dynamical trajectory. +The translational Killing vector Xa becomes null at these points; thus as in the classical +theory, this boundary represents the horizon. In the classical theory, the singularity is +characterized by the vanishing of the radius of the metric 2-spheres, i.e., of pc. In the +effective theory, however, pc has a non-zero minimum, pmin +c += 1 +2γ(L◦δc)m which occurs +at T = 1 +2 ln(γL◦δc)/8m. Note that this minimum radius is of Planck scale but depends on +the mass of the initial black hole: rmin ∼ (mℓ2 +Pl)1/3. This is the surface that replaces the +classical singularity and the space-time metric (2.14) can be smoothly extended across +this 3-manifold. +One can explore the causal structure around this surface by calculating the expansions +Θ± of the two null normals to the metric 2-spheres. To the past of this surface one finds +that both expansions are negative. Thus this is a trapped region just as the entire region II +is in the classical theory. Interestingly, both null-expansions vanish on this surface. This is +a novel situation that is not encountered in classical GR. Since the metric is smooth across +this surface, space-time is well-defined across it and one can analyze the two expansions +to the future of this surface. They are both positive, so the region to the future is anti- +trapped. Thus in the quantum-extended effective space-time, the surface neatly separates +a trapped region and an anti-trapped region. Therefore it is called a transition surface, +denoted by T . It is analogous to the ‘bounce surface’ in LQC (that replaces the big- +bang), to the past of which the expansion of the universe is negative and to the future of + +10 +which it is positive. However, now the term ‘expansion’ refers to changes in the areas +of metric 2-spheres along its two null normals. How far into the future is the space-time +extended by this procedure? The metric is well defined in the open region bounded by the +surface T = −(4/bo)tanh−1 (1/bo) where (δb b) = π and pb = 0. The Killing field Xa is +again null on the boundary so it again represents a horizon that bounds the anti-trapped +region to the future. In summary, effective dynamics extends the open region II (of Fig. +1) to the diamond shaped open region (shown in Fig. 2) bounded by Killing horizons. +The region is separated by a transition surface T , to the past of which one has a trapped +region and to the future of which, an anti-trapped region. This extension is often referred +to as the black hole to white hole transition. +In LQC, space-time curvature attains the maximum value on the bounce surface and, +furthermore, this upper bound is universal. Does the quantum corrected geometry exhibit +the same feature at transition surface T ? The answer is in the affirmative. One has: +R2 |T ≈ 256π2 +γ4∆2 +... +RabRab |T = 256π2 +γ4∆2 +... +(2.15) +CabcdCabcd |T = 1024π2 +3γ4∆2 +... +RabcdRabcd = 768π2 +γ4∆2 +... +(2.16) +where all the correction terms ... have the same form O +� +(∆/m2))1/3 ln(m2/∆) +� +. Recall, +first, that the classical limit corresponds to ∆ → 0 (keeping γ > 0.) Hence in this limit +T +AT +T +Σ +FIG. 2: Quantum extension of region II of Fig. 1 in the effective theory. The singularity is replaced by +the transition surface T . It separates the trapped and anti-trapped regions. The past boundary T is a null +(black hole type) trapping horizon and the future boundary AT is a null (white-hole type) anti-trapping +horizon. The time-like 3-manifold Σ joining 2-spheres lying on the two horizons is used in Eq. (2.17). + +11 +all invariants diverge and T is replaced by the singularity. Secondly, since leading terms +are mass independent, the upper bounds are universal. (The numerical coefficients vary +simply because the invariants refer to distinct parts of the total curvature.) Third, as +one moves away from T , these curvature scalars rapidly approach their classical values +even for very small black holes. Thus quantum corrections to space-time geometry are +very small away from the transition surface. For instance, while the horizon radius of +the effective solution is always larger than that of its classical counterpart, even for m = +104ℓPl, the relative difference is ∼ 10−15 and for a solar mass black hole, it is ∼ 10−115! +Finally one can ask for the relation between the radius rT of the trapping horizon that +constitutes the past boundary of the diamond, and the radius rAT of the anti-trapping +horizon that constitutes the future boundary. Are they approximately the same? The +answer is in the affirmative for macroscopic black holes, even though the ‘bounce’ is not +exactly symmetric. For a stellar mass black hole for example, rT = 3km and rAT = (3 + +O(10−25))km. As we will see in Section II C, these consequences of effective dynamics +are non-trivial: it is surprisingly difficult to achieve the singularity resolution without, at +the same time, triggering unintended large effects away from the singularity. +Next, note that while the Ricci tensor vanishes identically in classical solutions, it is +non-zero in the effective solutions. One can simply set 8πGN T eff +ab := Rab − 1 +2Rgab and +interpret T eff +ab as the effective stress-energy tensor of the quantum corrected space-time. +As one would expect from the above discussion, for macroscopic black holes these quan- +tum corrections are negligible away from T . However, they become large and dominant +in the immediate vicinity of T . As one could have anticipated, although it is finite ev- +erywhere, the energy density defined by T eff +ab becomes large and negative in this region +thereby violating the energy conditions, as it must for the singularity resolution to occur. +Interestingly, this fact creates an apparent tension with considerations involving the +Komar mass MK. Recall that, in the classical theory, MK defined by the translational +Killing field Xa is given by (half the) horizon radius. As we saw, for macroscopic black +holes the radii rT and rAT are essentially the same. But the difference between the Ko- +mar mass evaluated at the anti-trapping horizon and the trapping horizon is given by the +integral over a 3-manifold Σ joining a cross-section of the trapping horizon with a cross- +section of the anti-trapping horizon (see Fig. 2), +MAT +K −M T +K = 2 +� +Σ +� +T eff +ab − 1 +2T eff geff +ab +� +XadΣb , +(2.17) +and for macroscopic black holes the integrand of the right is large and negative near T +(because it represents the effective energy density). How can the two Komar masses be +the same, then? It turns out that the integrand of (2.17) is indeed large and negative for +macroscopic black holes, but its numerical value is very close to −2MT +K. Therefore the +Komar mass associated with the anti-trapping horizon is given by MAT +K ≈ MT +K − 2MT +K = +−MT +K, and the minus sign is just right because while the translational Killing field is future +directed on the trapping horizon T, it is past directed on the anti-trapping horizon AT! +(See the (blue) arrows in Fig. 3.) This resolution is another example of the conceptually +subtle balance achieved with the choice of quantum parameters (2.13). + +12 +To summarize, the Schwarzschild singularity is naturally resolved in the effective the- +ory discussed in Section II A and region II of Fig. 1 bounded by the singularity to the fu- +ture is extended to the singularity free diamond-shaped region shown in Fig. 2, bounded +in the past by the trapping horizon and to the future by the anti-trapping horizon. The sin- +gularity is replaced by a space-like surface T that marks the transition between trapped +and anti-trapped regions. Curvature scalars achieve their maximum values on T which +are universal to the leading order. Although quantum corrections encoded in the area gap +∆ dominate near T , they decrease rapidly as one moves away and are completely neg- +ligible near horizons for macroscopic black holes. In particular, the radii of the trapping +and anti-trapping horizons are indistinguishable for macroscopic black holes. +C +Summary of LQG Investigations +As we already noted, the LQG investigations of the Schwarzschild singularity follow +the same general steps but differ in the selection of the quantum parameters δb, δc. Since +the Schwarzschild interior is isometric to the Kantowski-Sachs cosmological model, dis- +cussions have often focused on issues motivated by cosmological considerations such as +the behavior of ‘scalar factors’ and shears, rather than on considerations that are more +directly relevant to black holes, in particular properties of the effective geometry that lead +to trapping and anti-trapping. We focused on an approach that does [28, 29]. We will +now summarize various strategies that have been used to fix δb, δc and results they led +to. Since our goal is only to present a cohesive picture of the overall status through com- +parison of results, the discussion will be rather brief; details can be found in the original +papers listed in the bibliography. +By and large, these strategies fall into three categories: +(i) The parameters are chosen to be constants. These approaches are often referred to +as the µo-type schemes because they mimic the strategy of using constant values for the +quantum parameter µ used in LQC [46]. Here, the curvature operator is defined using +holonomies of the gravitational connection around plaquettes and shrinking them till the +coordinate area they enclose equals the area gap ∆; +(ii) The parameters are chosen to be phase space functions, using physical considerations. +These approaches are often referred to as the ¯µ-type schemes, named after the strategy of +selecting the quantum parameter µ in LQC [45] in which the curvature operator is defined +by shrinking the plaquettes till the physical area they enclose equals ∆; and, +(iii) The parameters are chosen to be phase space functions that are constants of motion +on the effective dynamical trajectories. The strategy used in the last two sub-sections falls +in this class. +The earliest investigations [12–14] used strategy (i); technically it is the simplest to +implement. Here the quantum parameters were set to δb = δc = 2 +√ +3 using ‘square’ +plaquettes in coordinates adapted the symmetries. Predictions of the resulting effective +theory were analyzed in detail in [19]. The singularity is again resolved and replaced by a +3-surface at which the symmetry 2-spheres attain the minimum area. However, physical +quantities such as the minimum value of the radius and the radius of the anti-trapping + +13 +horizon now depend on the infrared cutoff L◦. Another limitation is that quantum effects +can become significant even in the low curvature region near the horizons. +In the approaches [17, 18] based on strategy (ii), the quantum parameters were fixed +by mimicking the successful ¯µ strategy from LQC. One again adapts the plaquettes to +the symmetries of the problem, but shrinks them till the physical area they enclose is ∆. +Therefore the plaquettes themselves now depend on the phase space point under consider- +ations and change under time evolution. As a consequence, quantum parameters are spe- +cific phase space functions that are not constant along dynamical trajectories: δb = ∆/pc +and L2 +◦ δ 2 +c = (L2 +◦ pc∆)/p2 +b. In these definitions, the dependence of L◦ is exactly the one +that is needed to assure that physical results are independent of the fiducial choice of +L◦. This is a significant improvement over results from strategy (i). However, a techni- +cal complication arises because δb depends on pc and δc on pc: the equations in the b +and c sectors no longer decouple. Consequently, it has not been possible to write down +analytic solutions and all explorations to date have been performed numerically. These +calculations show that the framework has two types of limitations. First, as in (i), there +are large deviations from the classical theory even when the curvature is low. Second, +when one evolves beyond the transition surface, the dynamical trajectory enters a region +of the phase space where the metric 2-spheres have area that is less than the area gap ∆, +making the scheme internally inconsistent. Perhaps not surprisingly, then, some of the +properties of the extended space-time are difficult to understand physically. +Strategy (iii) was first adopted to improve on this situation by making δb, δc phase +space functions that remain constant along dynamical trajectories [19, 20]. Then the con- +siderations of the first part of Section II A are applicable, the b and the c sectors separate, +dynamical trajectories can be written down analytically, and mb and mc are constants of +motion. In the first investigation, δb and δc were chosen by dimensional considerations +and by taking into account the fact that it is only the combination (L◦δc) that is invari- +ant under the change of the infrared cutoff L◦. The simplest expressions satisfying these +requirements were then selected, (δb)2 := ∆/4m2 and L◦(δc)2 = ∆, without the consid- +erations of plaquettes and holonomies of the gravitational connection around them [19]. +The physical results are now invariant under rescalings of L◦ as desired. There is again a +transition surface T that separates the trapped and anti-trapped regions, and the quantum +corrected space-time is a diamond bounded by a trapping horizon in the past and an anti- +trapping horizon in the future. Furthermore, unlike the µo and ¯µ-type schemes, quantum +corrections are small in regions near the horizons where the curvature is low. However, +detailed examination revealed two limitations. First, at the transition surface the Kretch- +mann scalar of (initially) macroscopic black holes now goes as 1/m; whence it decreases +as the mass of m increases. Therefore for astrophysical black holes, large quantum correc- +tions at the heart of the ‘bounce’ at T occur at low curvature. A second counter-intuitive +result is involves ‘mass inflation’ across T . The radius rAT of the horizon in the future of +T now goes as rAT = (rT)×(rT/ℓPl)3. Therefore, if the initial black hole has solar mass +with rT = 3km, one has rAT ≈ 1093Gpc! The physical mechanism responsible for this +huge magnification has remained unclear. Therefore, subsequently, more general choices +of the quantum parameters were explored by introducing new dimensionless constants α + +14 +and β, setting (δb)2 := (α2 ∆)/4m2 and L◦(δc)2 = β 2∆ and varying α and β to ensure +rAT ≈ (rT) for large black holes. Two choices satisfying this condition were found nu- +merically and one analytically. The analytic expression implies that the leading term in +Kretchmann scalar at T is not universal but grows rapidly with m as m4/∆4. +The expressions (2.13) used in Section II B to discuss results also fall under strategy +(iii). However, now the quantum parameters δb and δc are obtained using certain pla- +quettes, holonomies around which are used to define the curvature. These plaquettes are +tailored to the symmetries of the problem, and enclose physical area ∆ as in the strategy +(ii). The key difference is that these loops are restricted to lie on the transition surface T +[28, 29]. Since each dynamical trajectory intersects T once and only one, the prescrip- +tion is unambiguous and, by construction, makes δb and δc constants of motion. This +choice automatically leads to the result rAT ≈ (rT), without recourse to any additional +free parameters (such as α, β discussed above). Furthermore, now the transition surface +necessarily lies in the region where curvature is Planckian and the leading terms in the +expressions of all curvature invariant are universal. +As this discussion shows, the task of choosing appropriate quantum parameters δb,δc +is a very subtle. While is it not difficult to make a ‘reasonable’ choice that resolves the +singularity, the resulting quantum corrected geometry has to satisfy several non-trivial +constraints to be physically admissible. Over the years, several choices have been pro- +posed but the subsequent careful scrutiny by the LQG community showed that they lead +to results that are physically unsatisfactory in one way or the other. The choice discussed +in the last two subsections passes all the checks known to date. While this is satisfying, +the analysis is still incomplete in one respect. In LQC the effective equations could be +derived systematically starting from the operator equations of the quantum theory, show- +ing that there are states that remain sharply peaked even in the deep quantum regime, +and using expectation values of observables in these states [47, 48] (and Section V of +[9]). Thus the LQC effective equations encode the dynamics of the peaks of these wave +functions. For black holes, the successful LQC techniques have been used in conjunction +with an extended phase space framework (introduced in [29]) to arrive at the desired op- +erator equations and to select physical states in [37]. A systematic derivation of effective +equations from this quantum theory remains an interesting open issue in LQG. +III +The Schwarzschild exterior +For the discussion of singularity resolution, it suffices to consider just the region II of +Fig. 1. Therefore, initially the focus on LQG investigations was on this region. However, +for a complete understanding of the quantum corrected space-time, one also has to con- +nect the effective space-time geometry of region II to that of region I. In Section III A we +present an approach to carry out this task. Section III B summarizes the properties of the +near-horizon quantum corrected geometry it provides, and III C discusses the asymptotic +structure of the effective space-times. As expected, for macroscopic black holes the near +horizon geometry exhibits physically expected features because quantum corrections are + +15 +small there. In the asymptotic region, on the other hand, this effective geometry has an +unforeseen feature: while the quantum corrected metric is asymptotically flat in a precise +sense, the approach to flatness is weaker than what one might have a priori expected. We +will discuss this issue and summarize its current status in Section VI. +A +The underlying framework +Recall that the analysis of the Schwarzschild interior was greatly facilitated by the +fact that this region is foliated by homogeneous, space-like slices. The exterior region +on the other hand does not admit such a foliation. However, the four Killing fields do +provide a natural foliation of this region by homogeneous, time-like slices. Indeed the +textbook derivation of the classical Schwarzschild metric can be interpreted as solving +the ‘evolution equation’ in the r direction together with the ‘Hamiltonian’ constraint on +the r = const homogeneous slices, mirroring the procedure used in the Schwarzschild +interior (or, Kantowski-Sachs space-times). The main difference is that the signature of +the intrinsic 3-metric on the homogeneous slices is now -,+,+ rather than +,+,+. There- +fore in the connection framework one has to change the internal group that acts on the +orthonormal triads from SU(2) to SU(1,1).3 The generators τi that provide a basis for the +Lie algebra of SU(2) are now replaced by ˜τi that constitute a basis for the Lie algebra of +SU(1,1). The relation between the two is +˜τ1 = iτ1, +˜τ2 = iτ2, +and +˜τ3 = τ3. +(3.1) +Hence, for exterior region we can choose our basic variables to be +Ai +a ˜τi dxa = +˜c +Lo +τ3 dx+i˜bτ2dθ −i˜bτ1 sinθ dφ +τ3 cosθ dφ, +Ea +i ˜τi∂a = ˜pc τ3 sinθ ∂x + i ˜pb +Lo +τ2 sinθ ∂θ − i ˜pb +Lo +τ1 ∂φ. +(3.2) +Comparison with (2.1) reveals that one can arrive at solutions to the ‘constraint’ and +‘evolution’ equations in the exterior region simply by using the substitutions +b → i˜b, pb → i ˜pb +and +c → ˜c, pc → ˜pc +in the solutions of the interior region. Indeed, one can explicitly check that if one makes +these substitutions in the classical solutions (2.3) and (2.4), one obtains the Schwarzschild +metric in the exterior region. Therefore we can use these substitutions in the solutions +(2.9), (2.10) (2.11) to the effective equations in the interior to obtain the desired dynam- +ical trajectories in the exterior region, T > 0. They yield +3This strategy of using time-like 3-manifolds to specify fields and then ‘evolving’ them in space-like +directions was proposed and pursued in [39] for the Hamiltonian framework of full LQG. As discussed +there, in the full theory one encounters certain non-trivial technical difficulties associated with the fact that +SU(1,1) is non-compact. These issues do not arise in the homogeneous context discussed here. + +16 +tan +�δc ˜c(T) +2 +� += γLoδc +8m e−2T, +˜pc(T) = 4m2� +e2T + γ2L2 +oδ 2 +c +64m2 e−2T� +, +(3.3) +cosh +� +δb ˜b(T) +� += ˜bo tanh +�1 +2 +� +˜boT +2tanh−1 � 1 +˜bo +��� +, +(3.4) +where ˜bo = (1+γ2δ 2 +b )1/2 δb, δc are given by (2.13) as in Section II, and, +˜pb(T) = −2 sin(δc ˜c(T)) +δc +sinh(δb ˜b(T)) +δb +| ˜pc(T)| +γ2 − sinh2(δb ˜b(T)) +δ 2 +b +. +(3.5) +Thus, the explicit solutions in the c-sector have the same form as their counterparts (2.9) +in the interior region (T < 0) while in the b-sector the trigonometric functions of (bδb) +are replaced by their hyperbolic analogs. Details of derivations and a discussion of the +comparison between the classical and effective descriptions of the exterior region can be +found in [29]. +Let us conclude by specifying space-time geometry in the exterior region. The trans- +lational Killing field –which is time-like in the exterior region– is still given by ∂/∂x and +T is a radial coordinate that vanishes on the horizon and is positive in the exterior region. +For T > 0, the effective metric is given by +˜gabdxadxb = − +˜p2 +b +| ˜pc|L2o +dx2 + γ2| ˜pc|δ 2 +b +sinh2(δb˜b)dT 2 +| ˜pc|(dθ 2 +sin2 θdφ2). +(3.6) +The metric is well-defined in this region and has signature -,+,+,+. It fails to be well- +defined at T = 0 because b and pb vanish there. However, as we show below, this is just +a reflection of the breakdown of the coordinate system. In the limit ℓPl → 0 (or, ∆ → 0, +keeping γ positive), the quantum parameters δb and δc vanish and the metric (3.6) reduces +to the Schwarzschild metric in the exterior region. Properties of the geometry induced by +this effective metric are discussed in the next two subsections. +B +Quantum corrected, near horizon geometry +In this subsection we will briefly discuss two features of the near horizon geometry: +Matching of the effective metric across the horizon and corrections to the Hawking tem- +perature, computed using Euclidean (or rather, Riemannian) geometry. Further details +can be found in [30]. +Matching across horizon T = 0. Recall that in the classical theory, although the metric +appears to be ill-defined across the horizon, one can introduce Eddington-Finkelstein type +coordinates to make its regularity explicit. The same strategy can be adopted at the hori- +zon T = 0 of the effective metric. As in the classical case, one can ignore the angular part +of the metric. Then the relevant 2-metrics in interior and the exterior can be respectively + +17 +written in the form +dS2 +2 = f1(T)dx2 − f2(T)dT 2; +and +d ˜S2 +2 = − ˜f1(T)dx2 + ˜f2(T)dT 2, +(3.7) +where +f1(T) = +p2 +b +pcL2o +, f2(T) = γ2pc δ 2 +b +sin2(δbb) +and +˜f1(T) = +˜p2 +b +˜pc L2o +, ˜f2(T) = +γ2 ˜pc δ 2 +˜b +sinh2(δ˜b˜b) . (3.8) +As in the Eddington-Finkelstein extension in the classical case, one can approach the +horizon from the exterior region. Remembering that the coordinates (T,x) used for the +effective metric are the analogs, respectively, of the Schwarzschild coordinates (r,t), one +defines an advanced null coordinate v = x+ ˜T⋆ where +d ˜T⋆ = +� +˜f2/ ˜f1 +� 1 +2dT. +(3.9) +Then the metric in the exterior region becomes +dS2 +2 = ˜f1 dv2 −2( ˜f1 ˜f2) +1 +2 dvdT. +(3.10) +Since ˜f1 vanishes at T = 0, the space-time metric is well-defined at the horizon with +signature -,+,+,+ if and only if ˜f1 is smooth, and ˜f1 ˜f2 is smooth and positive in a neigh- +borhood of T = 0. This is indeed the case. In particular, +limT→0 ˜f1 ˜f2 = 4m2. In the +standard Schwarzschild coordinates (r, t) used in the classical theory, the product is 1, +and since r = 2meTclass, it is again 4m2 in the (Tclass, x) coordinates. In this sense the prod- +uct is the ‘same’ for the classical and the effective metric. The first derivative of ˜f1 differs +from its classical values by terms of the order 0(εm) where εm = (γ2L2 +0δ 2 +c )/64m2 and the +second derivative by terms of the order 0(εm, δ 2 +b ). For the metric coefficient ( ˜f1 ˜f2) +1 +2, they +are given by 2m and m +� +2 + γ2δ 2 +˜b +� +, respectively. If one approaches the horizon from the +interior, one finds that the limits of f1 and ( f1 f2) +1 +2 and their first two derivatives exist +and match with those coming from the exterior. Thus, the effective metric is (at least) +C2 across the horizon T = 0. Furthermore, the corrections to the metric coefficients are +negligible for macroscopic black holes. +In summary, although the effective 4-metric is constructed in the interior region T < 0 +using spatial homogeneity of a space-like foliation and in the exterior region T > 0 using +temporal homogeneity of a time-like foliation, and the x coordinates becomes ill-defined +at the horizon, as in the classical theory, there is a well-defined Eddington-Finkelstein +type chart (v,T) in which dS2 +2 is well-defined also at T = 0. Therefore the effective metric +can be extended across both the future and past horizons as in the classical Kruskal case +shown in Fig. 1. Furthermore, since the singularity is resolved, one can extend the metric +also across the new, anti-trapping horizons shown in Fig. 2. One can continue these +extensions to arrive at the Penrose diagram of 3 which extends indefinitely to the future + +18 +I +III +I′ +III′ +B +B +W +W +i0 +J + +J − +i0 +J + +J − +i0 +J + +J − +i0 +J + +J − +T +T +T +FIG. 3: The Penrose digram of the quantum extended Kruskal space-time in the x,T plane. Arrows show +the orientation of the static Killing fields. Since the effective metric is at least C2 across Killing horizons, +space-time continues indefinitely into the future and the past. Successive Killing horizons are trapping and +anti-trapping. Each diamond shaped region they bound is divided by a space-like transition surface T that +separates a trapping region (that lies to the past of T ) and an anti-trapping region (that lies to its future). +Thus, the region B immediately to the past of T resembles a black hole interior, and the region W +immediately to its future resembles a white hole interior. The area-radii of successive horizons are very +nearly equal for macroscopic black holes. Only a part of this extension is relevant to black holes formed +dynamically through collapse. + +19 +and to the past. +Quantum corrections to the Hawking temperature. In the classical theory one can +arrive at the Hawking temperature by passing to the Riemannian section via wick rotation +of the metric in the exterior region. In these considerations, it suffices to restrict oneself +to the r, t plane where the Riemannian metric ˜gab has the form +˜gabdxadxb = ˜f1(r)dt2 +E + ˜f2(r)dr2 +(3.11) +Since the norm of the translation Killing vector ˜f1(r) vanishes at the horizon in the +Lorentzian section and since the only vector that has vanishing norm is the zero vector in +Riemannian signature, the horizon shrinks to a point where the Killing vector vanishes. +In a neighborhood of this point, the static Killing field resembles a rotation, whence tE +becomes periodic with period P. This ‘rotational’ character of ta +E becomes manifest if +we set R = ( ˜f1(r))1/2 so that the metric on the r −tE plane becomes +˜gabdxadxb = R2 dt2 +E +4 +˜f1 ˜f2 +( ˜f ′ +1)2 dR2 . +(3.12) +The requirement that the metric be free of a conical singularity at the point R = 0 (where +the Killing Field vanishes) constrains the period P of tE to be +P = lim +R→0 +4π( ˜f1 ˜f2) +1 +2 +˜f ′ +1 += lim +R→0 +4π( ˜f1) +1 +2 +||D ˜f1|| +(3.13) +where the last step brings out the invariant nature of P since it involves only the norm +˜f1 of the Killing field and the norm of its covariant derivative. This periodicity implies +that Green’s functions satisfying standard boundary conditions in the Riemannian sector +have the same periodicity, which is used to endow the temperature TH = ℏ/(KP) to the +black hole through the relation between Lorentzian field theories and their Wick rotated +versions [49, 50]. For the classical Schwarzschild solution, we have P = 8πm, which +yields TH = ℏ/(8π K m) +This strategy can be directly applied to the effective metric (3.6) in the exterior region. +The Wick rotated, positive-definite metric in the (r, t) plane –i.e., now in the (T, x) plane– +becomes: +˜gabdxadxb = ˜f1(T)dx2 + ˜f2(T)dT 2 +with +˜f1 = +˜p2 +b +˜pc L2o +and +˜f2 = +γ2 ˜pc δ 2 +˜b +sinh2(δ˜b˜b) . +The horizon is at T = 0, where ˜pb and ˜b vanish in the effective solution. Regularity of +the metric follows from the properties of ˜f1 and ˜f1 ˜f2 discussed above. The period P +of (3.13) is now given by P = 8πm(1 + εm) where, as before, εm = (γ2L2 +0δ 2 +c )/64m2. +Therefore, the Hawking temperature of the quantum corrected black hole horizon is + +20 +TH = +ℏ +8πKm +1 +(1+εm) +(3.14) +The mass dependent correction 1/(1+εm) due to quantum geometry effects is very small +for macroscopic black holes. For a solar mass black hole it is of the order of ∼ 4×10−106. +Indeed, even for a black hole of ∼ 106MPl, the correction is of the order 10−21. (Because +there are inherent approximations in arriving at the effective theory, further extrapolation +to even smaller black holes would not be appropriate.) +As discussed in Section II, the quantum corrections to various curvature invariants are +very small near the horizon of macroscopic black holes. The correction εm to the Hawking +temperature provides another facet of that general phenomenon. +C +Asymptotic properties of the effective geometry +As we saw, the quantum gravity corrections are very small near horizons of macro- +scopic black holes. +Exact calculations have been done using MATHEMATICA in a +(large) neighborhood of the horizon as one recedes outwards and they show that quan- +tum corrections to the geometry become even smaller, as one would expect. However, +as one recedes further to asymptotic regions r ≫ 2m, the trend does not continue. The +main issue is tied with certain subtleties related to asymptotic flatness and the associated +Arnowitt, Deser, Misner (ADM) energy that are not widely appreciated and can lead to +confusion (for details, see [30]). +Let us therefore begin by recalling the elementary notion of asymptotic flatness. A +given metric gab is said to be asymptotically flat at spatial infinity if there exists a flat +metric ˚ηab such that in a Cartesian chart defined by ˚ηab, components of gab approach +the components of ˚ηab at least as fast as 1/r as r → ∞, keeping t,θ,ϕ constant (where +(t,r,θ,ϕ) refer to ˚ηab ). However, ˚ηab may not be the ‘obvious’ flat metric suggested +by the coordinates in which gab is presented. An obvious example is the 2-dimensional +metric ¯gab with the line element d¯s2 = −r2dt2 + dr2. The fact that ∂/∂t is the Killing +vector of the metric suggests that the coordinates t, r are ‘natural’, whence one may be +led to consider the flat metric ¯ηab with the line element ¯ηabdxadxb = −dt2 + dr2. One +would then conclude that the given metric ¯gab is not asymptotically flat because it does +not approach ¯ηab. Indeed, this conclusion may be further re-enforced by the fact that the +norm of the static Killing field diverges as r → ∞. But not only is ¯gab asymptotically +flat, it is in fact flat because ¯gab is just the Minkowski metric in the Rindler wedge. This +example brings out the fact that even a flat metric is generically not asymptotically flat +w.r.t. other flat metrics even in the elementary sense! Note, however, that for a given +metric gab to be asymptotically flat, it suffices to find one flat metric, say ˚ηab, to which it +approaches; it need not approach a pre-selected flat metric, like ¯ηab in the above example. +A more subtle example is provided by the Levi-Civita solution to Einstein’s equation +(known as the ‘c-metric’) [51] that, it turned out, represents the gravitational field of +two accelerating black holes [52]. In this solution, the norm of the Killing field ∂/∂t also +diverges at spatial infinity, and it too seems not to be asymptotically flat in the coordinates + +21 +it is normally presented in. (This feature led to considerable confusion on whether this +space-time admits gravitational radiation.) But the c-metric is in fact asymptotically flat +in the standard sense [53] (and does admit radiation); the form of the flat metric ˚ηab it +approaches at infinity is not obvious in the coordinates the c-metric is presented in. +With these preliminaries out of the way, let us return to the effective metric ˜gab of Eq. +(3.6) in the asymptotic region and ask if it asymptotically flat, keeping in mind the sub- +tleties discussed above. Now, b,c, pb, pc that enter the expression of ˜gab are complicated +functions of T. To make the asymptotic structure transparent, let us first set +rS := 2m, +r := rS eT, +and +ε := 1−b0 ≡ 1−(1+γ2δ 2 +b ) +1 +2 +(3.15) +and replace x by t so the translational Killing field is now ∂/∂t (rather than ∂/∂x). +For macroscopic black holes the dimensionless parameter ε is very small; for exam- +ple ε = 10−26 for a star mass black hole. Let us therefore assume that ε ≪ 1. Then +in the asymptotic region, where rS/r ≪ 1 and (γ2 rS ∆)1/3/2r ≪ 1, the exact expres- +sion (3.6) of the quantum corrected metric simplifies significantly: ˜gab ≈ ˜g◦ +abdxadxb = +˜g◦ +ttdt2 + ˜g◦ +rrdr2 +r2 dω2 , where, +˜g◦ +tt = − +� r +rS +�2ε � +1− +�rS +r +�1+ε � +and +˜g◦ +rr = +� +1− +�rS +r +�1+ε �−1 +. +(3.16) +Now, since ˜g◦ +tt –and hence ˜gtt– diverges as r → ∞, it is clear that the ‘obvious’ metric +does not approach the flat metric ˜ηabdxadxb = −dt2 + dr2 + r2dω2. Therefore, one may +be tempted to conclude that ˜g◦ +ab –and hence ˜g◦ +ab– is not asymptotically flat [54]. However, +as the examples of the Rindler and the c-metric show, the conclusion does not follow. +Rather, the question is whether there exists a flat metric ˜η◦ +ab to which ˜gab approaches as +r → ∞; this ˜η◦ +ab need not be the ‘obvious’ flat metric ˜ηab. +The answer turns out to be +in the affirmative [30]. To display its form, one has to replace t with τ = t(r/rS)ε (note +that τ agrees with t for ε = 0). Then, setting ˜η◦ +abdxadxb = −dτ2 +dr2 +r2dω2 one finds +that components of ˜gab approach those of ˜η◦ +ab as 1/r, ensuring asymptotic flatness of ˜gab. +As one would expect from this property, all curvature invariants of gab vanish as r → ∞. +Furthermore, this fall-off is sufficient to ensure that the ADM energy is well-defined. It +can be computed using the spatial Ricci tensor ˜ +Rab using an expression [55] that is often +used in the recent geometric analysis literature on the subject (see, e.g.,[56]). One finds +ERicci := lim +r→∞ +1 +8πG +� +r d2V rN ˜ +Rabˆraˆrb ≡ M(1+ε), +(3.17) +where d2V is the area element of the r = const 2-sphere of integration, ˆra a unit radial +vector, and M is the Schwarzschild mass of the classical solution. Thus there is a quantum +correction to the Schwarzschild mass, but it is minuscule for macroscopic black holes. +However, the fact that ˜gab does not approach the ‘obvious’ flat metic ˜ηab reflects a +limitation of its asymptotic behavior: the approach to flatness is not as strong as assumed +in the standard treatments of asymptotics (see, e.g. [55]) because, while the metric com- + +22 +ponents approach their flat space values as 1/r, not all components of the connection ˜∇ +defined by ˜gab fall-off as 1/r2. As a consequence several components of the space-time +curvature have weaker fall-offs than in the standard context. In particular, the curvature +invariants fall off only as 1/r4 rather than 1/r6. These deviations from standard asymp- +totic behavior have some subtle consequences. +Let us illustrate these subtleties with examples. As we just saw, the expression ERicci +of the ADM energy continues to be well-defined, and yields ERicci = M(1 + ε). One +can also carry out the calculation using the more familiar expression involving the 3- +metric, paying attention to the lapse defined by the Killing field [57]. One then finds +E3−metric = M, without any corrections. Similarly one can also evaluate the mass at the +horizon using its area Ahor, Mhor = (Ahor/16π)1/2 to find Mhor = M(1 + εm) where εm +is the mass dependent term that enters the expression (3.14) of the corrected Hawking +temperature we found in Section III B. For a solar mass black hole εm ≈ 10−106, much +smaller than the correction ε ≈ 10−26 that enters (3.17). All these quantities agree for +the classical Schwarzschild solution because the asymptotic fall-off is the standard one +[55]. Now, it often happens that notions that agree in a limiting theory (e.g., Newtonian +gravity) become ambiguous in a more complete theory (e.g., GR) and are thus replaced by +several different notions. It remains to be seen whether these findings associated with the +notion of energy are conceptually similar for the transition from GR to quantum gravity, +or if they are blemishes that point to a genuine limitation of the effective metric ˜gab in the +exterior region, that will be cured by a better candidate. As we will discuss in Section VI, +this issue is under active investigation in LQG. +IV +Quantum geometric effects in gravitational collapse: illustra- +tions +In Section II, we saw that the isometry between the Kantowski-Sachs space-time and +the Schwarzschild interior allows one to apply tools from LQC to the Schwarzschild +spacetime and permits one to study detailed physical implications. However, these stud- +ies have an inherent limitation: they can not capture the dynamics of a gravitational col- +lapse, resulting in a black hole. Models of gravitational collapse are significantly richer: +in contrast to eternal black holes, one now has a field theory, in which the time evolu- +tion of geometry and matter is coupled and governed by non-linear equations [33, 58– +68]. In this class of models, several investigations have been carried out to understand +the resolution of singularities associated with the dynamical collapse of homogeneous +dust in Oppenheimer-Snyder scenarios, in which the interior is modeled by a Friedmann, +Lemaˆıtre, Robertson, Walker (FLRW) cosmology [68–75]. This allows the application +of LQC techniques for the study of the fate of the classical singularity and yields similar +results on non-viability of certain quantization schemes. In particular, it turns out that +the ‘µo scheme’ on which early LQC was based –but subsequently ruled out on cosmo- +logical viability criteria [45, 76]– has novel limitations in the black hole sector: it does +not permit formation of trapped surfaces unless one chooses rather unnatural features of + +23 +quantum geometry [75]. This is an illustration of the fact that these models can provide +valuable insights, despite the limitations associated with their simplicity. +Another category of investigations considers dynamics of shells where the interior re- +gions is usually a patch of Minkowski spacetime, while the exterior is a Schwarzschild +geometry. They allow for the study of black hole formation, modeling the interior of the +star as a simple, empty, flat spacetime. At the quantum level, there is considerable litera- +ture on this topic (see for eg. [77] for a review). To understand quantum geometry effects +in this setting, a reduced phase space quantization of thin shells has been performed [78– +80]. One of these works shows that the classical singularity is eliminated, where the shell +either emerges through a white hole type geometry or tunnels into a baby universe inside +the black hole [78]. Another work proposes an effective semiclassical description moti- +vated by LQC quantization techniques for the study of a Lemaˆıtre-Tolman-Bondi (LTB) +spacetime, focusing on the dynamics of the outermost shell of matter [80]. Here, the sin- +gularity inside the black hole is resolved. Moreover, after black hole formation, matter +bounces, eventually ‘evaporating’ the black hole and dispersing towards infinity. There +are also studies that focus their attention to the search of an effective constraint algebra +that is free of anomalies, and include the so-called ‘inverse triad corrections’ [81, 82], +and ‘holonomy corrections’ [83, 84]. Finally, there have been studies to understand quan- +tum geometric effects on critical phenomena in the scalar field collapse discovered by +Choptuik [85] in classical GR [86–92]. +Given the richness and complexities of the underlying physics, at the present stage +these attempts aim at providing insights on specific aspects of the problem, rather than a +complete picture. To illustrate the overall status we will discuss two concrete examples in +some detail: the dust collapse scenario, and the critical collapse of a scalar field. The first +category of results focus on singularity resolution and therefore use horizon penetrating +coordinates. On the other hand, in the second category the focus is primarily on the +exterior region, whence it suffices to use coordinates that cover only that part of the space- +time. These examples are complementary in the following sense. In the first category, +geometry is treated quantum mechanically to start with, and induces quantum effects on +matter via field equations. In the second category, to begin with only matter is treated +quantum mechanically, and subsequently quantum features descend on geometry from +matter, again through field equations. +A +Dust field collapse models +In this subsection, we consider a few recent investigations [61, 62, 65, 67] that il- +lustrate the quantum modifications of classical dynamics. They use a reduced phase +space quantization with certain gauge fixing conditions in spherically symmetric space- +times, minimally coupled to an inhomogeneous dust field. The focus is on the family of +spherically symmetric Lemaˆıtre–Tolman-Bondi (LTB) spacetimes, and its sub-family of +Oppenheimer-Snyder (OS) models where the dust field is homogeneous. The approach is +inspired by the ‘improved dynamics’ strategy of LQC. In these models the matter sector +–dust– is not quantized but its dynamics is deeply influenced by the quantum nature of + +24 +underlying geometry, once it enters the high curvature regime. +The metric of LTB space-times is given by [65, 67] +ds2 +cl = −N2dt2 + Eϕ(t,x))2 +Ex +(dx+Nx +cl(t,x)dt)2 +x2dΩ2, +(4.1) +where x ∈ [0,∞) is the radial coordinate and ϕ is the azimuthal coordinate in spatial slices. +Let us restrict ourselves to the ‘marginally bound case’ where the spatial slices are flat. +In the Hamiltonian framework, one can gauge fix the momentum (or, diffeomorphism) +constraint by setting Ex = x2. Preservation of this gauge-fixing condition in time deter- +mines the shift Nx +cl in terms of the canonical variables: Nx +cl(t,x) = −N(Kϕ(t,x)/γ) where +Kϕ is the momentum conjugate to Eϕ. (Because the spatial slices are flat, Kϕ equals the +connection component Aϕ.) One can fix the lapse function N without loss of generality; +let us set N = 1 so that t represents proper time. The Hamiltonian constraint relates these +geometric variables to the matter density and determines evolution equations for Eϕ,Kϕ +through Poisson brackets [67]. +To pass to the effective theory, one sets β(t,x) = ( +√ +∆/x)Kϕ(t,x) and, motivated by +known results in LQC, one makes the ansatz: +Nx(t,x) = − +x +γ +√ +∆ +sin(β(t,x)) cos(β(t,x)) +(4.2) +(so that, in the limit area gap ∆ → 0 (keeping γ > 0), we recover the classical shift Nx +cl). +In the Painlev´e-Gullstrand like coordinates (for unit lapse), Eϕ is time independent, given +by Eϕ(x,t) = x. The Hamiltonian constraint and the evolution equation for Kϕ –which is +now encoded in β– are non-trivial: +ρ = +1 +8πGγ2∆x2 ∂x +� +x3 sin2 β +� +and +∂tβ = −4πGγ +√ +∆ ρ,. +(4.3) +As mentioned earlier, ρ is a classical field throughout this analysis; nonetheless it now +acquires an upper bound because of its coupling to quantum geometry. +Within this family of LTB spacetimes, it is interesting to analyze the subfamily of +OS solutions, those in which the energy density is homogeneous. The star is bounded +by the surface x = L(t), outside of which ρ(t) vanishes and inside of which ρ(t) is a +positive constant for each t. Thus, there is a finite discontinuity in ρ all along the boundary +x = L(t). Eq. (4.3) implies that β is continuous across the boundary but its time derivative +has a finite discontinuity there. One can now solve for the function ρ(t) to obtain +ρ(t) = +3GM +4π +� +L(t) +�3 +for x < L(t), +and +ρ(t) = 0 +for x > L(t). +(4.4) +The form of ρ(t) inside the star immediately implies an interesting relation that is remi- +niscent of the quantum corrected Friedmann equation of LQC [45]: + +25 +� ˙L +L +�2 += 8πG +3 ρ +� +1− ρ +ρc +� +, +(4.5) +for x(t) < L(t), where ρc = 3/8πGγ2∆, is again a universal constant. (A similar equation +of motion for the homogenous dust collapse was obtained in Refs. [68, 71, 75, 80].) At +the bounce, one has Lbounce = (2GMγ2∆)1/3; the value of the radius at which the bounce +occurs grows linearly with the mass of the star. In particular, while the density at the +bounce is of Planck scale irrespective of the mass of the star, for macroscopic black holes, +the radius at the bounce is not. For a solar mass black hole, for example, Lbounce ≈ 1013ℓPl. +This distinction is a robust feature of LQG. +Since the bounce of the effective theory replaces the classical singularity, one might +expect the subsequent dynamics to display richer structure. This is indeed the case. Soon +after the bounce, β(x,t) develops a discontinuity at the boundary. Therefore, it follows +from (4.3) that ρ(x,t) acquires a new term that is proportional to the delta distribution +δ(L(t)−x). Consequently, after the bounce the evolution equations have to be solved in +the distributional sense; one has weak solutions that solve integral equations obtained by +integrating the evolution equation w.r.t. x. When the shock wave meets the dynamical +horizon [41–43], it ceases to be a trapping horizon. Taking this instant of the time as the +end of the black hole, one can calculate its life time as the proper time interval, measured +by a distant observer, between the instant of formation of the dynamical horizon and its +disappearance. One finds: +Tlifetime ∼ 8πG2M2 +3γ +√ +∆ +. +(4.6) +Although in the above discussion we used the OS solutions to obtain this result, the scal- +ing Tlifetime ∝ M2 is more general in LQG. For example, it holds also for shell collapse +and the collapse of inhomogeneous dust (up to corrections linear in M) [67]. This life- +time contrasts with the suggestions of Tlifetime ∝ M that have appeared in the literature +[93–96], motivated by general quantum gravity considerations but based on less detailed +arguments. This possibility is ruled out by the LIGO discoveries of black hole mergers. +However, even with the M2 scaling, one is led to the some surprising conclusions. Recall +first that the life time of the black hole due to Hawking radiation goes as M3. Therefore, +if Tlifetime ∝ M2 were to be a firm prediction of a fully developed quantum gravity theory, +one would have to conclude that the Hawking evaporation process is physically unimpor- +tant since the black hole would disappeared before there is significant Hawking radiation. +Secondly, from an astrophysical standpoint, one knows that black holes were formed quite +early in the history of the universe. If there were any that formed with, say, lunar mass, +they would have disappeared and left us a signature of the shock wave accompanying the +bounce. It is more likely that the M2 scaling will be modified by more complete analyses +in the future. For example, the shift is chosen using an educated prescription and not ar- +rived at using some fundamental principles. In fact, recent investigations indicate that this +prescription differs from the one that arises from considerations of dynamical stability of +the effective gauge fixing conditions under the effective dynamics generated by the ‘poly- + +26 +merized’ canonical Hamiltonian [97]. The usefulness of the current LQG investigations +lies precisely in the fact they provide strong and concrete motivation to make the models +more and more realistic. +B +Quantum geometric effects in the critical phenomena +In the classical theory, there are two possible fates for the gravitational collapse of a +spherically-symmetric, minimally coupled, massless scalar field depending on the initial +data. One possible end state is that the field collapses to form a black hole, and the other +is that the field disperses to infinity. One can label each family of initial data of the field +by suitable parameters p, such that for p > p∗ the collapse leads to a black hole, and for +p < p∗ no black hole forms, i.e., the collapsing scalar field eventually disperses towards +infinity. For p ≃ p∗, it is possible to form black holes through a second order phase +transition with masses as close to zero as desired [85]. +More precisely, Choptuik demonstrated that the mass of the black hole depends on the +difference (p− p∗) via a universal power law mBH ∝ +��p− p∗��β, and there exists a discrete +self-similar behavior for p = p∗. It turns out that β ≈ 0.37 is a universal exponent which +is independent of the initial data. Further investigations have brought out a finer structure +over and above this power law relation [98]. Due to the discrete self-similarity one can +numerically observe echoes with a period whose ratio with β determines the periodicity +in the fine structure. Due to the scale invariance of the underlying equations there is no +mass gap for the formation of black holes in the classical theory; black holes can form +with arbitrarily small mass. +It is natural to ask: How does this universal phenomenon change when modifications +due to quantum geometric effects are included? In LQG investigations of such models, +the quantum modifications to the gravitational sector have different origins. The first pos- +sibility is to replace the inverse powers of triads using a classical identity to write them as +Poisson brackets between holonomies of the gravitational connection and the triads, and +then passing to the quantum theory by replacing the Poisson brackets with commutators +[99]. These quantum corrections are often referred to as ‘inverse triad modifications’. The +second possibility, explained in SectionII, is to express the field strength of the connec- +tion using holonomies around closed loops. These modifications are the ones responsible +for the bounce of the background effective geometry. In addition one can also treat the +matter sector using a polymer quantization [100]. While a complete treatment to study +the critical behavior of the scalar field including all these effects is yet to be performed, +explorations have been carried out to understand the modifications of the critical behavior +by including only the inverse triad modifications in Refs. [86–89], and by considering +LQG quantization of the scalar field in Refs. [90–92]. In all these models one assumes +the validity of the effective spacetime description resulting in dynamical equations en- +coding quantum geometry modifications. Due to inverse triad effects, the behavior of +matter-energy modifies the geometry in such a way that there is no divergence and, as a +result, the singularity is tamed [101]. Since inclusion of these modifications inevitably in- + +27 +troduces a length scale, the scale-invariance is broken. With these modifications, critical +phenomena is recovered albeit with a mass gap, below which a black hole can not form. +The value of this gap is determined by the discreteness scale in quantum geometry [87]. +The existence of mass gap on inclusion of inverse triad modifications can also be seen in +a more general collapse of the scalar field [101]. +In contrast, if one considers a quantum scalar field ´a la LQG, one obtains a set of +scale-invariant effective equations of motion [90–92]. Then the mass gap disappears, +allowing one to study of the effects of ‘polymer quantization’ of the scalar field during +the formation of black holes of very small masses. Since this treatment closely mirrors +the classical theory and, at the same time, captures ‘polymerization’effects in the matter +sector, we discuss it in some detail. The spacetime line element studied in [91] is given +by +ds2 = −N2(t,x)dt2 + +� +Eϕ(t,x) +�2 +Ex(t,x) +dx2 +Ex(t,x)dΩ2, +(4.7) +where one gauge fixes Ex = x2 to parallel the classical treatment by Choptuik. Its con- +jugate variable Kx(t,x) is fixed by the diffeomorphism constraint. The shift vector is +determined by demanding preservation of the gauge fixing condition in time. One also +uses the gauge freedom to set Kϕ(t,x) = 0 to maintain the diagonal form of the metric. +The dynamical variables are the triad Eϕ(t,x) and the lapse function N. With the matter +content as a scalar field (φ(t,x),Pφ(t,x)), the effective equations of motion are obtained +by ‘polymerizing’ the scalar field via φ → sin(kφ) +k +: +N′ +N − (Eϕ)′ +Eϕ ++ 2 +x − (Eϕ)2 +x3 += 0, +(4.8) +(Eϕ)′ +Eϕ +− 3 +2x + (Eϕ)2 +2x3 +−2πx +�� +Pφ +�2 +x4 ++ +� +φ′�2 cos2(kφ) +� += 0, +(4.9) +˙φ = 4πN +Eϕx Pφ, +(4.10) +˙Pφ = 4πx2 +Eϕ +��3NEϕ −xN (Eϕ)′ +N′Eϕx +Eϕ +� +φ′ cos2(kφ) ++xNφ′′ cos2(kφ)−xNk +� +φ′�2 cos(kφ)sin(kφ) +� +, +(4.11) +The lapse function can be determined from Eq. (4.8) (which is obtained by imposing +preservation in time of the gauge fixing condition Kϕ(t,x) = 0.) Finally, the Hamiltonian +constraint Eq. (4.9) determines the triad Eϕ(t,x). (Note that for k → 0, (and expressing +Eϕ(t,x) = xa(t,x)), one obtains the classical equations of motion of [85].) One can see +that these effective equations remain invariant under the transformation x → cx and t → ct +for constant c. Hence, there will be no mass gap, as in the classical theory. The coordinate + +28 +system used here cannot penetrate the horizon. Instead, the collapse of the lapse function, +namely N(t,x) → 0, is used to signal the formation of a black hole horizon. Numerical +simulations with these equations reveal existence of “wiggles” and “echoes” as in the +classical description [90, 91]. One finds that this effective theory shares the universality +of the scaling of the mass observed in the classical theory, up to small departures for large +values of the ‘polymer parameter ’k. The period of the discrete self-similarity seems to be +independent of the ‘polymerization parameter’ which indicates that the polymer effective +theory has a critical solution with the same periodicity as in the classical theory. +Let us conclude this section with a few remarks. In the investigations of the Kruskal +space-time reported in Sections II and III, detailed analysis of quantum corrections to the +geometry and their physical implications was made possible, thanks to the presence of a +4-dimensional symmetry group. Dynamical problems discussed in this section have only +spherical symmetry and therefore are much more difficult. Thus, various questions remain +unexplored. For instance, the quantization scheme (called ‘K-quantization’ [102, 103]), +used in [61, 62, 65, 67, 68] to arrive at effective equations governing the dust collapse, +is only valid for marginally bound cases. Secondly, there are indications [97] that one +may have to revisit the assumptions made while ‘polymerizing’ the Hamiltonian con- +straint, choices made in ‘polymerization’ of lapse and shift, and the issue of consistency +of gauge fixing conditions. Further, the choice of shift vector made in [61, 62, 65, 67] +and also in [33] seems to be problematic from the covariance of the effective geometries +[33]. Finally, there are also studies where another (‘non-polymeric’) quantization of these +classical models has been studied [104–111]. A detailed comparison of both quantization +schemes could add clarity on the physical viability and mathematical consistency of these +two complementary approaches. Similarly, in the investigations of the critical collapse of +scalar field, the role of quantum geometry in the gravitational sector is yet to be included +[90, 91]. If one were to introduce ‘polymerization’of the gravitational connection as in +the models for dust collapse, one will very likely introduce a length scale, breaking the +scale invariance and a mass gap would appear as in other works incorporating inverse triad +modifications [87, 101]. In explorations of the critical collapse, a more complete picture, +including quantum geometric effects in the gravitational sector, is not yet available. This +is an important gap as it is these quantum geometry effects that lead to singularity resolu- +tion. Despite such limitations, it is encouraging that these models have already provided +new perspectives on how quantum effects can manifest themselves in the dynamical pro- +cess of black hole formation and evolution, in the resolution of the classical singularity, +and in critical phenomena. +V +Black hole evaporation +Investigations reported in Section IV provide interesting insights into the nature of +quantum effects in dynamical situations leading to gravitational collapse. However, be- +cause of their underlying assumptions, they cannot address the issue of black hole evapo- +ration. In this section we turn to the LQG investigations of the Hawking process and the + +29 +associated issue of ‘information loss’. +In his original discussion [2] Hawking considered a test, scalar quantum field on a +classical space-time depicting gravitational collapse of a spherical star. Heuristic consid- +erations of the inclusion of the back reaction on space-time geometry led to the Penrose +diagram of Fig. 4 that is still widely used. In this diagram I + fails to be the complete +future boundary since the singularity is also a part of this boundary. One is then led to +the startling conclusion that quantum gravity considerations would force us to generalize +quantum physics by abandoning unitarity [112]. However, this line of reasoning has im- +portant limitations. The first comes from an elementary observation. For a self-consistent +discussion of unitarity, one needs a closed system. Thus, the incoming collapsing matter +in the distant past has to be represented by quantum fields, and the outgoing quantum state +in the distant future should refer to the same fields. This rather basic point is overlooked +in space-time diagram of Fig. 4 because the asymptotic Hilbert spaces do not include +the quantum state of matter in the star. The second issue is more subtle. In much of +the discussion on the subject, challenges and paradoxes arise because one assumes that +the quantum corrected space-time has an event horizon that encloses a trapped region +which is causally disconnected from the asymptotic region. This seems natural from the +perspective of the traditional Penrose diagram of Fig. 4. However, event horizons are tele- +ological and, as Hajicek pointed out already in 1987 [113], they can be shifted arbitrarily, +i− +i+ +uEH +i0 +I + +I − +Σi +Σf +FIG. 4: Commonly used Penrose diagram to depict black hole evaporation, including back reaction. +Modes are created in pairs, one escaping to I + and its partner falling into the black hole. The dashed line +is the continuation of the Event Horizon that meets I + at retarded time uEH. If this were an accurate +depiction, the evolution from I − to I + would fail to be unitary because the future singularity would act +as a ‘sink of information’. + +30 +and even completely removed, by changing the space-time geometry in a Planck scale +neighborhood of the singularity. Now, there is general consensus that classical GR cannot +be trusted in such neighborhoods. Therefore the assumption that the event horizon will +persist in quantum gravity has no obvious support. Indeed, LQG considerations suggest +that it will not. +In Section V A we explain how these two issues are addressed in the LQG literature. +In Section V B we summarize the current status of LQG investigations in semi-classical +gravity and expectations in full quantum gravity. In broad terms these investigations pro- +vide closely related avenues to realize the paradigm introduced in [40] based on singular- +ity resolution. Thus, from LQG perspective, non-singular black holes play a central role +in the discussion of the information loss issue. To anchor the discussion we will use the +approach developed in [114, 115]. A complementary discussion can be found in [139]. +A +Setting the stage +i− +i0 +uLR +I + +I − +T-DH +Σi +Σf +� +� +u0 +u +Σ +flat +FIG. 5: Semiclassical space-time: Black hole is formed by gravitational collapse of a pulse of scalar +field, depicted by the (gray) shaded region, incident from I −. A trapping Dynamical horizon T-DH is +formed. During the collapse, it is space-like and its area increases (in the outward direction). It becomes +time-like during evaporation and its area decreases (in the future direction). Hawking radiation starts in +earnest at u = u0. The dashed line with scissors that includes the last ray u = uLR represents the future +boundary of the semi-classical region. + +31 +A precise formulation of the issue of ‘information loss’ is provided by the question of +whether the S-matrix from I − to I + is unitary which, as we discussed, is relevant only +for closed systems. The simplest such system is a massless Klein-Gordon field coupled to +gravity. Consider, then, gravitational collapse of a spherically symmetric, massless scalar +field φ from I −. In the classical theory, if the infalling pulse of φ is narrow, the collapse +is prompt and analysis is not overly contaminated by the details of the pulse profile. The +solution has Minkowski metric ηab to the past of this narrow pulse and a Schwarzschild +black hole to its future. It is clear from the lower portion of Fig. 5 that the event horizon +first forms and grows in the flat portion of space-time. The actual collapse could occur +billions of years to the future! This is a concrete illustration of the teleological nature of +the event horizon (EH). In particular, it brings out the fact that the growth of the area of +the EH is not tied to any local physical process. +In the quantum theory, the pulse is replaced by a coherent state of the field ˆφ on +I +. In the semi-classical regime –which is expected to be valid in the region in which +space-time curvature is much smaller than the Planck scale– one can continue to describe +the quantum corrected geometry using a smooth metric. This portion of space-time is +depicted in Fig. 5, the region with Planck scale curvature in the future being excised. Let +us first focus on this region. The Hawking quanta of the quantum field ˆφ are emitted in +pairs; one escapes to I + and its partner falls into the black hole. The quantum state on a +Cauchy surface Σ of the semi-classical portion of space-time continues to be pure but there +is entanglement between the infalling and outgoing quanta. As for geometry, the space- +time metric to the past of the infalling pulse continues to be ηab. But to the future, it is no +longer given by the static Schwarzschild solution. The metric is dynamical not only within +the pulse but also to its future. Because of its dynamical nature, new structures emerge +that are directly relevant to the evaporation process: dynamical horizons. These are the +dynamical analogs of the trapping and anti-trapping horizons of the quantum corrected +Kruskal space-time discussed in Section II. They turn out to be more relevant than EHs +in discussions of black hole formation and mergers in numerical simulations in classical +GR and for the evaporation process in the quantum theory [41–43]. +Let us therefore briefly recall this notion. A dynamical horizon (DH) is a 3-dimensional +space-like or time-like submanifold that is foliated by 2-dimensional, surfaces S with 2- +sphere topology, such that the expansion of one of the null normals to each leaf S is zero +and that of the other null normal is either positive or negative everywhere. Thus, each +S is a marginally trapped surface (MTS). In an asymptotically flat space-time, we can +distinguish between the two null normals to S. Let us denote by la the outgoing null +normal and by na the ingoing null normal. On a black hole type DH, the expansion Θ(ℓ) +of the outgoing null normal vanishes (it is positive immediately outside and negative +immediately inside the MTS), while the expansion Θ(n) of the ingoing null normal is +negative (both outside and inside). Thus, immediately inside a black hole type DH, both +expansions are negative and we have a trapped region. Therefore, these DHs are called +trapping dynamical horizons, T-DHs. A T-DH is space-like when the area of the MTS +increases along the projection of la on DH, i.e. in the outward direction. In fact, there +is an explicit, precise relation between the growth of the area of a DH and the flux of + +32 +energy (carried by matter and/or gravitational waves) flowing into it [41]. Thus, not only +does the second law of black hole mechanics hold on T-DHs but the growth of the horizon +area is directly related to local physical processes. This is in striking contrast with the +situation for EHs, where we only have a qualitative statement of growth in classical GR: +Area of EHs cannot decrease. Indeed, it is not possible to directly trace the growth back +to the infall of energy locally because, as we just saw, EHs can form and grow in flat +space-time where there is nothing at all falling across it. +During the evaporation process, by contrast, the MTSs on the T-DH shrink, now in +response to the local negative energy flux across it, and the T-DH is time-like. Recall +that in the quantum corrected Kruskal space-time, we also have (white hole type) anti- +trapping horizons. But they emerge only when the space-time is extended across the +transition surface T on which curvature is of Plank scale. Therefore one would expect +that in dynamical situations, anti-trapping dynamical horizons AT-DH would also emerge +only when one extends space-time across a transition surface that replaces the classical +singularity. This expectation is correct. There is no AT-DH in the semi-classical space- +time Fig. 5 where the region with Planck scale curvature was excised by hand. However, +it is present in the quantum extended space-time depicted in Fig. 6. +On an AT-DH it is the expansion Θ(n) of the ingoing null normal that vanishes and the +expansion Θ(ℓ) of the outgoing null normal is positive. Thus, immediately inside these +horizons, both expansions are positive: we have an anti-trapped region. Since it is Θ(n) +that vanishes on any AT-DH, it is natural to investigate what happens to the area of the +MTSs as one moves along the projection of na on the AT-DH. If the AT-DH is space-like, +its area decreases (now in the inward direction) and if it is time-like its area increases +(now in the future direction). +Thus, the key differences between EHs and DHs can be summarized as follows. First, +EHs are teleological and can be located only after one has evolved the metric to infinite +future. DHs by contrast can be located quasi-locally and their properties have direct re- +lation to physical processes at their location. Second, EHs are null while DHs can be +space-like or time-like, and become null only when they become ‘isolated’ i.e. there is +no flux of energy across them. Third, nothing can ever escape to the ‘exterior’ region +from the trapped region enclosed by a black hole type EH and nothing can ever enter the +anti-trapped region bounded by a white hole type EH. While there are trapped surfaces +immediately inside a T-DH, one can send causal signals across a T-DH from inside to +outside (see Fig. 5). Similarly, there are future directed causal curves that traverse an AT- +DH from outside to inside (see Fig. 6). Finally, while there is no natural notion of mass +and angular momentum for cross-sections of EHs, there is one for the canonically defined +marginally trapped surfaces on DHs which, furthermore lead to the first and second laws +of black hole mechanics [41]. Discussions of quantum dynamics in LQG focus on DHs . +Much of the confusion about the evaporation process and ‘purification’ of the quantum +state melts away once EHs are deemphasized. + +33 +B +Black hole evaporation in LQG +LQG investigations of the semi-classical part of the quantum corrected space-time are +based on Fig. 5 and, although some of the detailed calculations are still in progress, the +overall understanding of structures in this space-time is quite satisfactory at a conceptual +level. To understand the structure of the future of this region one needs full quantum +gravity and, as in every other approach, several questions remain. But there is a general +consensus on a majority of issues. In this subsection we summarize this status. +Semi-classical Regime: Consider a coherent state Ψ of a quantum scalar field ˆφ, +peaked around an infalling classical pulse on I − and undergoing a prompt collapse. +Let us suppose that the ADM energy in the incoming state is of a solar mass, M⊙. When +the radius of the pulse has become sufficiently small, a trapping dynamical horizon T-DH +forms. In classical GR, this T-DH would only have a space-like component that grows +from zero radius till it has radius of 3km and then joins on to the null event horizon of +the same radius. Once the Hawking radiation starts and the back reaction is included, the +black hole shrinks. Initially the process is very slow because the ingoing negative energy +flux is extremely small. It takes some 1064 years for the black hole to shrink to lunar +mass Mmoon. However, even at the end of this long, adiabatic process, the black hole is +macroscopic. Therefore, from our discussion in Section II one would expect the quantum +gravity corrections to be sufficiently small for semi-classical considerations to suffice. Let +us focus on this phase of evaporation. In this phase, dynamics should be well-described +by equations +Gsc +ab = 8πGN ⟨ ˆTab ⟩ren +and +□ ˆφ = 0, +(5.1) +where Gsc +ab is the Einstein tensor of the semi-classical metric gsc +ab and the expectation value +of the renormalized stress-energy tensor is computed using the Heisenberg state Ψ. The +metric gsc +ab does include quantum corrections but they are induced by quantum matter +(rather than being dictated by the area gap considerations of quantum geometry). +These corrections to geometry are adiabatic and small. But the infalling negative en- +ergy flux introduces a qualitative difference in the horizon structure: Now the expanding, +space-like branch T-DH of the dynamical horizon joins on, not to a null event horizon as +in the classical case, but to the outer, time-like branch whose area decreases to the future +due to the negative energy flux carried by the Hawking ‘infalling partner modes’. These +two branches of the T-DH serve as the past boundary of a trapped region of the semi- +classical space-time (Msc, gsc +ab) of Fig. 5. During this long adiabatic process of ∼ 1064 +years, pairs of Hawking quanta are continually created, one going to I + and its partner +falling into the trapped region. These modes will be entangled whence, if one uses the +usual observable algebra based just at I +, the state would seem mixed, close to a thermal +state. +The issue of unitarity leads one to ask: When will the correlations be restored? For this +to happen, the partner modes would have to emerge from the trapped region and propagate +outward, restoring correlations at I + and ‘purifying’ the state there. Since the outer part +of the boundary T-DH of the trapped region is time-like, there is no causal obstruction + +34 +for these modes to continuously exit the trapped region across T-DH throughout the long +evaporation process; the standard causal obstructions associated with EHs do not apply. +This fact has been used in the literature to argue that there is no information loss issue +at all [116, 117]; purification could have occurred all along the evaporation process. But +this seemingly easy explanation is flawed. Examination of the renormalized energy flux +shows that throughout this process, there is only infall across T-DH in semi-classical +gravity. Thus, the lack of a causal obstruction for the partner modes to exit the trapped +region is not sufficient for the purification to occur in the semi-classical space-time. +In fact there is an apparent puzzle associated with the issue of information loss that +is already relevant in the semi-classical regime. Since Mmoon ∼ 10−7 M⊙, at the end of +this long evaporation process most of the initial ADM mass is carried away to I + by the +Hawking quanta. A back of the envelope calculation shows that a very large number N +(∼ 1075) of quanta escape to I + and all of them are correlated with the ones that fell +across T-DH . Therefore, at the end of the semi-classical process under consideration, one +would have to have a huge number N of quanta both at I + and in the trapped region, +but the mass associated with the trapped region is only 10−7 times that carried away by +the N quanta going out to I +. Furthermore, the radius of the outer part of T-DH has +shrunk to only 0.1mm – the Schwarzschild radius of a lunar mass black hole. How can +a T-DH with just a 0.1mm radius accommodate all these N quote? Even if we allowed +each mode to have the (apparently maximum) wavelength of 0.1mm, heuristically one +would need the horizon to have a huge mass –some 1022 times the lunar mass! While +these considerations are quite heuristic, one needs to face the conceptual tension: At the +end of the process under consideration, the trapped region has simply too many quanta +to accommodate, with a tiny energy budget. Such considerations have led to suggestions +that somehow ‘purification’ must begin already by Page time [118] when the T-DH has +lost only half its original mass of M⊙, and essentially completed by the time the T-DH has +shrunk down to the lunar mass. But this would imply that semi-classical considerations +must fail in apparently tame regimes due to unforeseen quantum gravity effects that are +relevant outside the horizons of astrophysical black holes! As we discussed in previous +sections, in LQG quantum gravity corrections are completely negligible near horizons of +macroscopic black holes. +The way out of the apparent paradox is that semi-classical theory itself predicts that the +geometry of the trapped region has some rather extraordinary features that had not been +noticed until relatively recently and not fully appreciated by the wider community even +now. Calculations of the stress-energy tensor on the Schwarzschild space-times confirm +the idea that, in semi-classical gravity there is a negative energy flux across the time- +like portion of T-DH such that MT-DH would decrease according to the standard Hawking +formula: dMT-DH/dv = −ℏ/(GMT-DH)2. (Indeed, this has been the basis of the standard +estimate that the evaporation time goes as ∼ M3 +ADM.) One can then argue that, in the phase +of evaporation from the solar mass to the lunar mass, the form of the space-time metric in +the trapped region of Fig. 5 is well approximated by the Vaidya metric: +ds2 = − +� +1− 2m(v) +r +� +dv2 +2dvdr +r2� +dθ 2 +sin2 θ dϕ2� +, +(5.2) + +35 +with m(v) = GMT-DH(v) decreasing very slowly. As we saw in Section II, corrections +due to quantum geometry are completely negligible in Schwarzschild interior until one +reaches Planck curvature and, as Fig. 5 shows, that region is excluded in the semi-classical +space-time under consideration. Thus, in the metric (5.2) quantum corrections are all in- +duced by quantum matter and encoded in m(v). To probe the geometry of the trapped +region, it is convenient to foliate it and two natural foliations have been used by the LQG +community: One defined by constancy of the Kretchmann scalar and the other by con- +stancy of the radius of metric 2-spheres [30, 114]. Each space-like slice is topologically +S2 × R and is itself foliated by round 2-spheres which can be labelled by values of the +advanced time coordinate v. Let us set v = 0 when MT-DH = 1M⊙ and v = v0 when +MT-DH = Mmoon. During this long process, the radius of the MTSs on the outer, time-like +part of AT-DH decreases from r|v=0 = 3km to r|v=v0 = 0.1mm. The surprising fact is +that as v increases the leaves develop longer and longer necks of length ℓN along the R +directions [30, 119, 120]. The ‘final leaf’ for the process under consideration starts at the +right end with v = v0. The length ℓN of this final leaf is astonishingly large: ℓN ≈ 1064 +light years for the first foliation and ℓN ≈ 1062 light years for the second! These astro- +nomically large lengths can result because the time the process takes is huge; 1064years +corresponds to ∼ 1053 times our cosmic history! +This enormous stretching is analogous to expansion in (an anisotropic) cosmology. +Recall that during the cosmic expansion –e.g. during inflation– the wavelengths of modes +get stretched enormously. This suggests that partner modes that fall into the trapped +region will also get enormously stretched during evolution from v = 0 to v = v0, as in +quantum field theory on an expanding cosmological space-time, and become infrared. +Can this phenomenon resolve the quandary of ‘so many quanta with so little energy’? The +answer is in the affirmative. With such infrared wavelengths, it is easy to accommodate +them in the trapped region with the energy budget only of Mmoon. Thus, even though +the outgoing modes carry away almost all of the initial mass M⊙ to I +, there is no +obstruction to housing all their partners in the trapped region on a slice Σ of Fig. 5 with +the small energy budget of just 10−7M⊙. This argument removes the necessity of starting +purification by Page time. In the LQG perspective, purification can be postponed to a +much later stage. +To summarize, in the semi-classical regime, there are apparent paradoxes associated +with the process of ‘purification’ that is necessary for dynamics to be described by a uni- +tary process. These disappear when one shifts the focus from event horizons to trapping +dynamical horizons and takes into account the time evolving geometry of the trapped +region. By and large the LQG community has adopted this view. +Beyond the semi-classical regime: When do quantum geometry effects become signif- +icant making the semi-classical approximation inadequate? The viewpoint in LQG is that +this happens when physically observable quantities such as curvature scalars and matter +density enter the Planck regime. This expectation was borne out in the investigation of +the Schwarzschild interior in Section II. Therefore, one would expect semi-classical con- +siderations to be valid well beyond the time when T-DH has shrunk to MT-DH = Mmoon +we considered in the above discussion, all the way till the curvature is, say, 10−6 times + +36 +the Planck curvature which corresponds to MT-DH ≈ 103MPl. LQG explorations of the +evaporation process beyond this stage are being carried out by different groups. The main +ingredients are: results on causal structure of the Schwarzschild interior summarized in +Section II, intuition derived from simpler models such as CGHS [121], conclusions drawn +from a long series of works (see, e.g., [122–131]) that posit a space-time structure for the +entire process and work out its consequences, strong consistency requirements on the +ensuing space-time geometry (see, e.g., [96]), and calculations based on the Vaidya met- +ric for the structure of space-time in the distant future [127, 132]. While there is broad +consensus on the overall picture, many open issues remain. We will now summarize the +current status. +To the future of the semi-classical region, curvature can exceed 10−6ℓ−2 +Pl , whence we +need full quantum gravity. This region with Planck scale curvature is depicted by the +shaded (pink) region in Fig. 6. (To the past and future of this region, semi-classical grav- +ity should yield a reasonable approximation.) In the shaded (pink) region geometry is +described by a quantum state Ψgeo and the difficult task is to evolve the quantum field +ˆφ on this quantum geometry. Fortunately, prior experience with other systems –such as +the propagation cosmological perturbations on the quantum FLRW geometry– suggests +a strategy that is applicable during the adiabatic phase of the Planck regime. The evapo- +ration process is adiabatic so long as mass-loss does not occur too rapidly, i.e., until the +radius of T-DH is ≈ 103ℓPl. At the end of this process, one enters a neighborhood of the +future endpoint of the T-DH depicted by a (red) blob, where the curvature is Planckian +and the process speeds up very rapidly, violating the adiabatic approximation. Let us first +discuss the adiabatic phase and then return to the (red) blob. +Prior results in LQC strongly suggest that the problem of propagating a quantum field +ˆφ on a quantum geometry represented by Ψgeo can be greatly simplified during the adi- +abatic phase: One can construct a smooth ˜gab that carries all the information in Ψgeo +that the dynamics of quantum fields ˆφ is sensitive to (see, e.g., [133]). ˜gab is called the +dressed metric. Thus, the difficult task of evolving quantum fields on quantum geometry +is reduced to that of evolving them on the space-time of the dressed metric ˜gab. Next, +the expectation from results of Section II is that the shaded (pink) region will contain a +transition surface T (w.r.t. ˜gab) that replaces the classical singularity and separates the +trapped region that lies to its past and the untapped region that lies to its future. The met- +ric ˜gab will capture two distinct effects: those that originate from quantum geometry and +feature the area gap ∆ (as in Section II), and those that are induced on ˜gab by the falling +quantum matter, dominated by the incident pulse of the scalar field at the left end of the +(pink) shaded region, and by the infalling Hawking quanta carrying negative energy as +one moves towards the right end. +Discussion of Section II strongly suggest that the first set of effects will decay rapidly +as we move away from Planck curvature into the semi-classical region. Therefore in the +semi-classical region, ˜gab will be well approximated by gsc +ab used there. As we move to +the future of the (pink) shaded region, one would encounter an anti-trapping dynamical +horizon AT-DH (see Fig. 6). The region enclosed by the transition surface T to the past +and AT-DH to the future would be anti-trapped as in Fig. 2. But now AT-DH would be + +37 +space-like rather than null and its area would not be constant, but decrease as one moves +left. Qualitatively this change in the structure of AT-DH from the one of Fig. 2 is parallel +to the change in the structure of the trapping horizon T-DH that we already discussed in +some detail. +Finally, the region to the future of AT-DH would also be well approximated by a +Vaidya metric, but now the outgoing one, expressed in terms of the retarded time coor- +dinate u in place of the advanced time coordinate v of Eq. (5.2). It will describe the +propagation of the infrared modes that will emerge from the AT-DH and arrive at I + at +very late times. (The metric in this region will be nearly flat because the total energy in +the scalar field is small and dispersed over very large spatial regions.) Recall that these +are the partner modes that fell into the horizon and were therefore entangled with the +outgoing modes that carried away most of the initial ADM mass. In the LQG scenario, +then, correlations are finally restored at I + where, in the end, the infalling modes also +arrive. The total energy carried by the two sets of modes is very different. But this is not +an obstruction for restoring correlations, i.e., for the ‘purification’ to occur. The timescale +of this purification process is very long, 0(M4) [127, 130, 132]. Purification can occur +much later than the page time because of the LQG singularity resolution. +The final picture is rather similar to the process of burning a piece of coal that is of- +i− +i0 +I + +I − +T-DH +i+ +u0 +u1 +u2 +LNS +Σ +τ +flat +AT-DH +FIG. 6: Quantum extension of the space-time in LQG. The classical singularity is replaced by a +Transition surface τ, to the past of which we have a trapped region, bounded in the past by a trapping +dynamical horizon T-DH, and to the future of which we have an anti-trapped region bounded by an +anti-trapping dynamical horizon AT-DH. Cauchy surfaces Σ develop astronomically long necks already in +the semi-classical region. The dark (red) blob at the right end of τ is a genuinely quantum region. + +38 +ten invoked in the discussion of black hole evaporation. Initially the piece of coal is in +a pure state. When lit, it emits photons and the energy they carry is well described by a +thermal state at high temperature. But the total state is pure because there are correlations +between the quantum state of outgoing photons and the left-over coal. As the fire extin- +guishes, there is very little energy left and the ashes emits photons with lower and lower +frequencies for a very long time. At the end of the process the cold ashes are in a pure state +and all photons have escaped. The late time, long wavelength photons are able to restore +the correlations that were apparently lost in the middle of the process (when the photon +spectrum seemed approximately thermal) even though the total energy they carry is small +compared to the energy carried by high frequency photons that were emitted earlier. +Note, however, that this scenario is incomplete because one still has to deal with the +very last part of the evaporation process, depicted by the red blob and the associated null +rays u = u1 and u = u2 of Fig. 6. In this region, not only is the curvature of Planck +scale, but it is varying extremely rapidly because it lies at the end point of the evaporation +process. It is this combination that makes the problem difficult; if we had only one of these +features, we could have used known approximation methods. Independent considerations +suggest that something very non-trivial must happen in this region. We will conclude +with an example. During the semi-classical phase, as the T-DH shrinks, the temperature +associated with the radiation at I + grows. Consequently, the modes become increasingly +ultraviolet as one approaches the point u = u1 on I +. On the other hand, the radiation that +emerges from the anti-trapped region is infrared and received at I + to the future of u = +u2. This is a dramatic transition, strongly suggesting that the physics of the region which +is highly dynamical and has Planck scale curvature will be very subtle and interesting. +For example, it has been suggested that Planck scale ‘seeds’ may be left behind, scattered +in this region [134]. Understanding the nature of this quantum geometry remains an +attractive challenge in LQG. +Let us summarize the current status of LQG investigations of black-evaporation. They +are distinguished by their emphasis on two features that are generally ignored in other +approaches: (i) A shift away from the teleological event horizons EH to quasi-locally +defined trapping and anti-trapping DHs T-DH and AT-DH ; and, (ii) replacement of the +classical singularity by the transition surface T . As a consequence, the traditionally used +Penrose diagram of Fig. 4 is replaced by the Penrose diagram of Fig. 6. +VI +Discussion +As the bibliography indicates, LQG literature on regular black holes is very rich. In- +deed, even this long list is far from being exhaustive! To make the material accessible to +non-experts, we focused on four lines along which advances have occurred, and in each +case built the discussion around a few of the mainstream developments. Much of this dis- +cussion is based effective equations, motivated by the fact that high performance compu- +tations have shown that effective space-time metrics provide an excellent approximations +to the quantum geometry in LQC, also in Bianchi models where the Weyl curvature is +non-zero and diverges at the singularity [135, 136]. + +39 +The first area, discussed in Section II, focuses on the ‘Schwarzschild interior’ that +contains the singularity. Since the resolution of this singularity is central to the theme of +‘regular black holes’ of this Volume, we included an account of several different effec- +tive descriptions. These investigations bring out two features: (i) singularity resolution +due to the underlying quantum geometry effects of LQG is robust, and does not depend +on details of the quantization methods; but, (ii) the precise manner in which quantiza- +tion is carried out can unleash unintended and physically undesirable effects that are not +apparent until a detailed examination is carried out. We summarized a scheme that is +free of these drawbacks. The resulting quantum corrected geometry exhibits interesting +causal structures: the singularity is replaced by a transition surface, T , to the past of +which there is a trapped region, and to the future, an anti-trapped region. Each is bounded +by null horizons and, for macroscopic black holes, the area of the future horizon is ap- +proximately equal to that of the past (see Fig. 2). In each of these effective geometries, +curvature scalars attain their maxima at the transition surface which, furthermore, have +universal values, independent of the mass of the black hole. This universality seems to be +a general feature of the singularity resolution due to quantum geometry effects of LQG. +Section III extended the quantum corrected geometry of Section II to the exterior, +asymptotic region of the Schwarzschild space-time by exploring the homogeneity of time- +like surfaces rsch = const. For macroscopic black holes (i.e., those with m ≫ ℓPl), the near +horizon geometry of this exterior has the expected and physically desired features: the +quantum corrected, effective metric is smooth across the horizon and corrections to the +Hawking temperature, computed using methods from Euclidean quantum field theory, are +tiny. More generally, for macroscopic black holes there is excellent agreement between +the effective geometry and that of the classical Schwarzschild metric in a vast neighbor- +hood of the horizon in the exterior region. Unfortunately, there is some confusion in the +literature on this point arising from the simplified form (3.16) of the metric that holds +in the far-asymptotic region, i.e., only on ignoring terms O(rs/r). If one overlooks this +key approximation and uses (3.16) in the entire exterior region –as was done in [137]– +one obtains “unsettling features”, such as non-trivial corrections to the innermost circular +orbit. These are consequences not of the actual effective geometry, but of the incorrect +use of its simplified form. (Nonetheless, unfortunately, incorrect conclusions of [137] +have been repeated in some of the subsequent literature, e.g. [138]). More generally, the +near horizon quantum corrections to astrophysical black holes will be very small to have +observable relevance in the foreseeable future (at least in the non-rotating case on which +most LQG investigations have focused so far).4 +The full metric in the exterior region is also asymptotically flat with curvature decay +4In this review, we did not touch on the issue of black hole entropy that arises in LQG by counting +microstates of the area operator that are compatible with parameters characterizing a given macroscopic +black hole (see. e.g., [139]. The possibility of testing discreteness of area using gravitational waves has +drawn considerable attention in the literature. It has been argued that the simplest area spectrum with +area eigenvalues given by knℓ2 +Pl (where n is an integer and k a constant), considered by Bekenstein and +Mukhanov [140], could be ruled out using data from a sufficiently large number of compact binary mergers. +But in LQG the area spectrum is not equidistant, it crowds exponentially, making the continuum an excellent +approximation very quickly. However, for small black holes the area eigenvalues are grouped, exhibiting a +band structure, and the separation between bands is O(ℓ2 +Pl). If this structure were to persist for large rotating +black holes, each band would serve as a proxy of the Bekenstein-Mukhanov eigenvalues and gravitational +observations would then lead to non-trivial constraints [141]. However, currently there is no evidence that +points to the persistence of bands for macroscopic areas. + +40 +that is sufficient for the ADM mass to be well defined (e.g., if one uses the expression +(3.17) in terms of the spatial Ricci tensor). For macroscopic black holes, quantum correc- +tions to the classical value are very small. However, the decay is slower than that in the +standard notion of asymptotic flatness. Consequently, different expressions of the ADM +mass, that must agree with one another exactly if the standard asymptotic conditions hold +[55], now differ by quantum corrections. Much more surprising is the feature that the +norm of the time-translation Killing field of the effective metric diverges at spatial infin- +ity! One’s first reaction would be that such deviations from standard asymptotic flatness +must lead to a plethora of physically inadmissible consequences. One test is provided by +quasi-normal modes. Do they exhibit a pathological behavior? A detailed investigation +[142] has shown that the potential which enters the quasi-normal mode analysis con- +tinues to be well-defined everywhere. One can then compute quasi-normal frequencies +using an approximation tailored to improving accuracy. The corrections to the classi- +cal result are found to be negligibly small. An independent investigation [143] provided +expressions for axial and polar perturbations, computed their quasi-normal frequencies +and found departures with respect to the classical theory; in particular, isospectrality is +broken. However, all these relative deviations from the classical predictions are only a +small-percent effect even for black holes as small as rS ∼ 103ℓPl, and they decrease with +the mass of the black hole, becoming completely negligible for macroscopic black holes. +These investigations also show that the metric passes the stability criterion for tensor and +massless scalar field perturbations. However, the infrared behavior of the potential is dif- +ferent from that in the classical Schwarzschild case, leading to a qualitative difference +in the power-law tails. These tails play an important role in the mathematical literature +but are not astrophysical significant because they occur after the waves are exponentially +damped in the quasi-normal ringing phase. In summary, at present it is not clear whether +counter-intuitive features associated with the asymptotic behavior of the effective metric +of [29] are indications that it may be inadmissible in the asymptotic region rS/r ≪ 1, or +if they are physically harmless. In view of this uncertainty, several investigations are ex- +ploring alternate ways of arriving at an effective metric that has the standard asymptotic +behavior (see, e.g., [33, 38]). +Another conceptual issue concerns covariance. There is a 4-metric in the full quantum +extended Kruskal space-time and results on singularity resolution, for example, refer to +curvature invariants; these considerations are all 4-dimensionally covariant. But to arrive +at the effective Einstein’s equations with quantum geometry modifications, one uses sym- +metry reduction. The question is whether there is a covariant action for the full theory +without symmetry reduction whose equations of motion reduce to those in Sections II and +III in its static, spherically symmetric sector [144]. This is a technically difficult issue that +is still open. Indeed, it took some time to show that the much simpler effective equations +of the homogeneous, isotropic sector of LQC [29] can be obtained in this way, but finally +the answer turned out to be in the affirmative [145]. A similar situation arose also in string +theory where it seemed for quite some time that the exact 1+1 dimensional stringy black +hole [146] did not arise from the symmetry reduction of a covariant action [147]. In the +end, it was shown that there is such an action but it requires inclusion of additional fields + +41 +[148]. There are some concrete indications that the situation is likely to be similar in the +LQG black hole sector we discussed; see e.g., [38] that introduces a covariant action in +the ‘mimetic gravity’ setting that adds a scalar field with a specific potential and uses the +same time-like homogeneous slices as in Section III in the symmetry reduced sector, and +[84] that uses a reasoning based on the constraint algebra to argue for covariance. +Discussion in Sections II and III was confined to the LQG treatment of the eternal black +hole and arrived at the Penrose diagram of Fig. 3 for the quantum extension of the Kruskal +space-time. This entire space-time is non-singular. In classical relativity as well as in +the discussion of black hole evaporation, Kruskal space-time of Fig. 1 provides useful +mathematical tools as well physical intuition. The same is true of its quantum extension. +However, realistic and more interesting situations involve formation of black holes by +gravitational collapse (for which only a part of the full Kruskal space-time is relevant). +In Section IV we focused on two complementary issues that have been investigated in +dynamical situations featuring gravitational collapse. The first involves the resolution of +singularity for collapsing dust models. Here, the emphasis is on quantum geometry effects +because the matter is characterized by dust rather than a fundamental quantum field. Thus, +non-classical features associated with matter –such as boundedness of the dust density– +are induced on matter by the quantum nature of geometry. These investigations show +that, as in cosmology, the singularity is replaced by a bounce; this is a robust result. The +second class of investigations focuses on critical phenomena. Now the strategy is the +opposite in that it is matter that is represented by a quantum scalar field of LQG [100] +while geometry is classical to begin with; corrections to classical effects on geometry +are induced by quantum matter through field equations. The overall finding is that the +quantum corrections to the classical results are small for macroscopic black holes, just as +one would hope. However, while there is no ‘mass-gap’ in the classical theory –i.e. a +black hole can be formed with arbitrarily small mass– a mass gap can develop if one uses +a quantization scheme that leads to effective equations violating scale invariance. +While dynamics is at the heart of these investigations, they do not encompass the +Hawking process because, in the first set of analyses matter is classical, and the second +focuses on critical behavior in gravitational collapse, rather than on the scalar field quanta +going out to I +, or the issue of entanglement. LQG investigations of the evaporation +process –including the issue of back reaction on geometry– were discussed in Section +V. They reflect a broad consensus that the arguments that lead to the traditional Penrose +diagram of Fig. 4 are flawed in two important respects. First, they assume that a part of +classical singularity persists in the quantum theory while it is resolved in LQG. +Second, the event horizon plays a key role in Fig. 4 even though it is teleological and +can be made to disappear by changing space-time geometry in a Planck scale neighbor- +hood of the singularity [113]. The traditional Penrose diagram is replaced by a new LQG +Penrose diagram shown in Fig. 6. There is consensus that there is no information loss: +The S-matrix from I − to I + is unitary provided, of course, we consider a closed system +in which the black hole forms by the gravitational collapse of a quantum field from I − +and we use the quantum state of the same field at I +. That the singularity would be +resolved by quantum geometry effects is motivated by two considerations: (i) Quantum + +42 +geometry effects discussed in Section II that provide universal upper bounds to curvature +scalars because of a non-zero value of the area gap; and, (ii) detailed numerical simula- +tions in the CGHS case that show that even in the semi-classical theory, the singularity is +significantly weakened when back reaction effects are included, which already suffice to +make the metric continuous there [149]. There is no EH in the final picture; what forms +classically in the gravitational collapse and evaporates through quantum processes is +a DH. +The evaporation process of LQG can be described as follows. The Hawking quanta are +created in pairs, the outgoing quanta go out to I + as in Hawking’s original paradigm, and +their partner quanta fall across the trapping dynamical horizon T-DH. In the semi-classical +regime depicted in Fig. 5, the outgoing quantum state is well approximated by a thermal +state at I + (at sufficiently late times), and the partner modes carry a negative energy flux +into the trapped region that is bounded by T-DH in this figure. When the back reaction +is included, the geometry in the trapped region changes adiabatically, and space-like sur- +faces Σ of Fig. 6 get stretched and become long necked surfaces (LNS). Suppose that at its +formation, the black hole has solar mass M⊙. Although the process of elongation of necks +is very slow, it continues for a very long time since the semi-classical phase lasts some +1064 years. At the end of this phase, the necks become astronomically long, stretched to +some 1062 light years! Therefore the modes that have fallen in the trapped region also +get enormously stretched (as they do during inflation) and become infrared. They can +continue to be entangled with their partner modes that went out to I + during this long +semi-classical phase –even though the total energy carried by the outgoing modes is al- +most M⊙ and that carried by the trapped modes is tiny– precisely because the trapped +modes are infrared. Thus, the quantum state on a surface such as Σ continues to be pure. +Since the singularity is resolved and replaced by a transition surface T that lies in the +shaded (pink) region, these modes can evolve across the transition surface T and emerge +on the other side. Then they propagate to the approximately flat region that lies to the +future of the anti-trapped dynamical horizon AT-DH, and arrive at I + restoring the cor- +relations with the partner modes that reached I + much earlier, during the semi-classical +phase. (As explained towards the end of section V B, this situation is qualitatively similar +to that of burning a piece of coal where correlations are restored at late times when the +large wavelength modes emerge from ashes as they cool down, restring correlations with +short wavelength modes emitted earlier.) +What remains largely unexplored so far is the (red) ‘blob’ at the right end to the shaded +(pink) region in Fig. 6 and how it affects the physics at I +. As discussed at the end of +Section V, the problem is hard because one simultaneously encounters two difficulties: +Planck scale curvature and rapid changes that make adiabatic approximation inadequate. +But feasible calculations may suffice to reveal whether most, if not all, of the correlations +are restored when the infrared modes traverse the anti-trapped region and emerge at I +. +If they are restored, then the fully quantum ‘blob’ would not be that relevant for the issue +of information loss and the S-matrix would be unitary. Although the consensus in LQG +favors this possibility, this issue is open. There are arguments involving a fully quantum +evolution from past of the blob to its future, but so far they are inconclusive because of the + +43 +underlying assumptions. At this stage, one cannot, for example, rule out the possibility +that the ‘blob’ joins on to a baby universe whose states are inaccessible from I + of +Fig. 6. If this were ti happen, from the perspective of I ± of this figure, information may +be lost, although the ‘total’ S-matrix would be unitary. Showing that this does not happen +remains a fascinating challenge in the LQG community. +Let us summarize. The LQG community has explored different aspects of the many +fascinating properties of black holes. The distinguishing feature of these investigations +is their emphasis on quantum geometry that is directly responsible for replacement of +the singularity by a transition surface with interesting causal properties, and boundedness +of physical observables such as curvature scalars and matter density. As discussed in +Section I, these features are not shared by other approaches: Using the AdS/CFT corre- +spondence as motivation, it is sometimes argued that singularities should persist also in +quantum gravity, and indeed, much of the literature uses the Penrose diagram of Fig. 4 +in which a singularity features as part of the future boundary of space-time. On the other +hand, because quantum geometry effects become important only in the Planck regime, +LQG corrections to the classical results are very small near the horizons of astrophysical +black holes; examples we discussed include corrections to the Hawking temperature using +the near horizon geometry and the machinery of Euclidean quantum field theory, correc- +tions to quasi-normal frequencies of astrophysical black holes, and to results associated +with critical collapse. This is also in striking contrast to some other approaches, e.g., +the ‘firewall scenario’ that emerged from string theory considerations. More generally, +LQG does not lead to violations of semi-classical expectations of physics near horizons +of astrophysical black holes that had been advocated before the LIGO discoveries showed +that predictions of classical GR, without such major corrections, are realized in compact +binary mergers. As mentioned in Section V B, there is also a large body of investigations +that posit a space-time structure for the entire process and work out its consequences. By +and large one solves classical Einstein’s equations (with suitable stress-energy tensors) in +various patches, and joins them consistently. Some of these space-time diagrams resem- +ble Fig. 6. While these investigations do pay close attention to consistency conditions, +and often also to energy considerations, the issue of quantum correlations and unitarity +received little attention in these works. The LQG line of reasoning of Section V fills this +conceptually important gap that is key to the issue of ‘information loss’. +There is also considerable discussion on the issue of young versus old black holes, +and long lived remnants. In LQG, there is indeed an important difference between a +young and an old black hole. As a concrete example, let us consider two lunar mass +black holes – a young one that is freshly formed from gravitational collapse, and an old +one what started out as a solar mass black hole and then evaporated down to the lunar +mass, as discussed in Section V B. While their dynamical horizons will have the same +radius, 0.1mm, and mass MT-DH = Mmoon, their external environment as well as internal +structure will be very different. In the second case, the evaporation process would have +gone on for some 1064 years. Therefore, there will be a very large number of outgoing +Hawking quanta in the exterior region, and an equal number of ingoing quanta in the +trapped region, the two being entangled. Therefore, the small area of T-DH will not be a + +44 +measure of the entropy of what is in the interior (or exterior). However, in LQG, in both +cases the area is a measure of the surface degrees of freedom of the horizon, i.e., degrees +of freedom that can communicate both the outside and inside regions. But it is sometimes +argued that there is a potential problem with this scenario: because old black holes can +have small energy but an enormous number of modes, it should be easy to produce them +in particle accelerators. But these arguments use only the conservation laws normally +used in computing scattering amplitudes; since old black holes have astronomically long +necks, it is hard to imagine how such changes in space-time structure can occur on time +scales of accelerator physics [150]. +Finally, let us discuss some of the limitations of the current LQG investigations. The +analyses we summarized make a strong use of symmetry reduced models and effective +equations that capture the leading order quantum corrections. However, there have been a +number of interesting investigations that aim at arriving at these effective equations start- +ing from full LQG (see, e.g., [151, 152]). But they are still in a rather preliminary stage, +and further and more detailed investigations are needed. Another key limitation is that so +far the LQG investigations have focused primarily on non-rotating black holes, where the +classical singularity is space-like. But for rotating black holes the inner horizons would +be unstable and therefore the singularity would be null. So far quantum geometry consid- +erations have not been applied to null singularities. This is an outstanding open problem. +Indeed, inclusion of rotating black holes in the discussion of ‘information loss’ during +evaporation remains a fascinating problem in all approaches to quantum gravity. +Acknowledgements +This work was supported in part by the NSF grant PHY-1806356, PHY-1912274 and +PHY-2110207, Penn State research funds associated with the Eberly Chair and Atherton +professorship, and by Projects PID2020-118159GB-C43, PID2019-105943GB-I00 (with +FEDER contribution), by the Spanish Government, and also by the “Operative Program +FEDER2014-2020 Junta de Andaluc´ıa-Consejer´ıa de Econom´ıa y Conocimiento” under +project E-FQM-262-UGR18 by Universidad de Granada. We would like to thank Eugenio +Bianchi, Kristina Giesel, Muxin Han, Bao-Fei Li, Guillermo Mena, Sahil Saini and Ed +Wilson-Ewing for discussions, and Tommaso De Lorenzo for Figures 4-6. +[1] N. Englehardt and G. T. Horowitz, Int. J. Mod. Phys. D25, 1643002 (2016). +[2] S. W. Hawking, [erratum: Commun. 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D 101, 026002 (2020). + diff --git a/QNAzT4oBgHgl3EQfW_wb/content/tmp_files/load_file.txt b/QNAzT4oBgHgl3EQfW_wb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..22f7e2485b37806720b4d1a6ff3f83f7b60dddc7 --- /dev/null +++ b/QNAzT4oBgHgl3EQfW_wb/content/tmp_files/load_file.txt @@ -0,0 +1,2109 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf,len=2108 +page_content='Regular black holes from Loop Quantum Gravity Abhay Ashtekar1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='* Javier Olmedo2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='† and Parampreet Singh3‡ 1 Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Institute for Gravitation & the Cosmos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Penn State,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' PA 16802,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' USA 2 Departamento de F´ısica Te´orica y del Cosmos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Universidad de Granada,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Granada-18071,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Spain 3 Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Louisiana State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Baton Rouge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' LA 70803,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' USA There is rich literature on regular black holes from loop quantum gravity (LQG),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' where quantum geometry effects resolve the singularity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' leading to a quantum extension of the classical space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As we will see, the mechanism that resolves the singularity can also trigger conceptually undesirable features that can be subtle and are often uncovered only after a detailed examination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, the quantization scheme has to be chosen rather astutely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We illustrate the new physics that emerges first in the context of the eternal black hole represented by the Kruskal space-time in classical general relativity, then in dynami- cal situations involving gravitational collapse, and finally, during the Hawking evaporation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The emphasis is on novel conceptual features associated with the causal structure, trapping and anti-trapping horizons and boundedness of invariants associated with curvature and matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This Chapter is not intended to be an exhaustive account of all LQG results on non-singular black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Rather, we have selected a few main-stream thrusts to anchor the discussion, and provided references where further details as well as discussions of related developments can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the spirit of this Volume, the goal is to present a bird’s eye view that is accessible to a broad audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='a I Introduction There is general agreement in the gravity community that black hole singularities of classical general relativity (GR) offer excellent opportunities to probe physics beyond Einstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, as of now, there is no consensus on the fate of black hole singularities in full quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Indeed, there is an ongoing debate even on a central question in the subject: Will singularities of classical GR be naturally resolved in full quantum gravity, or will they persist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As the very name of this Volume suggests, in many circles an affirmative answer is taken to be a necessary condition for the viability of a proposed quantum gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But this is not an universally accepted viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For example, it has been argued that taming of black hole singularities in asymptotically anti-deSitter aInvited Chapter for the book Regular Black Holes: Towards a New Paradigm of Gravitational Col- lapse, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Bambi, Springer Singapore (2023) ashtekar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='gravity@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='com † javolmedo@ugr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='es ‡ psingh@lsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='01309v1 [gr-qc] 3 Jan 2023 2 space-times would violate a “No Transmission Principle” motivated by the AdS/CFT cor- respondence [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' More generally, discussions of the black hole evaporation process are often based on the assumption that there is a singularity also in quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These expectations are based on the Penrose diagram of an evaporating black hole that Hawking drew over 40 years ago [2], where the singularity persists as part of the future boundary of space-time even after the black hole has completely disappeared (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' How- ever, this feature of the diagram was not arrived at from a calculation, and indeed such a calculation is not available even today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Furthermore, some forty years later Hawking him- self changed his mind: A new Penrose diagram was proposed to represent an evaporating black hole in which there is no singularity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 2 of [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Nonetheless, interestingly, Hawking’s first paradigm continues to feature prominently in discussions on the issue of information loss: see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [4] where the persistence of this singularity leads to a non-unitary evolution from I − to I +, and Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [5–7] where proposals are made on how unitarity could be rescued in spite of this singularity, thereby preventing information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Loop quantum gravity (LQG) provides a systematic avenue to investigate the fate of singularities of classical GR because it is based on quantum Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Con- sequently, new physics arises in the Planck regime where the continuum space-time of classical GR becomes inadequate (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Implications of this new physics have been analyzed in detail in the commonly used cosmological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Non-perturbative quantum corrections to Einstein’s equations imply that, once a curvature invariant ap- proaches the Planck scale, quantum geometry modifications of Einstein dynamics intro- duce strong ‘repulsive corrections’ that dilute that invariant, preventing a blow-up (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', [9–11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, the big-bang/big-crunch singularity is replaced by a quantum bounce in loop quantum cosmology (LQC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Once the curvature drops to about ∼ 10−4 Planck scale, quantum corrections can be neglected and classical GR becomes a good approxi- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A natural question then is whether the same phenomenon occurs at the black hole singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Results to date provide considerable evidence that it does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, tech- nically, the situation is more complicated than that in cosmological models for two rea- sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' First, even in the Schwarzschild solution, although space-time is homogeneous in the vicinity of the singularity, it is not isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Second, the nature of the blow up of curvature is different from that in the commonly used cosmological models: As Pen- rose has emphasized, while the Weyl curvature vanishes identically at the big-bang in homogeneous isotropic cosmologies, it diverges at the Schwarzschild singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As a result, although the singularity is resolved in all LQG investigations, as of now, results in the black hole sector are not as strong as they are in LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Nonetheless, a large num- ber of investigations, carried out since 2004, have provided conceptual insights as well as detailed technical results on the nature of the resolution of the Schwarzschild singu- larity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Our goal is to convey an overall picture at a technical level that is accessible to beginning researchers, emphasizing conceptual issues, novel elements, and problems that remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We also provide references where details can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Also for convenience of non-experts, throughout the Chapter, we pause to summarize the main points after each 3 technical discussion and also at the end of subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In Sections II and III we focus on the quantum extension of the Kruskal space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Because the static Killing field is space-like in the ‘interior’ region –bounded by the sin- gularity in the future and the horizon in the past– the space-time metric is spatially ho- mogeneous (but not isotropic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As is well-known, this portion of Kruskal space-time is isometric with the vacuum Kantowski-Sachs cosmological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore techniques from LQC have been used to analyze the fate of the Schwarzschild singularity in a number of investigations within LQG (See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=',[12–38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' While some of these analyses present us with the equations that dictate the evolution of the quantum state of the system, the detailed results are based on the so-called ‘effective equations’ whose goal is to incor- porate the leading order quantum corrections to the classical geometry in sharply peaked quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1 At a conceptual level, all these investigations follow the same strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, the technical implementation of this procedure differs, leading to different ef- fective geometries in the interior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Nonetheless, in all these cases, the singularity is resolved due to quantum corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We will discuss the strategy and compare and contrast various results in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Singularity resolution in the Kruskal space-time provides several sharp results on the causal structure of its quantum extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In particu- lar, the singularity is replaced by a ‘transition surface’ to the immediate past of which we have a trapped region and to the immediate future, an anti-trapped region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This geometry is sometimes referred to as depicting ‘a black hole to white hole transition’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We will avoid this terminology because it has other connotations that are not realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In particular, the terms ‘black hole’ and ‘white hole’ normally go hand in hand with singularities and event horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In LQG, singularities are absent and, in dynamical situations, there are also no event horizons either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In Section III we consider the Schwarzschild exterior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' the region bounded by the horizon and I ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Space-time is again foliated by homogeneous 3-dimensional surfaces but they are now time-like rather than space-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We discuss a possible extension of the ‘interior’ geometry to this exterior region, following [28, 29, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This extension has several attractive properties [30], but it also has some puzzling features: while the quantum corrected metric is again asymptotically flat in a precise sense (that suffices to define the ADM mass, for example), the approach to the flat metric is weaker than the one generally used in the physics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' There are alternate proposals to arrive at effective metrics with the standard asymptotic behavior (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', [33, 38]) but a definitive picture is yet to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Now, the Kruskal space-time itself is an idealization since it represents an ‘eternal black hole’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' black holes encountered in nature are formed dynamically, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', via a gravita- tional collapse, or compact binary mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Nonetheless, one would expect the qualitative features of the causal structure that arises from taming of the singularity due to quan- tum effects would be robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In Section IV we discuss models of dynamical situations 1For the conceptual framework underlying effective equations see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', Section V of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Note that the term ‘effective equations’ has a very different connotation here than in standard quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This has caused occasional confusion in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In LQG one does not integrate out ‘high energy modes’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Planck scale effects are retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In LQC, for example, there are states that remain sharply peaked even in the Planck regime and the effective equations capture the evolution of the peak of the quantum wave function in these states, ignoring the fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 4 that have been analyzed within LQG and summarize the current status, focusing on the Lemaˆıtre-Tolman-Bondi type models of collapse and critical phenomena discovered by Choptuik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In Section V we turn to the issue of black hole evaporation and ‘information loss’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The LQG discussion of these issues is characterized by two key features [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' First, as discussed above, in contrast to the Penrose diagram in Hawking’s seminal paper [2], there is no singularity in the space-time interior which can serve as a ‘sink of informa- tion’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Second, as the LQG Penrose diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6 shows, there is no event horizon: what forms and evaporates is a dynamical horizon [41–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Much of the discussion in the literature assumes that there is an event horizon which serves as a boundary of an ‘inte- rior’ region from which no causal signal can ever be sent to the asymptotic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One is then led one to either conclude that information is lost, or, to introduce ‘exotic’ ideas such as quantum Xerox machines, firewalls and fast scramblers to restore unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As we discuss, there is a more direct pathway to unitarity once it is realized that there is no event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, as in every other approach, important issues remain: the precise nature quantum radiation at the final stages of the evaporation process require full LQG and this analysis has only begun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We summarize the current status in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In Section VI we collect the key features of regular black holes in LQG compare and contrast the regular LQG black holes with this in other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Our conventions are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Space-time metric gab has signature -,+,+,+ and the curvature tensors are defined by Rabcdkd = 2∇[a∇b]kc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Rac = Rabcb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' and R = gabRab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' By macroscopic black holes we mean those for which GM =: m ≫ ℓPl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' II The Schwarzschild interior Denote by (M,gab) the Kruskal extension of the Schwarzschild metric (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 1) and by (MII, gab) the quadrant of this space-time that represents the (open) ‘interior region’ II, bounded by the black hole singularity and future horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This region is foliated by the rsch = const space-like manifolds, with topology S2 × R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Each leaf admits 3 rotational Killing fields tangential to its 2-dimensional spherical cross sections that are mapped to one another by the translational Killing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Consequently, (MII, gab) is spatially ho- mogeneous, but not isotropic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' it is isometric to the (vacuum) Kantowski-Sachs cosmolog- ical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore LQG approaches use the procedure from homogeneous cosmolo- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Now, while the big-bang and big-crunch singularities persist in the Wheeler-DeWitt (WDW) theory based on metric variables, they are naturally resolved in LQC because of the quantum geometry resulting from the use of connection variables (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For the Schwarzschild interior, then, LQG investigations also begin with a 3+1 decomposition of Einstein’s equations using connection variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the classical theory, components of the curvature tensor that features in these equations can be obtained by first evaluating holonomies of the gravitational connections around suitable closed loops (called plaque- ttes) and then taking the limit as the area enclosed by these plackets tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In LQG, the corresponding quantum operator is obtained by shrinking these plaquettes till the area they enclose reaches the smallest non-zero eigenvalue of the area operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This eigenvalue is called the area gap and denoted by ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As a consequence, information about 5 quantum geometry gets encoded in the dynamical equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Observables such as curva- ture scalars can acquire finite upper bounds on entire dynamical trajectories, whence the singularity is resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' ∆ appears in the denominator of the expressions of these upper bounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' classical singularities emerge as ∆ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For black holes, while operator equations have been written down [12–15, 34, 37], detailed investigations of the singularity resolution and ensuing quantum corrected geom- etry have been obtained using ‘effective equations’ discussed in Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Solutions to effective equations show that the central singularity is resolved due to quantum correc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, different investigations within LQG have made different choices to arrive at the quantum corrected curvature operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Intuitively these choices represent quanti- zation ambiguities that then affect detailed predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For brevity, in Sections II A and II B we will present the general framework and results following a recent approach that is free of limitations of the earlier investigations and in Section II C we will briefly compare and contrast other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Due to space limitation, by and large we will only include motivations behind various constructions and summarize the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For detailed derivations and other details, see in particular [12, 13, 19, 20, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A The framework In connection-dynamics, the initial data for space-time geometry consists of an SU(2)- valued connection Ai a and its conjugate ‘electric field’ Ea i as in Yang-Mills theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In I II III IV J J J J i i o 0 + + FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 1: The Penrose diagram of the Kruskal space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In this section we discuss the quantum extension of part II, bounded to the past by future horizons and the future by the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The quantum corrected effective geometry of region I is discussed in Section II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6 the final solutions to Einstein’s equations, Ai a has the interpretation of the gravitational connection that parallel transports SU(2) spinors, and Ea i , represent the ortho-normal spa- tial triads (with density weight 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Because of spatial homogeneity of the model, various spatial integrals in the Hamiltonian framework have a trivial divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, one introduces an ‘infrared cut-off’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus one truncates the homogeneous slices to be finite (rather than infinite) cylinders, with coordinates (θ,φ,x) with x ∈ (0, L◦) (rather than x ∈ (0, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One has to make sure, of course, that none of the final results depend on L◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One can solve the ‘kinematical’ constraint equations and use gauge-fixing to cast the basic variables in the form Ai a τi dxa = c/L◦ τ3 dx+b(τ2dθ −τ1 sinθ dφ)+τ3 cosθ dφ, Ea i τi∂a = pc τ3 sinθ ∂x +(pb/L◦)τ2 sinθ ∂θ −(pb/L◦) τ1 ∂φ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1) where τi are SU(2) generators related to Pauli spin matrices σi via τi = −iσi/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Real valued connection components b,c and the triad components pb, pc are functions only of time and serve as conjugate coordinates on the 4-dimensional phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It is conve- nient to choose an orientation of the triads so that b, c, pc are positive and pb is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1) that physical quantities can only depend on b, (pb/L◦), (c/L◦), pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Given a time coordinate τ that labels the spatially homogeneous surfaces and the corre- sponding lapse Nτ, in region II the space-time metric has the form gabdxadxb ≡ ds2 = −N2 τ dτ2 + p2 b pcL2◦ dx2 + pc(dθ 2 +sin2 θdφ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='2) At the horizon, b, pb vanish and the translation Killing field X = ∂/∂x becomes null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' When pc vanishes, the radius of the metric 2-spheres shrinks to zero, making the curvature scalars diverge there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is Schwarzschild singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It turns out that Einstein’s equations that govern the dynamics of the basic variables simplify significantly if one uses the lapse Ncl = (γ √pc)/b (which is different from the standard lapse in the Schwarzschild coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') The γ in this expression is the dimen- sionless Barbero-Immirzi parameter of LQG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It is analogous to the θ-parameter of QCD in that it represents a quantization ambiguity: classical physics is insensitive to the precise value of γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' we only need γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In terms of the corresponding time-coordinate Tcl, the dynamical trajectories are given by: b(Tcl) = γ � e−Tcl −1 �1/2 and pb(Tcl) = p(◦) b eTcl � e−Tcl −1 �1/2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='3) and c(Tcl) = c(◦) e−2Tcl and pc(Tcl) = p(◦) c e2Tcl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='4) Here c(◦), p(◦) b , p(◦) c are integration constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Comparison with the standard form of the Schwarzschild solution yields p(◦) c = 4m2, p(◦) b /L◦ = −2m, and c(◦)/L◦ = γ/4m, where m is related to the mass of the Schwarzschild solution via m = GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' At the horizon Tcl = 0 and at the singularity Tcl = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 7 The dynamical variables are subject to the Hamiltonian constraint Hcl[Ncl] ≡ − 1 2Gγ �� b+ γ2 b � pb + 2c pc � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='5) It is easy to verify that the terms in the b and c sectors on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='5) are separately conserved in time, and equal −m and m respectively on solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, if the constraint (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='5) is satisfied at one instant Tcl, then it holds for all T ∈ (−∞,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As explained above, in the passage to quantum theory the spatial curvature is expressed using the holonomoly of the gravitational connection Ai a around appropriately chosen pla- quettes that enclose the minimum non-zero area, ∆ = 4 √ 3πγℓ2 Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (Thus, while classical physics is insensitive of the value of the Barbero-Immirzi parameter γ, quantum physics is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Its value is generally taken to be γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='2375 via black hole entropy calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') As a consequence, the effective equations that capture the leading quantum corrections inherit new ‘quantum parameters’, denoted by δb and δc, that refer to edge lengths of these plackets, and go to zero in the classical limit, ℓPl → 0 (or, ∆ → 0, keeping γ fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Different choices of these quantum parameters represent quantization ambiguities men- tioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In this section we will use a strategy [28–30] that is free of the physically undesirable features encountered in other approaches (discussed in Section II C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A key idea behind this strategy is to use δb and δc that are ‘Dirac observables’ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' phase space functions that are constant along dynamical trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='2 Let us restrict ourselves to such δb, δc from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Then, again, the evolution equations simplify if we include the appropriate quantum corrections in the choice of the lapse, defining it as N := (γ √pc)δb/sin(δbb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (Note that as the area gap ∆ goes to zero, so does δb and N reduces to Ncl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') Denote by T the corresponding time parameter and by ‘dot’ the derivative with respect to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Then, as in the classical theory, the effective evolution equations b and the c sectors separate: ˙b = −1 2 �sin(δbb) δb + γ2δb sin(δbb) � , ˙pb = pb 2 cos(δbb) � 1− γ2δ 2 b sin2(δbb) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='6) and ˙c = −2 sin(δcc) δc , ˙pc = 2 pc cos(δcc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='7) But, again as in the classical theory, the two sectors are linked by the (now, effective) Hamiltonian constraint: Heff[N] ≡ − 1 2Gγ ��sin(δbb) δb + γ2δb sin(δbb) � pb +2sin(δcc) δc pc � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='8) 2Because the spatial curvature features on the right side of Einstein’s evolution equations, the quantum corrected version of the classical dynamical trajectories (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='4) along which δb and δc are to remain constant themselves feature δb and δc (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='9), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore the issue of finding δb and δc that are Dirac observables is rather subtle conceptually and quite intricate technically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These subtleties has led to some concerns [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This issue is analyzed in detail [35–37, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Consistency of the final results directly follows from the effective equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='6) - (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 8 A direct calculation shows that the constraint (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='8) is preserved in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To summarize, conditions ˙δb = 0, ˙δc = 0, the evolution equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='7) and the constraint equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='8) constitute a set of consistent equations that generalize the clas- sical constraint and evolution equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A notable difference from the classical theory arises because in LQG there is a well-defined operator in the quantum theory correspond- ing only to the holonomy defined by the gravitational connection Ai a, rather than Ai a itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As a consequence, only trigonometric functions of δb b and δc c appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Hence the do- main of these variables is compactified (just as in LQC [45]): they take values in the open interval (0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The momenta pb, pc, by contrast, continue to assume values pb < 0 and pc > 0 as in the classical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To solve the evolution equations, it is convenient to first obtain solutions c(T), pc(T) and b(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Now, in the c sector, equations of motion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='7) immediately imply that mc = (sin(δcc) pc)/(γL◦δc) is a constant of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This fact simplifies the form of the solu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One obtains tan �δc c(T) 2 � = γLoδc 8mc e−2T, pc(T) = 4m2 c � e2T + γ2L2 oδ 2 c 64m2c e−2T� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='9) cos � δb b(T) � = bo tanh �1 2 � boT +2tanh−1 � 1 bo ��� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='10) where there constant bo is given by bo = (1+γ2δb 2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One then uses the Hamiltonian constraint to determine pb(T): pb(T) = −2sin(δc c(T)) δc sin(δb b(T)) δb pc(T) sin2(δb b(T)) δ 2 b +γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='11) Eqs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='9) - (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='10) provide the dynamical trajectories of the effective theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It is easy to verify that in the limit δb → 0, δc → 0, one recovers the classical trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To sum- marize, the quantum corrected, effective trajectories are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='9) - (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='11) for any choice of constants of motion δb, δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since these equations only involve the combina- tions b, (pb/L◦), δb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (c/L◦), pc, and L◦δc, the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='2) and all physical results are insensitive to the choice of the infrared cut-off L◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' So far δb, δc could be any quantum parameters satisfying ˙δb = 0 and ˙δc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The following considerations provide a natural avenue to determine them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Recall that on classical solutions, the c part of Hcl[Ncl] equals m, and the b part equals −m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore in the effective theory, one is led to set 1 2γ �sin(δbb) δb + γ2δb sin(δbb) � pb L◦ = −mb and sin(δcc) γL◦δc = mc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='12) Equations of motion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='7) imply that both mb and mc are constants of motion and the effective Hamiltonian constraint reads mb = mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' On solutions, we will drop the suffix and set mb = mc = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The fact that mb and mc are constants of motion suggests a 9 natural strategy to restrict the form of δb,δc: Require that δb be a function only of mb, and δc be a function only of mc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To constrain the functional form requires additional input, summarized in Section II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Here we only note that the final answer has a rather simple form for large black holes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' for solutions for which m ≫ ℓPl): δb and δc are extremely well-approximated by δb = � √ ∆ √ 2πγ2mb �1/3 , and Loδc = 1 2 � γ∆2 4π2mc �1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='13) (Recall that physical results can only depend on the combination Loδc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') To summarize, the effective metric in the interior region is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='14), where c, pc are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='9), b, pb by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='11), and δb, δc by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' By inspection we see that as the area gap ∆ goes to zero, δb and δc both go to zero and the effective theory reduces to the classical GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' B Singularity Resolution, Causal Structure and Curvature Bounds Let us explore properties of the space-time metric gabdxadxb ≡ ds2 = −N2dT 2 + p2 b pcL2◦ dx2 + pc(dθ 2 +sin2 θdφ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='14) of the effective theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The past boundary of the open region under consideration is again given by b = 0, pb = 0 which occurs at T = 0 on every dynamical trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The translational Killing vector Xa becomes null at these points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' thus as in the classical theory, this boundary represents the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the classical theory, the singularity is characterized by the vanishing of the radius of the metric 2-spheres, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', of pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the effective theory, however, pc has a non-zero minimum, pmin c = 1 2γ(L◦δc)m which occurs at T = 1 2 ln(γL◦δc)/8m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Note that this minimum radius is of Planck scale but depends on the mass of the initial black hole: rmin ∼ (mℓ2 Pl)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is the surface that replaces the classical singularity and the space-time metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='14) can be smoothly extended across this 3-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One can explore the causal structure around this surface by calculating the expansions Θ± of the two null normals to the metric 2-spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To the past of this surface one finds that both expansions are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus this is a trapped region just as the entire region II is in the classical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Interestingly, both null-expansions vanish on this surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is a novel situation that is not encountered in classical GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since the metric is smooth across this surface, space-time is well-defined across it and one can analyze the two expansions to the future of this surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' They are both positive, so the region to the future is anti- trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus in the quantum-extended effective space-time, the surface neatly separates a trapped region and an anti-trapped region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore it is called a transition surface, denoted by T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It is analogous to the ‘bounce surface’ in LQC (that replaces the big- bang), to the past of which the expansion of the universe is negative and to the future of 10 which it is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, now the term ‘expansion’ refers to changes in the areas of metric 2-spheres along its two null normals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' How far into the future is the space-time extended by this procedure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The metric is well defined in the open region bounded by the surface T = −(4/bo)tanh−1 (1/bo) where (δb b) = π and pb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The Killing field Xa is again null on the boundary so it again represents a horizon that bounds the anti-trapped region to the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In summary, effective dynamics extends the open region II (of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 1) to the diamond shaped open region (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 2) bounded by Killing horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The region is separated by a transition surface T , to the past of which one has a trapped region and to the future of which, an anti-trapped region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This extension is often referred to as the black hole to white hole transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In LQC, space-time curvature attains the maximum value on the bounce surface and, furthermore, this upper bound is universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Does the quantum corrected geometry exhibit the same feature at transition surface T ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The answer is in the affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One has: R2 |T ≈ 256π2 γ4∆2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' RabRab |T = 256π2 γ4∆2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='15) CabcdCabcd |T = 1024π2 3γ4∆2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' RabcdRabcd = 768π2 γ4∆2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='16) where all the correction terms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' have the same form O � (∆/m2))1/3 ln(m2/∆) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Recall, first, that the classical limit corresponds to ∆ → 0 (keeping γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') Hence in this limit T AT T Σ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 2: Quantum extension of region II of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 1 in the effective theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The singularity is replaced by the transition surface T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It separates the trapped and anti-trapped regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The past boundary T is a null (black hole type) trapping horizon and the future boundary AT is a null (white-hole type) anti-trapping horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The time-like 3-manifold Σ joining 2-spheres lying on the two horizons is used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 11 all invariants diverge and T is replaced by the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Secondly, since leading terms are mass independent, the upper bounds are universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (The numerical coefficients vary simply because the invariants refer to distinct parts of the total curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') Third, as one moves away from T , these curvature scalars rapidly approach their classical values even for very small black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus quantum corrections to space-time geometry are very small away from the transition surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For instance, while the horizon radius of the effective solution is always larger than that of its classical counterpart, even for m = 104ℓPl, the relative difference is ∼ 10−15 and for a solar mass black hole, it is ∼ 10−115!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Finally one can ask for the relation between the radius rT of the trapping horizon that constitutes the past boundary of the diamond, and the radius rAT of the anti-trapping horizon that constitutes the future boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Are they approximately the same?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The answer is in the affirmative for macroscopic black holes, even though the ‘bounce’ is not exactly symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For a stellar mass black hole for example, rT = 3km and rAT = (3 + O(10−25))km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As we will see in Section II C, these consequences of effective dynamics are non-trivial: it is surprisingly difficult to achieve the singularity resolution without, at the same time, triggering unintended large effects away from the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Next, note that while the Ricci tensor vanishes identically in classical solutions, it is non-zero in the effective solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One can simply set 8πGN T eff ab := Rab − 1 2Rgab and interpret T eff ab as the effective stress-energy tensor of the quantum corrected space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As one would expect from the above discussion, for macroscopic black holes these quan- tum corrections are negligible away from T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, they become large and dominant in the immediate vicinity of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As one could have anticipated, although it is finite ev- erywhere, the energy density defined by T eff ab becomes large and negative in this region thereby violating the energy conditions, as it must for the singularity resolution to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Interestingly, this fact creates an apparent tension with considerations involving the Komar mass MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Recall that, in the classical theory, MK defined by the translational Killing field Xa is given by (half the) horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As we saw, for macroscopic black holes the radii rT and rAT are essentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But the difference between the Ko- mar mass evaluated at the anti-trapping horizon and the trapping horizon is given by the integral over a 3-manifold Σ joining a cross-section of the trapping horizon with a cross- section of the anti-trapping horizon (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 2), MAT K −M T K = 2 � Σ � T eff ab − 1 2T eff geff ab � XadΣb , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='17) and for macroscopic black holes the integrand of the right is large and negative near T (because it represents the effective energy density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' How can the two Komar masses be the same, then?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It turns out that the integrand of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='17) is indeed large and negative for macroscopic black holes, but its numerical value is very close to −2MT K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore the Komar mass associated with the anti-trapping horizon is given by MAT K ≈ MT K − 2MT K = −MT K, and the minus sign is just right because while the translational Killing field is future directed on the trapping horizon T, it is past directed on the anti-trapping horizon AT!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (See the (blue) arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') This resolution is another example of the conceptually subtle balance achieved with the choice of quantum parameters (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 12 To summarize, the Schwarzschild singularity is naturally resolved in the effective the- ory discussed in Section II A and region II of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 1 bounded by the singularity to the fu- ture is extended to the singularity free diamond-shaped region shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 2, bounded in the past by the trapping horizon and to the future by the anti-trapping horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The sin- gularity is replaced by a space-like surface T that marks the transition between trapped and anti-trapped regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Curvature scalars achieve their maximum values on T which are universal to the leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Although quantum corrections encoded in the area gap ∆ dominate near T , they decrease rapidly as one moves away and are completely neg- ligible near horizons for macroscopic black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In particular, the radii of the trapping and anti-trapping horizons are indistinguishable for macroscopic black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' C Summary of LQG Investigations As we already noted, the LQG investigations of the Schwarzschild singularity follow the same general steps but differ in the selection of the quantum parameters δb, δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since the Schwarzschild interior is isometric to the Kantowski-Sachs cosmological model, dis- cussions have often focused on issues motivated by cosmological considerations such as the behavior of ‘scalar factors’ and shears, rather than on considerations that are more directly relevant to black holes, in particular properties of the effective geometry that lead to trapping and anti-trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We focused on an approach that does [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We will now summarize various strategies that have been used to fix δb, δc and results they led to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since our goal is only to present a cohesive picture of the overall status through com- parison of results, the discussion will be rather brief;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' details can be found in the original papers listed in the bibliography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' By and large, these strategies fall into three categories: (i) The parameters are chosen to be constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These approaches are often referred to as the µo-type schemes because they mimic the strategy of using constant values for the quantum parameter µ used in LQC [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Here, the curvature operator is defined using holonomies of the gravitational connection around plaquettes and shrinking them till the coordinate area they enclose equals the area gap ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (ii) The parameters are chosen to be phase space functions, using physical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These approaches are often referred to as the ¯µ-type schemes, named after the strategy of selecting the quantum parameter µ in LQC [45] in which the curvature operator is defined by shrinking the plaquettes till the physical area they enclose equals ∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' and, (iii) The parameters are chosen to be phase space functions that are constants of motion on the effective dynamical trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The strategy used in the last two sub-sections falls in this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The earliest investigations [12–14] used strategy (i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' technically it is the simplest to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Here the quantum parameters were set to δb = δc = 2 √ 3 using ‘square’ plaquettes in coordinates adapted the symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Predictions of the resulting effective theory were analyzed in detail in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The singularity is again resolved and replaced by a 3-surface at which the symmetry 2-spheres attain the minimum area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, physical quantities such as the minimum value of the radius and the radius of the anti-trapping 13 horizon now depend on the infrared cutoff L◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Another limitation is that quantum effects can become significant even in the low curvature region near the horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the approaches [17, 18] based on strategy (ii), the quantum parameters were fixed by mimicking the successful ¯µ strategy from LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One again adapts the plaquettes to the symmetries of the problem, but shrinks them till the physical area they enclose is ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore the plaquettes themselves now depend on the phase space point under consider- ations and change under time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As a consequence, quantum parameters are spe- cific phase space functions that are not constant along dynamical trajectories: δb = ∆/pc and L2 δ 2 c = (L2 pc∆)/p2 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In these definitions, the dependence of L◦ is exactly the one that is needed to assure that physical results are independent of the fiducial choice of L◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is a significant improvement over results from strategy (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, a techni- cal complication arises because δb depends on pc and δc on pc: the equations in the b and c sectors no longer decouple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Consequently, it has not been possible to write down analytic solutions and all explorations to date have been performed numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These calculations show that the framework has two types of limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' First, as in (i), there are large deviations from the classical theory even when the curvature is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Second, when one evolves beyond the transition surface, the dynamical trajectory enters a region of the phase space where the metric 2-spheres have area that is less than the area gap ∆, making the scheme internally inconsistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Perhaps not surprisingly, then, some of the properties of the extended space-time are difficult to understand physically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Strategy (iii) was first adopted to improve on this situation by making δb, δc phase space functions that remain constant along dynamical trajectories [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Then the con- siderations of the first part of Section II A are applicable, the b and the c sectors separate, dynamical trajectories can be written down analytically, and mb and mc are constants of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the first investigation, δb and δc were chosen by dimensional considerations and by taking into account the fact that it is only the combination (L◦δc) that is invari- ant under the change of the infrared cutoff L◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The simplest expressions satisfying these requirements were then selected, (δb)2 := ∆/4m2 and L◦(δc)2 = ∆, without the consid- erations of plaquettes and holonomies of the gravitational connection around them [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The physical results are now invariant under rescalings of L◦ as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' There is again a transition surface T that separates the trapped and anti-trapped regions, and the quantum corrected space-time is a diamond bounded by a trapping horizon in the past and an anti- trapping horizon in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Furthermore, unlike the µo and ¯µ-type schemes, quantum corrections are small in regions near the horizons where the curvature is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, detailed examination revealed two limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' First, at the transition surface the Kretch- mann scalar of (initially) macroscopic black holes now goes as 1/m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' whence it decreases as the mass of m increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore for astrophysical black holes, large quantum correc- tions at the heart of the ‘bounce’ at T occur at low curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A second counter-intuitive result is involves ‘mass inflation’ across T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The radius rAT of the horizon in the future of T now goes as rAT = (rT)×(rT/ℓPl)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, if the initial black hole has solar mass with rT = 3km, one has rAT ≈ 1093Gpc!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The physical mechanism responsible for this huge magnification has remained unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, subsequently, more general choices of the quantum parameters were explored by introducing new dimensionless constants α 14 and β, setting (δb)2 := (α2 ∆)/4m2 and L◦(δc)2 = β 2∆ and varying α and β to ensure rAT ≈ (rT) for large black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Two choices satisfying this condition were found nu- merically and one analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The analytic expression implies that the leading term in Kretchmann scalar at T is not universal but grows rapidly with m as m4/∆4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The expressions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='13) used in Section II B to discuss results also fall under strategy (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, now the quantum parameters δb and δc are obtained using certain pla- quettes, holonomies around which are used to define the curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These plaquettes are tailored to the symmetries of the problem, and enclose physical area ∆ as in the strategy (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The key difference is that these loops are restricted to lie on the transition surface T [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since each dynamical trajectory intersects T once and only one, the prescrip- tion is unambiguous and, by construction, makes δb and δc constants of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This choice automatically leads to the result rAT ≈ (rT), without recourse to any additional free parameters (such as α, β discussed above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Furthermore, now the transition surface necessarily lies in the region where curvature is Planckian and the leading terms in the expressions of all curvature invariant are universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As this discussion shows, the task of choosing appropriate quantum parameters δb,δc is a very subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' While is it not difficult to make a ‘reasonable’ choice that resolves the singularity, the resulting quantum corrected geometry has to satisfy several non-trivial constraints to be physically admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Over the years, several choices have been pro- posed but the subsequent careful scrutiny by the LQG community showed that they lead to results that are physically unsatisfactory in one way or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The choice discussed in the last two subsections passes all the checks known to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' While this is satisfying, the analysis is still incomplete in one respect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In LQC the effective equations could be derived systematically starting from the operator equations of the quantum theory, show- ing that there are states that remain sharply peaked even in the deep quantum regime, and using expectation values of observables in these states [47, 48] (and Section V of [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus the LQC effective equations encode the dynamics of the peaks of these wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For black holes, the successful LQC techniques have been used in conjunction with an extended phase space framework (introduced in [29]) to arrive at the desired op- erator equations and to select physical states in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A systematic derivation of effective equations from this quantum theory remains an interesting open issue in LQG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' III The Schwarzschild exterior For the discussion of singularity resolution, it suffices to consider just the region II of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, initially the focus on LQG investigations was on this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, for a complete understanding of the quantum corrected space-time, one also has to con- nect the effective space-time geometry of region II to that of region I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In Section III A we present an approach to carry out this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Section III B summarizes the properties of the near-horizon quantum corrected geometry it provides, and III C discusses the asymptotic structure of the effective space-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As expected, for macroscopic black holes the near horizon geometry exhibits physically expected features because quantum corrections are 15 small there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the asymptotic region, on the other hand, this effective geometry has an unforeseen feature: while the quantum corrected metric is asymptotically flat in a precise sense, the approach to flatness is weaker than what one might have a priori expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We will discuss this issue and summarize its current status in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A The underlying framework Recall that the analysis of the Schwarzschild interior was greatly facilitated by the fact that this region is foliated by homogeneous, space-like slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The exterior region on the other hand does not admit such a foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, the four Killing fields do provide a natural foliation of this region by homogeneous, time-like slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Indeed the textbook derivation of the classical Schwarzschild metric can be interpreted as solving the ‘evolution equation’ in the r direction together with the ‘Hamiltonian’ constraint on the r = const homogeneous slices, mirroring the procedure used in the Schwarzschild interior (or, Kantowski-Sachs space-times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The main difference is that the signature of the intrinsic 3-metric on the homogeneous slices is now -,+,+ rather than +,+,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' There- fore in the connection framework one has to change the internal group that acts on the orthonormal triads from SU(2) to SU(1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='3 The generators τi that provide a basis for the Lie algebra of SU(2) are now replaced by ˜τi that constitute a basis for the Lie algebra of SU(1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The relation between the two is ˜τ1 = iτ1, ˜τ2 = iτ2, and ˜τ3 = τ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1) Hence, for exterior region we can choose our basic variables to be Ai a ˜τi dxa = ˜c Lo τ3 dx+i˜bτ2dθ −i˜bτ1 sinθ dφ +τ3 cosθ dφ, Ea i ˜τi∂a = ˜pc τ3 sinθ ∂x + i ˜pb Lo τ2 sinθ ∂θ − i ˜pb Lo τ1 ∂φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='2) Comparison with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1) reveals that one can arrive at solutions to the ‘constraint’ and ‘evolution’ equations in the exterior region simply by using the substitutions b → i˜b, pb → i ˜pb and c → ˜c, pc → ˜pc in the solutions of the interior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Indeed, one can explicitly check that if one makes these substitutions in the classical solutions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='4), one obtains the Schwarzschild metric in the exterior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore we can use these substitutions in the solutions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='9), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='10) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='11) to the effective equations in the interior to obtain the desired dynam- ical trajectories in the exterior region, T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' They yield 3This strategy of using time-like 3-manifolds to specify fields and then ‘evolving’ them in space-like directions was proposed and pursued in [39] for the Hamiltonian framework of full LQG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As discussed there, in the full theory one encounters certain non-trivial technical difficulties associated with the fact that SU(1,1) is non-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These issues do not arise in the homogeneous context discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 16 tan �δc ˜c(T) 2 � = γLoδc 8m e−2T, ˜pc(T) = 4m2� e2T + γ2L2 oδ 2 c 64m2 e−2T� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='3) cosh � δb ˜b(T) � = ˜bo tanh �1 2 � ˜boT +2tanh−1 � 1 ˜bo ��� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='4) where ˜bo = (1+γ2δ 2 b )1/2 δb, δc are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='13) as in Section II, and, ˜pb(T) = −2 sin(δc ˜c(T)) δc sinh(δb ˜b(T)) δb | ˜pc(T)| γ2 − sinh2(δb ˜b(T)) δ 2 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='5) Thus, the explicit solutions in the c-sector have the same form as their counterparts (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='9) in the interior region (T < 0) while in the b-sector the trigonometric functions of (bδb) are replaced by their hyperbolic analogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Details of derivations and a discussion of the comparison between the classical and effective descriptions of the exterior region can be found in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us conclude by specifying space-time geometry in the exterior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The trans- lational Killing field –which is time-like in the exterior region– is still given by ∂/∂x and T is a radial coordinate that vanishes on the horizon and is positive in the exterior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For T > 0, the effective metric is given by ˜gabdxadxb = − ˜p2 b | ˜pc|L2o dx2 + γ2| ˜pc|δ 2 b sinh2(δb˜b)dT 2 +| ˜pc|(dθ 2 +sin2 θdφ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='6) The metric is well-defined in this region and has signature -,+,+,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It fails to be well- defined at T = 0 because b and pb vanish there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, as we show below, this is just a reflection of the breakdown of the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the limit ℓPl → 0 (or, ∆ → 0, keeping γ positive), the quantum parameters δb and δc vanish and the metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='6) reduces to the Schwarzschild metric in the exterior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Properties of the geometry induced by this effective metric are discussed in the next two subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' B Quantum corrected, near horizon geometry In this subsection we will briefly discuss two features of the near horizon geometry: Matching of the effective metric across the horizon and corrections to the Hawking tem- perature, computed using Euclidean (or rather, Riemannian) geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Further details can be found in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Matching across horizon T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Recall that in the classical theory, although the metric appears to be ill-defined across the horizon, one can introduce Eddington-Finkelstein type coordinates to make its regularity explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The same strategy can be adopted at the hori- zon T = 0 of the effective metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As in the classical case, one can ignore the angular part of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Then the relevant 2-metrics in interior and the exterior can be respectively 17 written in the form dS2 2 = f1(T)dx2 − f2(T)dT 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' and d ˜S2 2 = − ˜f1(T)dx2 + ˜f2(T)dT 2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='7) where f1(T) = p2 b pcL2o , f2(T) = γ2pc δ 2 b sin2(δbb) and ˜f1(T) = ˜p2 b ˜pc L2o , ˜f2(T) = γ2 ˜pc δ 2 ˜b sinh2(δ˜b˜b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='8) As in the Eddington-Finkelstein extension in the classical case, one can approach the horizon from the exterior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Remembering that the coordinates (T,x) used for the effective metric are the analogs, respectively, of the Schwarzschild coordinates (r,t), one defines an advanced null coordinate v = x+ ˜T⋆ where d ˜T⋆ = � ˜f2/ ˜f1 � 1 2dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='9) Then the metric in the exterior region becomes dS2 2 = ˜f1 dv2 −2( ˜f1 ˜f2) 1 2 dvdT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='10) Since ˜f1 vanishes at T = 0, the space-time metric is well-defined at the horizon with signature -,+,+,+ if and only if ˜f1 is smooth, and ˜f1 ˜f2 is smooth and positive in a neigh- borhood of T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In particular, limT→0 ˜f1 ˜f2 = 4m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the standard Schwarzschild coordinates (r, t) used in the classical theory, the product is 1, and since r = 2meTclass, it is again 4m2 in the (Tclass, x) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In this sense the prod- uct is the ‘same’ for the classical and the effective metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The first derivative of ˜f1 differs from its classical values by terms of the order 0(εm) where εm = (γ2L2 0δ 2 c )/64m2 and the second derivative by terms of the order 0(εm, δ 2 b ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For the metric coefficient ( ˜f1 ˜f2) 1 2, they are given by 2m and m � 2 + γ2δ 2 ˜b � , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' If one approaches the horizon from the interior, one finds that the limits of f1 and ( f1 f2) 1 2 and their first two derivatives exist and match with those coming from the exterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, the effective metric is (at least) C2 across the horizon T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Furthermore, the corrections to the metric coefficients are negligible for macroscopic black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In summary, although the effective 4-metric is constructed in the interior region T < 0 using spatial homogeneity of a space-like foliation and in the exterior region T > 0 using temporal homogeneity of a time-like foliation, and the x coordinates becomes ill-defined at the horizon, as in the classical theory, there is a well-defined Eddington-Finkelstein type chart (v,T) in which dS2 2 is well-defined also at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore the effective metric can be extended across both the future and past horizons as in the classical Kruskal case shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Furthermore, since the singularity is resolved, one can extend the metric also across the new, anti-trapping horizons shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One can continue these extensions to arrive at the Penrose diagram of 3 which extends indefinitely to the future 18 I III I′ III′ B B W W i0 J + J − i0 J + J − i0 J + J − i0 J + J − T T T FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 3: The Penrose digram of the quantum extended Kruskal space-time in the x,T plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Arrows show the orientation of the static Killing fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since the effective metric is at least C2 across Killing horizons, space-time continues indefinitely into the future and the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Successive Killing horizons are trapping and anti-trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Each diamond shaped region they bound is divided by a space-like transition surface T that separates a trapping region (that lies to the past of T ) and an anti-trapping region (that lies to its future).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, the region B immediately to the past of T resembles a black hole interior, and the region W immediately to its future resembles a white hole interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The area-radii of successive horizons are very nearly equal for macroscopic black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Only a part of this extension is relevant to black holes formed dynamically through collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 19 and to the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Quantum corrections to the Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the classical theory one can arrive at the Hawking temperature by passing to the Riemannian section via wick rotation of the metric in the exterior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In these considerations, it suffices to restrict oneself to the r, t plane where the Riemannian metric ˜gab has the form ˜gabdxadxb = ˜f1(r)dt2 E + ˜f2(r)dr2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='11) Since the norm of the translation Killing vector ˜f1(r) vanishes at the horizon in the Lorentzian section and since the only vector that has vanishing norm is the zero vector in Riemannian signature, the horizon shrinks to a point where the Killing vector vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In a neighborhood of this point, the static Killing field resembles a rotation, whence tE becomes periodic with period P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This ‘rotational’ character of ta E becomes manifest if we set R = ( ˜f1(r))1/2 so that the metric on the r −tE plane becomes ˜gabdxadxb = R2 dt2 E +4 ˜f1 ˜f2 ( ˜f ′ 1)2 dR2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='12) The requirement that the metric be free of a conical singularity at the point R = 0 (where the Killing Field vanishes) constrains the period P of tE to be P = lim R→0 4π( ˜f1 ˜f2) 1 2 ˜f ′ 1 = lim R→0 4π( ˜f1) 1 2 ||D ˜f1|| (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='13) where the last step brings out the invariant nature of P since it involves only the norm ˜f1 of the Killing field and the norm of its covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This periodicity implies that Green’s functions satisfying standard boundary conditions in the Riemannian sector have the same periodicity, which is used to endow the temperature TH = ℏ/(KP) to the black hole through the relation between Lorentzian field theories and their Wick rotated versions [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For the classical Schwarzschild solution, we have P = 8πm, which yields TH = ℏ/(8π K m) This strategy can be directly applied to the effective metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='6) in the exterior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The Wick rotated, positive-definite metric in the (r, t) plane –i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', now in the (T, x) plane– becomes: ˜gabdxadxb = ˜f1(T)dx2 + ˜f2(T)dT 2 with ˜f1 = ˜p2 b ˜pc L2o and ˜f2 = γ2 ˜pc δ 2 ˜b sinh2(δ˜b˜b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The horizon is at T = 0, where ˜pb and ˜b vanish in the effective solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Regularity of the metric follows from the properties of ˜f1 and ˜f1 ˜f2 discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The period P of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='13) is now given by P = 8πm(1 + εm) where, as before, εm = (γ2L2 0δ 2 c )/64m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, the Hawking temperature of the quantum corrected black hole horizon is 20 TH = ℏ 8πKm 1 (1+εm) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='14) The mass dependent correction 1/(1+εm) due to quantum geometry effects is very small for macroscopic black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For a solar mass black hole it is of the order of ∼ 4×10−106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Indeed, even for a black hole of ∼ 106MPl, the correction is of the order 10−21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (Because there are inherent approximations in arriving at the effective theory, further extrapolation to even smaller black holes would not be appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') As discussed in Section II, the quantum corrections to various curvature invariants are very small near the horizon of macroscopic black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The correction εm to the Hawking temperature provides another facet of that general phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' C Asymptotic properties of the effective geometry As we saw, the quantum gravity corrections are very small near horizons of macro- scopic black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Exact calculations have been done using MATHEMATICA in a (large) neighborhood of the horizon as one recedes outwards and they show that quan- tum corrections to the geometry become even smaller, as one would expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, as one recedes further to asymptotic regions r ≫ 2m, the trend does not continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The main issue is tied with certain subtleties related to asymptotic flatness and the associated Arnowitt, Deser, Misner (ADM) energy that are not widely appreciated and can lead to confusion (for details, see [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us therefore begin by recalling the elementary notion of asymptotic flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A given metric gab is said to be asymptotically flat at spatial infinity if there exists a flat metric ˚ηab such that in a Cartesian chart defined by ˚ηab, components of gab approach the components of ˚ηab at least as fast as 1/r as r → ∞, keeping t,θ,ϕ constant (where (t,r,θ,ϕ) refer to ˚ηab ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, ˚ηab may not be the ‘obvious’ flat metric suggested by the coordinates in which gab is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' An obvious example is the 2-dimensional metric ¯gab with the line element d¯s2 = −r2dt2 + dr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The fact that ∂/∂t is the Killing vector of the metric suggests that the coordinates t, r are ‘natural’, whence one may be led to consider the flat metric ¯ηab with the line element ¯ηabdxadxb = −dt2 + dr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One would then conclude that the given metric ¯gab is not asymptotically flat because it does not approach ¯ηab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Indeed, this conclusion may be further re-enforced by the fact that the norm of the static Killing field diverges as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But not only is ¯gab asymptotically flat, it is in fact flat because ¯gab is just the Minkowski metric in the Rindler wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This example brings out the fact that even a flat metric is generically not asymptotically flat w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' other flat metrics even in the elementary sense!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Note, however, that for a given metric gab to be asymptotically flat, it suffices to find one flat metric, say ˚ηab, to which it approaches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' it need not approach a pre-selected flat metric, like ¯ηab in the above example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A more subtle example is provided by the Levi-Civita solution to Einstein’s equation (known as the ‘c-metric’) [51] that, it turned out, represents the gravitational field of two accelerating black holes [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In this solution, the norm of the Killing field ∂/∂t also diverges at spatial infinity, and it too seems not to be asymptotically flat in the coordinates 21 it is normally presented in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (This feature led to considerable confusion on whether this space-time admits gravitational radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') But the c-metric is in fact asymptotically flat in the standard sense [53] (and does admit radiation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' the form of the flat metric ˚ηab it approaches at infinity is not obvious in the coordinates the c-metric is presented in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' With these preliminaries out of the way, let us return to the effective metric ˜gab of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='6) in the asymptotic region and ask if it asymptotically flat, keeping in mind the sub- tleties discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Now, b,c, pb, pc that enter the expression of ˜gab are complicated functions of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To make the asymptotic structure transparent, let us first set rS := 2m, r := rS eT, and ε := 1−b0 ≡ 1−(1+γ2δ 2 b ) 1 2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='15) and replace x by t so the translational Killing field is now ∂/∂t (rather than ∂/∂x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For macroscopic black holes the dimensionless parameter ε is very small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' for exam- ple ε = 10−26 for a star mass black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us therefore assume that ε ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Then in the asymptotic region, where rS/r ≪ 1 and (γ2 rS ∆)1/3/2r ≪ 1, the exact expres- sion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='6) of the quantum corrected metric simplifies significantly: ˜gab ≈ ˜g◦ abdxadxb = ˜g◦ ttdt2 + ˜g◦ rrdr2 +r2 dω2 , where, ˜g◦ tt = − � r rS �2ε � 1− �rS r �1+ε � and ˜g◦ rr = � 1− �rS r �1+ε �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='16) Now, since ˜g◦ tt –and hence ˜gtt– diverges as r → ∞, it is clear that the ‘obvious’ metric does not approach the flat metric ˜ηabdxadxb = −dt2 + dr2 + r2dω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, one may be tempted to conclude that ˜g◦ ab –and hence ˜g◦ ab– is not asymptotically flat [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, as the examples of the Rindler and the c-metric show, the conclusion does not follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Rather, the question is whether there exists a flat metric ˜η◦ ab to which ˜gab approaches as r → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' this ˜η◦ ab need not be the ‘obvious’ flat metric ˜ηab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The answer turns out to be in the affirmative [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To display its form, one has to replace t with τ = t(r/rS)ε (note that τ agrees with t for ε = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Then, setting ˜η◦ abdxadxb = −dτ2 +dr2 +r2dω2 one finds that components of ˜gab approach those of ˜η◦ ab as 1/r, ensuring asymptotic flatness of ˜gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As one would expect from this property, all curvature invariants of gab vanish as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Furthermore, this fall-off is sufficient to ensure that the ADM energy is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It can be computed using the spatial Ricci tensor ˜ Rab using an expression [55] that is often used in the recent geometric analysis literature on the subject (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=',[56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One finds ERicci := lim r→∞ 1 8πG � r d2V rN ˜ Rabˆraˆrb ≡ M(1+ε), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='17) where d2V is the area element of the r = const 2-sphere of integration, ˆra a unit radial vector, and M is the Schwarzschild mass of the classical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus there is a quantum correction to the Schwarzschild mass, but it is minuscule for macroscopic black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, the fact that ˜gab does not approach the ‘obvious’ flat metic ˜ηab reflects a limitation of its asymptotic behavior: the approach to flatness is not as strong as assumed in the standard treatments of asymptotics (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [55]) because, while the metric com- 22 ponents approach their flat space values as 1/r, not all components of the connection ˜∇ defined by ˜gab fall-off as 1/r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As a consequence several components of the space-time curvature have weaker fall-offs than in the standard context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In particular, the curvature invariants fall off only as 1/r4 rather than 1/r6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These deviations from standard asymp- totic behavior have some subtle consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us illustrate these subtleties with examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As we just saw, the expression ERicci of the ADM energy continues to be well-defined, and yields ERicci = M(1 + ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One can also carry out the calculation using the more familiar expression involving the 3- metric, paying attention to the lapse defined by the Killing field [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One then finds E3−metric = M, without any corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Similarly one can also evaluate the mass at the horizon using its area Ahor, Mhor = (Ahor/16π)1/2 to find Mhor = M(1 + εm) where εm is the mass dependent term that enters the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='14) of the corrected Hawking temperature we found in Section III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For a solar mass black hole εm ≈ 10−106, much smaller than the correction ε ≈ 10−26 that enters (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' All these quantities agree for the classical Schwarzschild solution because the asymptotic fall-off is the standard one [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Now, it often happens that notions that agree in a limiting theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', Newtonian gravity) become ambiguous in a more complete theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', GR) and are thus replaced by several different notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It remains to be seen whether these findings associated with the notion of energy are conceptually similar for the transition from GR to quantum gravity, or if they are blemishes that point to a genuine limitation of the effective metric ˜gab in the exterior region, that will be cured by a better candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As we will discuss in Section VI, this issue is under active investigation in LQG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' IV Quantum geometric effects in gravitational collapse: illustra- tions In Section II, we saw that the isometry between the Kantowski-Sachs space-time and the Schwarzschild interior allows one to apply tools from LQC to the Schwarzschild spacetime and permits one to study detailed physical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, these stud- ies have an inherent limitation: they can not capture the dynamics of a gravitational col- lapse, resulting in a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Models of gravitational collapse are significantly richer: in contrast to eternal black holes, one now has a field theory, in which the time evolu- tion of geometry and matter is coupled and governed by non-linear equations [33, 58– 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In this class of models, several investigations have been carried out to understand the resolution of singularities associated with the dynamical collapse of homogeneous dust in Oppenheimer-Snyder scenarios, in which the interior is modeled by a Friedmann, Lemaˆıtre, Robertson, Walker (FLRW) cosmology [68–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This allows the application of LQC techniques for the study of the fate of the classical singularity and yields similar results on non-viability of certain quantization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In particular, it turns out that the ‘µo scheme’ on which early LQC was based –but subsequently ruled out on cosmo- logical viability criteria [45, 76]– has novel limitations in the black hole sector: it does not permit formation of trapped surfaces unless one chooses rather unnatural features of 23 quantum geometry [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is an illustration of the fact that these models can provide valuable insights, despite the limitations associated with their simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Another category of investigations considers dynamics of shells where the interior re- gions is usually a patch of Minkowski spacetime, while the exterior is a Schwarzschild geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' They allow for the study of black hole formation, modeling the interior of the star as a simple, empty, flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' At the quantum level, there is considerable litera- ture on this topic (see for eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [77] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To understand quantum geometry effects in this setting, a reduced phase space quantization of thin shells has been performed [78– 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One of these works shows that the classical singularity is eliminated, where the shell either emerges through a white hole type geometry or tunnels into a baby universe inside the black hole [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Another work proposes an effective semiclassical description moti- vated by LQC quantization techniques for the study of a Lemaˆıtre-Tolman-Bondi (LTB) spacetime, focusing on the dynamics of the outermost shell of matter [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Here, the sin- gularity inside the black hole is resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Moreover, after black hole formation, matter bounces, eventually ‘evaporating’ the black hole and dispersing towards infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' There are also studies that focus their attention to the search of an effective constraint algebra that is free of anomalies, and include the so-called ‘inverse triad corrections’ [81, 82], and ‘holonomy corrections’ [83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Finally, there have been studies to understand quan- tum geometric effects on critical phenomena in the scalar field collapse discovered by Choptuik [85] in classical GR [86–92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Given the richness and complexities of the underlying physics, at the present stage these attempts aim at providing insights on specific aspects of the problem, rather than a complete picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To illustrate the overall status we will discuss two concrete examples in some detail: the dust collapse scenario, and the critical collapse of a scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The first category of results focus on singularity resolution and therefore use horizon penetrating coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' On the other hand, in the second category the focus is primarily on the exterior region, whence it suffices to use coordinates that cover only that part of the space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These examples are complementary in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the first category, geometry is treated quantum mechanically to start with, and induces quantum effects on matter via field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the second category, to begin with only matter is treated quantum mechanically, and subsequently quantum features descend on geometry from matter, again through field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A Dust field collapse models In this subsection, we consider a few recent investigations [61, 62, 65, 67] that il- lustrate the quantum modifications of classical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' They use a reduced phase space quantization with certain gauge fixing conditions in spherically symmetric space- times, minimally coupled to an inhomogeneous dust field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The focus is on the family of spherically symmetric Lemaˆıtre–Tolman-Bondi (LTB) spacetimes, and its sub-family of Oppenheimer-Snyder (OS) models where the dust field is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The approach is inspired by the ‘improved dynamics’ strategy of LQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In these models the matter sector –dust– is not quantized but its dynamics is deeply influenced by the quantum nature of 24 underlying geometry, once it enters the high curvature regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The metric of LTB space-times is given by [65, 67] ds2 cl = −N2dt2 + Eϕ(t,x))2 Ex (dx+Nx cl(t,x)dt)2 +x2dΩ2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1) where x ∈ [0,∞) is the radial coordinate and ϕ is the azimuthal coordinate in spatial slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us restrict ourselves to the ‘marginally bound case’ where the spatial slices are flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the Hamiltonian framework, one can gauge fix the momentum (or, diffeomorphism) constraint by setting Ex = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Preservation of this gauge-fixing condition in time deter- mines the shift Nx cl in terms of the canonical variables: Nx cl(t,x) = −N(Kϕ(t,x)/γ) where Kϕ is the momentum conjugate to Eϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (Because the spatial slices are flat, Kϕ equals the connection component Aϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') One can fix the lapse function N without loss of generality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' let us set N = 1 so that t represents proper time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The Hamiltonian constraint relates these geometric variables to the matter density and determines evolution equations for Eϕ,Kϕ through Poisson brackets [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To pass to the effective theory, one sets β(t,x) = ( √ ∆/x)Kϕ(t,x) and, motivated by known results in LQC, one makes the ansatz: Nx(t,x) = − x γ √ ∆ sin(β(t,x)) cos(β(t,x)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='2) (so that, in the limit area gap ∆ → 0 (keeping γ > 0), we recover the classical shift Nx cl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the Painlev´e-Gullstrand like coordinates (for unit lapse), Eϕ is time independent, given by Eϕ(x,t) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The Hamiltonian constraint and the evolution equation for Kϕ –which is now encoded in β– are non-trivial: ρ = 1 8πGγ2∆x2 ∂x � x3 sin2 β � and ∂tβ = −4πGγ √ ∆ ρ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='3) As mentioned earlier, ρ is a classical field throughout this analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' nonetheless it now acquires an upper bound because of its coupling to quantum geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Within this family of LTB spacetimes, it is interesting to analyze the subfamily of OS solutions, those in which the energy density is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The star is bounded by the surface x = L(t), outside of which ρ(t) vanishes and inside of which ρ(t) is a positive constant for each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, there is a finite discontinuity in ρ all along the boundary x = L(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='3) implies that β is continuous across the boundary but its time derivative has a finite discontinuity there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One can now solve for the function ρ(t) to obtain ρ(t) = 3GM 4π � L(t) �3 for x < L(t), and ρ(t) = 0 for x > L(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='4) The form of ρ(t) inside the star immediately implies an interesting relation that is remi- niscent of the quantum corrected Friedmann equation of LQC [45]: 25 � ˙L L �2 = 8πG 3 ρ � 1− ρ ρc � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='5) for x(t) < L(t), where ρc = 3/8πGγ2∆, is again a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (A similar equation of motion for the homogenous dust collapse was obtained in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [68, 71, 75, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') At the bounce, one has Lbounce = (2GMγ2∆)1/3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' the value of the radius at which the bounce occurs grows linearly with the mass of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In particular, while the density at the bounce is of Planck scale irrespective of the mass of the star, for macroscopic black holes, the radius at the bounce is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For a solar mass black hole, for example, Lbounce ≈ 1013ℓPl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This distinction is a robust feature of LQG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since the bounce of the effective theory replaces the classical singularity, one might expect the subsequent dynamics to display richer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Soon after the bounce, β(x,t) develops a discontinuity at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='3) that ρ(x,t) acquires a new term that is proportional to the delta distribution δ(L(t)−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Consequently, after the bounce the evolution equations have to be solved in the distributional sense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' one has weak solutions that solve integral equations obtained by integrating the evolution equation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' When the shock wave meets the dynamical horizon [41–43], it ceases to be a trapping horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Taking this instant of the time as the end of the black hole, one can calculate its life time as the proper time interval, measured by a distant observer, between the instant of formation of the dynamical horizon and its disappearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One finds: Tlifetime ∼ 8πG2M2 3γ √ ∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='6) Although in the above discussion we used the OS solutions to obtain this result, the scal- ing Tlifetime ∝ M2 is more general in LQG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For example, it holds also for shell collapse and the collapse of inhomogeneous dust (up to corrections linear in M) [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This life- time contrasts with the suggestions of Tlifetime ∝ M that have appeared in the literature [93–96], motivated by general quantum gravity considerations but based on less detailed arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This possibility is ruled out by the LIGO discoveries of black hole mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, even with the M2 scaling, one is led to the some surprising conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Recall first that the life time of the black hole due to Hawking radiation goes as M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, if Tlifetime ∝ M2 were to be a firm prediction of a fully developed quantum gravity theory, one would have to conclude that the Hawking evaporation process is physically unimpor- tant since the black hole would disappeared before there is significant Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Secondly, from an astrophysical standpoint, one knows that black holes were formed quite early in the history of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' If there were any that formed with, say, lunar mass, they would have disappeared and left us a signature of the shock wave accompanying the bounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It is more likely that the M2 scaling will be modified by more complete analyses in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For example, the shift is chosen using an educated prescription and not ar- rived at using some fundamental principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In fact, recent investigations indicate that this prescription differs from the one that arises from considerations of dynamical stability of the effective gauge fixing conditions under the effective dynamics generated by the ‘poly- 26 merized’ canonical Hamiltonian [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The usefulness of the current LQG investigations lies precisely in the fact they provide strong and concrete motivation to make the models more and more realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' B Quantum geometric effects in the critical phenomena In the classical theory, there are two possible fates for the gravitational collapse of a spherically-symmetric, minimally coupled, massless scalar field depending on the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One possible end state is that the field collapses to form a black hole, and the other is that the field disperses to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One can label each family of initial data of the field by suitable parameters p, such that for p > p∗ the collapse leads to a black hole, and for p < p∗ no black hole forms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', the collapsing scalar field eventually disperses towards infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For p ≃ p∗, it is possible to form black holes through a second order phase transition with masses as close to zero as desired [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' More precisely, Choptuik demonstrated that the mass of the black hole depends on the difference (p− p∗) via a universal power law mBH ∝ ��p− p∗��β, and there exists a discrete self-similar behavior for p = p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It turns out that β ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='37 is a universal exponent which is independent of the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Further investigations have brought out a finer structure over and above this power law relation [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Due to the discrete self-similarity one can numerically observe echoes with a period whose ratio with β determines the periodicity in the fine structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Due to the scale invariance of the underlying equations there is no mass gap for the formation of black holes in the classical theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' black holes can form with arbitrarily small mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It is natural to ask: How does this universal phenomenon change when modifications due to quantum geometric effects are included?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In LQG investigations of such models, the quantum modifications to the gravitational sector have different origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The first pos- sibility is to replace the inverse powers of triads using a classical identity to write them as Poisson brackets between holonomies of the gravitational connection and the triads, and then passing to the quantum theory by replacing the Poisson brackets with commutators [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These quantum corrections are often referred to as ‘inverse triad modifications’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The second possibility, explained in SectionII, is to express the field strength of the connec- tion using holonomies around closed loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These modifications are the ones responsible for the bounce of the background effective geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In addition one can also treat the matter sector using a polymer quantization [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' While a complete treatment to study the critical behavior of the scalar field including all these effects is yet to be performed, explorations have been carried out to understand the modifications of the critical behavior by including only the inverse triad modifications in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [86–89], and by considering LQG quantization of the scalar field in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [90–92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In all these models one assumes the validity of the effective spacetime description resulting in dynamical equations en- coding quantum geometry modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Due to inverse triad effects, the behavior of matter-energy modifies the geometry in such a way that there is no divergence and, as a result, the singularity is tamed [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since inclusion of these modifications inevitably in- 27 troduces a length scale, the scale-invariance is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' With these modifications, critical phenomena is recovered albeit with a mass gap, below which a black hole can not form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The value of this gap is determined by the discreteness scale in quantum geometry [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The existence of mass gap on inclusion of inverse triad modifications can also be seen in a more general collapse of the scalar field [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In contrast, if one considers a quantum scalar field ´a la LQG, one obtains a set of scale-invariant effective equations of motion [90–92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Then the mass gap disappears, allowing one to study of the effects of ‘polymer quantization’ of the scalar field during the formation of black holes of very small masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since this treatment closely mirrors the classical theory and, at the same time, captures ‘polymerization’effects in the matter sector, we discuss it in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The spacetime line element studied in [91] is given by ds2 = −N2(t,x)dt2 + � Eϕ(t,x) �2 Ex(t,x) dx2 +Ex(t,x)dΩ2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='7) where one gauge fixes Ex = x2 to parallel the classical treatment by Choptuik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Its con- jugate variable Kx(t,x) is fixed by the diffeomorphism constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The shift vector is determined by demanding preservation of the gauge fixing condition in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One also uses the gauge freedom to set Kϕ(t,x) = 0 to maintain the diagonal form of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The dynamical variables are the triad Eϕ(t,x) and the lapse function N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' With the matter content as a scalar field (φ(t,x),Pφ(t,x)), the effective equations of motion are obtained by ‘polymerizing’ the scalar field via φ → sin(kφ) k : N′ N − (Eϕ)′ Eϕ + 2 x − (Eϕ)2 x3 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='8) (Eϕ)′ Eϕ − 3 2x + (Eϕ)2 2x3 −2πx �� Pφ �2 x4 + � φ′�2 cos2(kφ) � = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='9) ˙φ = 4πN Eϕx Pφ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='10) ˙Pφ = 4πx2 Eϕ ��3NEϕ −xN (Eϕ)′ +N′Eϕx Eϕ � φ′ cos2(kφ) +xNφ′′ cos2(kφ)−xNk � φ′�2 cos(kφ)sin(kφ) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='11) The lapse function can be determined from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='8) (which is obtained by imposing preservation in time of the gauge fixing condition Kϕ(t,x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') Finally, the Hamiltonian constraint Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='9) determines the triad Eϕ(t,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (Note that for k → 0, (and expressing Eϕ(t,x) = xa(t,x)), one obtains the classical equations of motion of [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') One can see that these effective equations remain invariant under the transformation x → cx and t → ct for constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Hence, there will be no mass gap, as in the classical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The coordinate 28 system used here cannot penetrate the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Instead, the collapse of the lapse function, namely N(t,x) → 0, is used to signal the formation of a black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Numerical simulations with these equations reveal existence of “wiggles” and “echoes” as in the classical description [90, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One finds that this effective theory shares the universality of the scaling of the mass observed in the classical theory, up to small departures for large values of the ‘polymer parameter ’k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The period of the discrete self-similarity seems to be independent of the ‘polymerization parameter’ which indicates that the polymer effective theory has a critical solution with the same periodicity as in the classical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us conclude this section with a few remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the investigations of the Kruskal space-time reported in Sections II and III, detailed analysis of quantum corrections to the geometry and their physical implications was made possible, thanks to the presence of a 4-dimensional symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Dynamical problems discussed in this section have only spherical symmetry and therefore are much more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, various questions remain unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For instance, the quantization scheme (called ‘K-quantization’ [102, 103]), used in [61, 62, 65, 67, 68] to arrive at effective equations governing the dust collapse, is only valid for marginally bound cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Secondly, there are indications [97] that one may have to revisit the assumptions made while ‘polymerizing’ the Hamiltonian con- straint, choices made in ‘polymerization’ of lapse and shift, and the issue of consistency of gauge fixing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Further, the choice of shift vector made in [61, 62, 65, 67] and also in [33] seems to be problematic from the covariance of the effective geometries [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Finally, there are also studies where another (‘non-polymeric’) quantization of these classical models has been studied [104–111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A detailed comparison of both quantization schemes could add clarity on the physical viability and mathematical consistency of these two complementary approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Similarly, in the investigations of the critical collapse of scalar field, the role of quantum geometry in the gravitational sector is yet to be included [90, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' If one were to introduce ‘polymerization’of the gravitational connection as in the models for dust collapse, one will very likely introduce a length scale, breaking the scale invariance and a mass gap would appear as in other works incorporating inverse triad modifications [87, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In explorations of the critical collapse, a more complete picture, including quantum geometric effects in the gravitational sector, is not yet available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is an important gap as it is these quantum geometry effects that lead to singularity resolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Despite such limitations, it is encouraging that these models have already provided new perspectives on how quantum effects can manifest themselves in the dynamical pro- cess of black hole formation and evolution, in the resolution of the classical singularity, and in critical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' V Black hole evaporation Investigations reported in Section IV provide interesting insights into the nature of quantum effects in dynamical situations leading to gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, be- cause of their underlying assumptions, they cannot address the issue of black hole evapo- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In this section we turn to the LQG investigations of the Hawking process and the 29 associated issue of ‘information loss’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In his original discussion [2] Hawking considered a test, scalar quantum field on a classical space-time depicting gravitational collapse of a spherical star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Heuristic consid- erations of the inclusion of the back reaction on space-time geometry led to the Penrose diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 4 that is still widely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In this diagram I + fails to be the complete future boundary since the singularity is also a part of this boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One is then led to the startling conclusion that quantum gravity considerations would force us to generalize quantum physics by abandoning unitarity [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, this line of reasoning has im- portant limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The first comes from an elementary observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For a self-consistent discussion of unitarity, one needs a closed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, the incoming collapsing matter in the distant past has to be represented by quantum fields, and the outgoing quantum state in the distant future should refer to the same fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This rather basic point is overlooked in space-time diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 4 because the asymptotic Hilbert spaces do not include the quantum state of matter in the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The second issue is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In much of the discussion on the subject, challenges and paradoxes arise because one assumes that the quantum corrected space-time has an event horizon that encloses a trapped region which is causally disconnected from the asymptotic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This seems natural from the perspective of the traditional Penrose diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, event horizons are tele- ological and, as Hajicek pointed out already in 1987 [113], they can be shifted arbitrarily, i− i+ uEH i0 I + I − Σi Σf FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 4: Commonly used Penrose diagram to depict black hole evaporation, including back reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Modes are created in pairs, one escaping to I + and its partner falling into the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The dashed line is the continuation of the Event Horizon that meets I + at retarded time uEH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' If this were an accurate depiction, the evolution from I − to I + would fail to be unitary because the future singularity would act as a ‘sink of information’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 30 and even completely removed, by changing the space-time geometry in a Planck scale neighborhood of the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Now, there is general consensus that classical GR cannot be trusted in such neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore the assumption that the event horizon will persist in quantum gravity has no obvious support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Indeed, LQG considerations suggest that it will not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In Section V A we explain how these two issues are addressed in the LQG literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In Section V B we summarize the current status of LQG investigations in semi-classical gravity and expectations in full quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In broad terms these investigations pro- vide closely related avenues to realize the paradigm introduced in [40] based on singular- ity resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, from LQG perspective, non-singular black holes play a central role in the discussion of the information loss issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To anchor the discussion we will use the approach developed in [114, 115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A complementary discussion can be found in [139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A Setting the stage i− i0 uLR I + I − T-DH Σi Σf � � u0 u Σ flat FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 5: Semiclassical space-time: Black hole is formed by gravitational collapse of a pulse of scalar field, depicted by the (gray) shaded region, incident from I −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A trapping Dynamical horizon T-DH is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' During the collapse, it is space-like and its area increases (in the outward direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It becomes time-like during evaporation and its area decreases (in the future direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Hawking radiation starts in earnest at u = u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The dashed line with scissors that includes the last ray u = uLR represents the future boundary of the semi-classical region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 31 A precise formulation of the issue of ‘information loss’ is provided by the question of whether the S-matrix from I − to I + is unitary which, as we discussed, is relevant only for closed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The simplest such system is a massless Klein-Gordon field coupled to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Consider, then, gravitational collapse of a spherically symmetric, massless scalar field φ from I −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the classical theory, if the infalling pulse of φ is narrow, the collapse is prompt and analysis is not overly contaminated by the details of the pulse profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The solution has Minkowski metric ηab to the past of this narrow pulse and a Schwarzschild black hole to its future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It is clear from the lower portion of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 5 that the event horizon first forms and grows in the flat portion of space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The actual collapse could occur billions of years to the future!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is a concrete illustration of the teleological nature of the event horizon (EH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In particular, it brings out the fact that the growth of the area of the EH is not tied to any local physical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the quantum theory, the pulse is replaced by a coherent state of the field ˆφ on I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the semi-classical regime –which is expected to be valid in the region in which space-time curvature is much smaller than the Planck scale– one can continue to describe the quantum corrected geometry using a smooth metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This portion of space-time is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 5, the region with Planck scale curvature in the future being excised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us first focus on this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The Hawking quanta of the quantum field ˆφ are emitted in pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' one escapes to I + and its partner falls into the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The quantum state on a Cauchy surface Σ of the semi-classical portion of space-time continues to be pure but there is entanglement between the infalling and outgoing quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As for geometry, the space- time metric to the past of the infalling pulse continues to be ηab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But to the future, it is no longer given by the static Schwarzschild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The metric is dynamical not only within the pulse but also to its future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Because of its dynamical nature, new structures emerge that are directly relevant to the evaporation process: dynamical horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These are the dynamical analogs of the trapping and anti-trapping horizons of the quantum corrected Kruskal space-time discussed in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' They turn out to be more relevant than EHs in discussions of black hole formation and mergers in numerical simulations in classical GR and for the evaporation process in the quantum theory [41–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us therefore briefly recall this notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A dynamical horizon (DH) is a 3-dimensional space-like or time-like submanifold that is foliated by 2-dimensional, surfaces S with 2- sphere topology, such that the expansion of one of the null normals to each leaf S is zero and that of the other null normal is either positive or negative everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, each S is a marginally trapped surface (MTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In an asymptotically flat space-time, we can distinguish between the two null normals to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us denote by la the outgoing null normal and by na the ingoing null normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' On a black hole type DH, the expansion Θ(ℓ) of the outgoing null normal vanishes (it is positive immediately outside and negative immediately inside the MTS), while the expansion Θ(n) of the ingoing null normal is negative (both outside and inside).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, immediately inside a black hole type DH, both expansions are negative and we have a trapped region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, these DHs are called trapping dynamical horizons, T-DHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A T-DH is space-like when the area of the MTS increases along the projection of la on DH, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' in the outward direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In fact, there is an explicit, precise relation between the growth of the area of a DH and the flux of 32 energy (carried by matter and/or gravitational waves) flowing into it [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, not only does the second law of black hole mechanics hold on T-DHs but the growth of the horizon area is directly related to local physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is in striking contrast with the situation for EHs, where we only have a qualitative statement of growth in classical GR: Area of EHs cannot decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Indeed, it is not possible to directly trace the growth back to the infall of energy locally because, as we just saw, EHs can form and grow in flat space-time where there is nothing at all falling across it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' During the evaporation process, by contrast, the MTSs on the T-DH shrink, now in response to the local negative energy flux across it, and the T-DH is time-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Recall that in the quantum corrected Kruskal space-time, we also have (white hole type) anti- trapping horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But they emerge only when the space-time is extended across the transition surface T on which curvature is of Plank scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore one would expect that in dynamical situations, anti-trapping dynamical horizons AT-DH would also emerge only when one extends space-time across a transition surface that replaces the classical singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This expectation is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' There is no AT-DH in the semi-classical space- time Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 5 where the region with Planck scale curvature was excised by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, it is present in the quantum extended space-time depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' On an AT-DH it is the expansion Θ(n) of the ingoing null normal that vanishes and the expansion Θ(ℓ) of the outgoing null normal is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, immediately inside these horizons, both expansions are positive: we have an anti-trapped region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since it is Θ(n) that vanishes on any AT-DH, it is natural to investigate what happens to the area of the MTSs as one moves along the projection of na on the AT-DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' If the AT-DH is space-like, its area decreases (now in the inward direction) and if it is time-like its area increases (now in the future direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, the key differences between EHs and DHs can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' First, EHs are teleological and can be located only after one has evolved the metric to infinite future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' DHs by contrast can be located quasi-locally and their properties have direct re- lation to physical processes at their location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Second, EHs are null while DHs can be space-like or time-like, and become null only when they become ‘isolated’ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' there is no flux of energy across them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Third, nothing can ever escape to the ‘exterior’ region from the trapped region enclosed by a black hole type EH and nothing can ever enter the anti-trapped region bounded by a white hole type EH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' While there are trapped surfaces immediately inside a T-DH, one can send causal signals across a T-DH from inside to outside (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Similarly, there are future directed causal curves that traverse an AT- DH from outside to inside (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Finally, while there is no natural notion of mass and angular momentum for cross-sections of EHs, there is one for the canonically defined marginally trapped surfaces on DHs which, furthermore lead to the first and second laws of black hole mechanics [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Discussions of quantum dynamics in LQG focus on DHs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Much of the confusion about the evaporation process and ‘purification’ of the quantum state melts away once EHs are deemphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 33 B Black hole evaporation in LQG LQG investigations of the semi-classical part of the quantum corrected space-time are based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 5 and, although some of the detailed calculations are still in progress, the overall understanding of structures in this space-time is quite satisfactory at a conceptual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To understand the structure of the future of this region one needs full quantum gravity and, as in every other approach, several questions remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But there is a general consensus on a majority of issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In this subsection we summarize this status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Semi-classical Regime: Consider a coherent state Ψ of a quantum scalar field ˆφ, peaked around an infalling classical pulse on I − and undergoing a prompt collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us suppose that the ADM energy in the incoming state is of a solar mass, M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' When the radius of the pulse has become sufficiently small, a trapping dynamical horizon T-DH forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In classical GR, this T-DH would only have a space-like component that grows from zero radius till it has radius of 3km and then joins on to the null event horizon of the same radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Once the Hawking radiation starts and the back reaction is included, the black hole shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Initially the process is very slow because the ingoing negative energy flux is extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It takes some 1064 years for the black hole to shrink to lunar mass Mmoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, even at the end of this long, adiabatic process, the black hole is macroscopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, from our discussion in Section II one would expect the quantum gravity corrections to be sufficiently small for semi-classical considerations to suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us focus on this phase of evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In this phase, dynamics should be well-described by equations Gsc ab = 8πGN ⟨ ˆTab ⟩ren and □ ˆφ = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1) where Gsc ab is the Einstein tensor of the semi-classical metric gsc ab and the expectation value of the renormalized stress-energy tensor is computed using the Heisenberg state Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The metric gsc ab does include quantum corrections but they are induced by quantum matter (rather than being dictated by the area gap considerations of quantum geometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These corrections to geometry are adiabatic and small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But the infalling negative en- ergy flux introduces a qualitative difference in the horizon structure: Now the expanding, space-like branch T-DH of the dynamical horizon joins on, not to a null event horizon as in the classical case, but to the outer, time-like branch whose area decreases to the future due to the negative energy flux carried by the Hawking ‘infalling partner modes’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These two branches of the T-DH serve as the past boundary of a trapped region of the semi- classical space-time (Msc, gsc ab) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' During this long adiabatic process of ∼ 1064 years, pairs of Hawking quanta are continually created, one going to I + and its partner falling into the trapped region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These modes will be entangled whence, if one uses the usual observable algebra based just at I +, the state would seem mixed, close to a thermal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The issue of unitarity leads one to ask: When will the correlations be restored?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For this to happen, the partner modes would have to emerge from the trapped region and propagate outward, restoring correlations at I + and ‘purifying’ the state there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since the outer part of the boundary T-DH of the trapped region is time-like, there is no causal obstruction 34 for these modes to continuously exit the trapped region across T-DH throughout the long evaporation process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' the standard causal obstructions associated with EHs do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This fact has been used in the literature to argue that there is no information loss issue at all [116, 117];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' purification could have occurred all along the evaporation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But this seemingly easy explanation is flawed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Examination of the renormalized energy flux shows that throughout this process, there is only infall across T-DH in semi-classical gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, the lack of a causal obstruction for the partner modes to exit the trapped region is not sufficient for the purification to occur in the semi-classical space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In fact there is an apparent puzzle associated with the issue of information loss that is already relevant in the semi-classical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since Mmoon ∼ 10−7 M⊙, at the end of this long evaporation process most of the initial ADM mass is carried away to I + by the Hawking quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A back of the envelope calculation shows that a very large number N (∼ 1075) of quanta escape to I + and all of them are correlated with the ones that fell across T-DH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, at the end of the semi-classical process under consideration, one would have to have a huge number N of quanta both at I + and in the trapped region, but the mass associated with the trapped region is only 10−7 times that carried away by the N quanta going out to I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Furthermore, the radius of the outer part of T-DH has shrunk to only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1mm – the Schwarzschild radius of a lunar mass black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' How can a T-DH with just a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1mm radius accommodate all these N quote?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Even if we allowed each mode to have the (apparently maximum) wavelength of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1mm, heuristically one would need the horizon to have a huge mass –some 1022 times the lunar mass!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' While these considerations are quite heuristic, one needs to face the conceptual tension: At the end of the process under consideration, the trapped region has simply too many quanta to accommodate, with a tiny energy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Such considerations have led to suggestions that somehow ‘purification’ must begin already by Page time [118] when the T-DH has lost only half its original mass of M⊙, and essentially completed by the time the T-DH has shrunk down to the lunar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But this would imply that semi-classical considerations must fail in apparently tame regimes due to unforeseen quantum gravity effects that are relevant outside the horizons of astrophysical black holes!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As we discussed in previous sections, in LQG quantum gravity corrections are completely negligible near horizons of macroscopic black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The way out of the apparent paradox is that semi-classical theory itself predicts that the geometry of the trapped region has some rather extraordinary features that had not been noticed until relatively recently and not fully appreciated by the wider community even now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Calculations of the stress-energy tensor on the Schwarzschild space-times confirm the idea that, in semi-classical gravity there is a negative energy flux across the time- like portion of T-DH such that MT-DH would decrease according to the standard Hawking formula: dMT-DH/dv = −ℏ/(GMT-DH)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (Indeed, this has been the basis of the standard estimate that the evaporation time goes as ∼ M3 ADM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') One can then argue that, in the phase of evaporation from the solar mass to the lunar mass, the form of the space-time metric in the trapped region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 5 is well approximated by the Vaidya metric: ds2 = − � 1− 2m(v) r � dv2 +2dvdr +r2� dθ 2 +sin2 θ dϕ2� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='2) 35 with m(v) = GMT-DH(v) decreasing very slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As we saw in Section II, corrections due to quantum geometry are completely negligible in Schwarzschild interior until one reaches Planck curvature and, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 5 shows, that region is excluded in the semi-classical space-time under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, in the metric (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='2) quantum corrections are all in- duced by quantum matter and encoded in m(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To probe the geometry of the trapped region, it is convenient to foliate it and two natural foliations have been used by the LQG community: One defined by constancy of the Kretchmann scalar and the other by con- stancy of the radius of metric 2-spheres [30, 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Each space-like slice is topologically S2 × R and is itself foliated by round 2-spheres which can be labelled by values of the advanced time coordinate v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us set v = 0 when MT-DH = 1M⊙ and v = v0 when MT-DH = Mmoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' During this long process, the radius of the MTSs on the outer, time-like part of AT-DH decreases from r|v=0 = 3km to r|v=v0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The surprising fact is that as v increases the leaves develop longer and longer necks of length ℓN along the R directions [30, 119, 120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The ‘final leaf’ for the process under consideration starts at the right end with v = v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The length ℓN of this final leaf is astonishingly large: ℓN ≈ 1064 light years for the first foliation and ℓN ≈ 1062 light years for the second!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These astro- nomically large lengths can result because the time the process takes is huge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 1064years corresponds to ∼ 1053 times our cosmic history!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This enormous stretching is analogous to expansion in (an anisotropic) cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Recall that during the cosmic expansion –e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' during inflation– the wavelengths of modes get stretched enormously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This suggests that partner modes that fall into the trapped region will also get enormously stretched during evolution from v = 0 to v = v0, as in quantum field theory on an expanding cosmological space-time, and become infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Can this phenomenon resolve the quandary of ‘so many quanta with so little energy’?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The answer is in the affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' With such infrared wavelengths, it is easy to accommodate them in the trapped region with the energy budget only of Mmoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, even though the outgoing modes carry away almost all of the initial mass M⊙ to I +, there is no obstruction to housing all their partners in the trapped region on a slice Σ of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 5 with the small energy budget of just 10−7M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This argument removes the necessity of starting purification by Page time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the LQG perspective, purification can be postponed to a much later stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To summarize, in the semi-classical regime, there are apparent paradoxes associated with the process of ‘purification’ that is necessary for dynamics to be described by a uni- tary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These disappear when one shifts the focus from event horizons to trapping dynamical horizons and takes into account the time evolving geometry of the trapped region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' By and large the LQG community has adopted this view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Beyond the semi-classical regime: When do quantum geometry effects become signif- icant making the semi-classical approximation inadequate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The viewpoint in LQG is that this happens when physically observable quantities such as curvature scalars and matter density enter the Planck regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This expectation was borne out in the investigation of the Schwarzschild interior in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, one would expect semi-classical con- siderations to be valid well beyond the time when T-DH has shrunk to MT-DH = Mmoon we considered in the above discussion, all the way till the curvature is, say, 10−6 times 36 the Planck curvature which corresponds to MT-DH ≈ 103MPl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' LQG explorations of the evaporation process beyond this stage are being carried out by different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The main ingredients are: results on causal structure of the Schwarzschild interior summarized in Section II, intuition derived from simpler models such as CGHS [121], conclusions drawn from a long series of works (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', [122–131]) that posit a space-time structure for the entire process and work out its consequences, strong consistency requirements on the ensuing space-time geometry (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', [96]), and calculations based on the Vaidya met- ric for the structure of space-time in the distant future [127, 132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' While there is broad consensus on the overall picture, many open issues remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We will now summarize the current status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To the future of the semi-classical region, curvature can exceed 10−6ℓ−2 Pl , whence we need full quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This region with Planck scale curvature is depicted by the shaded (pink) region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (To the past and future of this region, semi-classical grav- ity should yield a reasonable approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') In the shaded (pink) region geometry is described by a quantum state Ψgeo and the difficult task is to evolve the quantum field ˆφ on this quantum geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Fortunately, prior experience with other systems –such as the propagation cosmological perturbations on the quantum FLRW geometry– suggests a strategy that is applicable during the adiabatic phase of the Planck regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The evapo- ration process is adiabatic so long as mass-loss does not occur too rapidly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', until the radius of T-DH is ≈ 103ℓPl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' At the end of this process, one enters a neighborhood of the future endpoint of the T-DH depicted by a (red) blob, where the curvature is Planckian and the process speeds up very rapidly, violating the adiabatic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us first discuss the adiabatic phase and then return to the (red) blob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Prior results in LQC strongly suggest that the problem of propagating a quantum field ˆφ on a quantum geometry represented by Ψgeo can be greatly simplified during the adi- abatic phase: One can construct a smooth ˜gab that carries all the information in Ψgeo that the dynamics of quantum fields ˆφ is sensitive to (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', [133]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' ˜gab is called the dressed metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, the difficult task of evolving quantum fields on quantum geometry is reduced to that of evolving them on the space-time of the dressed metric ˜gab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Next, the expectation from results of Section II is that the shaded (pink) region will contain a transition surface T (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' ˜gab) that replaces the classical singularity and separates the trapped region that lies to its past and the untapped region that lies to its future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The met- ric ˜gab will capture two distinct effects: those that originate from quantum geometry and feature the area gap ∆ (as in Section II), and those that are induced on ˜gab by the falling quantum matter, dominated by the incident pulse of the scalar field at the left end of the (pink) shaded region, and by the infalling Hawking quanta carrying negative energy as one moves towards the right end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Discussion of Section II strongly suggest that the first set of effects will decay rapidly as we move away from Planck curvature into the semi-classical region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore in the semi-classical region, ˜gab will be well approximated by gsc ab used there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As we move to the future of the (pink) shaded region, one would encounter an anti-trapping dynamical horizon AT-DH (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The region enclosed by the transition surface T to the past and AT-DH to the future would be anti-trapped as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But now AT-DH would be 37 space-like rather than null and its area would not be constant, but decrease as one moves left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Qualitatively this change in the structure of AT-DH from the one of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 2 is parallel to the change in the structure of the trapping horizon T-DH that we already discussed in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Finally, the region to the future of AT-DH would also be well approximated by a Vaidya metric, but now the outgoing one, expressed in terms of the retarded time coor- dinate u in place of the advanced time coordinate v of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It will describe the propagation of the infrared modes that will emerge from the AT-DH and arrive at I + at very late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (The metric in this region will be nearly flat because the total energy in the scalar field is small and dispersed over very large spatial regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') Recall that these are the partner modes that fell into the horizon and were therefore entangled with the outgoing modes that carried away most of the initial ADM mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the LQG scenario, then, correlations are finally restored at I + where, in the end, the infalling modes also arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The total energy carried by the two sets of modes is very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But this is not an obstruction for restoring correlations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', for the ‘purification’ to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The timescale of this purification process is very long, 0(M4) [127, 130, 132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Purification can occur much later than the page time because of the LQG singularity resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The final picture is rather similar to the process of burning a piece of coal that is of- i− i0 I + I − T-DH i+ u0 u1 u2 LNS Σ τ flat AT-DH FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6: Quantum extension of the space-time in LQG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The classical singularity is replaced by a Transition surface τ, to the past of which we have a trapped region, bounded in the past by a trapping dynamical horizon T-DH, and to the future of which we have an anti-trapped region bounded by an anti-trapping dynamical horizon AT-DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Cauchy surfaces Σ develop astronomically long necks already in the semi-classical region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The dark (red) blob at the right end of τ is a genuinely quantum region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 38 ten invoked in the discussion of black hole evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Initially the piece of coal is in a pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' When lit, it emits photons and the energy they carry is well described by a thermal state at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But the total state is pure because there are correlations between the quantum state of outgoing photons and the left-over coal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As the fire extin- guishes, there is very little energy left and the ashes emits photons with lower and lower frequencies for a very long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' At the end of the process the cold ashes are in a pure state and all photons have escaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The late time, long wavelength photons are able to restore the correlations that were apparently lost in the middle of the process (when the photon spectrum seemed approximately thermal) even though the total energy they carry is small compared to the energy carried by high frequency photons that were emitted earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Note, however, that this scenario is incomplete because one still has to deal with the very last part of the evaporation process, depicted by the red blob and the associated null rays u = u1 and u = u2 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In this region, not only is the curvature of Planck scale, but it is varying extremely rapidly because it lies at the end point of the evaporation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It is this combination that makes the problem difficult;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' if we had only one of these features, we could have used known approximation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Independent considerations suggest that something very non-trivial must happen in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We will conclude with an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' During the semi-classical phase, as the T-DH shrinks, the temperature associated with the radiation at I + grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Consequently, the modes become increasingly ultraviolet as one approaches the point u = u1 on I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' On the other hand, the radiation that emerges from the anti-trapped region is infrared and received at I + to the future of u = u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is a dramatic transition, strongly suggesting that the physics of the region which is highly dynamical and has Planck scale curvature will be very subtle and interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For example, it has been suggested that Planck scale ‘seeds’ may be left behind, scattered in this region [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Understanding the nature of this quantum geometry remains an attractive challenge in LQG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us summarize the current status of LQG investigations of black-evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' They are distinguished by their emphasis on two features that are generally ignored in other approaches: (i) A shift away from the teleological event horizons EH to quasi-locally defined trapping and anti-trapping DHs T-DH and AT-DH ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' and, (ii) replacement of the classical singularity by the transition surface T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As a consequence, the traditionally used Penrose diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 4 is replaced by the Penrose diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' VI Discussion As the bibliography indicates, LQG literature on regular black holes is very rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In- deed, even this long list is far from being exhaustive!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' To make the material accessible to non-experts, we focused on four lines along which advances have occurred, and in each case built the discussion around a few of the mainstream developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Much of this dis- cussion is based effective equations, motivated by the fact that high performance compu- tations have shown that effective space-time metrics provide an excellent approximations to the quantum geometry in LQC, also in Bianchi models where the Weyl curvature is non-zero and diverges at the singularity [135, 136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 39 The first area, discussed in Section II, focuses on the ‘Schwarzschild interior’ that contains the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since the resolution of this singularity is central to the theme of ‘regular black holes’ of this Volume, we included an account of several different effec- tive descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These investigations bring out two features: (i) singularity resolution due to the underlying quantum geometry effects of LQG is robust, and does not depend on details of the quantization methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' but, (ii) the precise manner in which quantiza- tion is carried out can unleash unintended and physically undesirable effects that are not apparent until a detailed examination is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We summarized a scheme that is free of these drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The resulting quantum corrected geometry exhibits interesting causal structures: the singularity is replaced by a transition surface, T , to the past of which there is a trapped region, and to the future, an anti-trapped region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Each is bounded by null horizons and, for macroscopic black holes, the area of the future horizon is ap- proximately equal to that of the past (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In each of these effective geometries, curvature scalars attain their maxima at the transition surface which, furthermore, have universal values, independent of the mass of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This universality seems to be a general feature of the singularity resolution due to quantum geometry effects of LQG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Section III extended the quantum corrected geometry of Section II to the exterior, asymptotic region of the Schwarzschild space-time by exploring the homogeneity of time- like surfaces rsch = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For macroscopic black holes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', those with m ≫ ℓPl), the near horizon geometry of this exterior has the expected and physically desired features: the quantum corrected, effective metric is smooth across the horizon and corrections to the Hawking temperature, computed using methods from Euclidean quantum field theory, are tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' More generally, for macroscopic black holes there is excellent agreement between the effective geometry and that of the classical Schwarzschild metric in a vast neighbor- hood of the horizon in the exterior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Unfortunately, there is some confusion in the literature on this point arising from the simplified form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='16) of the metric that holds in the far-asymptotic region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', only on ignoring terms O(rs/r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' If one overlooks this key approximation and uses (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='16) in the entire exterior region –as was done in [137]– one obtains “unsettling features”, such as non-trivial corrections to the innermost circular orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These are consequences not of the actual effective geometry, but of the incorrect use of its simplified form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (Nonetheless, unfortunately, incorrect conclusions of [137] have been repeated in some of the subsequent literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [138]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' More generally, the near horizon quantum corrections to astrophysical black holes will be very small to have observable relevance in the foreseeable future (at least in the non-rotating case on which most LQG investigations have focused so far).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='4 The full metric in the exterior region is also asymptotically flat with curvature decay 4In this review, we did not touch on the issue of black hole entropy that arises in LQG by counting microstates of the area operator that are compatible with parameters characterizing a given macroscopic black hole (see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', [139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The possibility of testing discreteness of area using gravitational waves has drawn considerable attention in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' It has been argued that the simplest area spectrum with area eigenvalues given by knℓ2 Pl (where n is an integer and k a constant), considered by Bekenstein and Mukhanov [140], could be ruled out using data from a sufficiently large number of compact binary mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But in LQG the area spectrum is not equidistant, it crowds exponentially, making the continuum an excellent approximation very quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, for small black holes the area eigenvalues are grouped, exhibiting a band structure, and the separation between bands is O(ℓ2 Pl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' If this structure were to persist for large rotating black holes, each band would serve as a proxy of the Bekenstein-Mukhanov eigenvalues and gravitational observations would then lead to non-trivial constraints [141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, currently there is no evidence that points to the persistence of bands for macroscopic areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 40 that is sufficient for the ADM mass to be well defined (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', if one uses the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='17) in terms of the spatial Ricci tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' For macroscopic black holes, quantum correc- tions to the classical value are very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, the decay is slower than that in the standard notion of asymptotic flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Consequently, different expressions of the ADM mass, that must agree with one another exactly if the standard asymptotic conditions hold [55], now differ by quantum corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Much more surprising is the feature that the norm of the time-translation Killing field of the effective metric diverges at spatial infin- ity!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One’s first reaction would be that such deviations from standard asymptotic flatness must lead to a plethora of physically inadmissible consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One test is provided by quasi-normal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Do they exhibit a pathological behavior?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A detailed investigation [142] has shown that the potential which enters the quasi-normal mode analysis con- tinues to be well-defined everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' One can then compute quasi-normal frequencies using an approximation tailored to improving accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The corrections to the classi- cal result are found to be negligibly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' An independent investigation [143] provided expressions for axial and polar perturbations, computed their quasi-normal frequencies and found departures with respect to the classical theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' in particular, isospectrality is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, all these relative deviations from the classical predictions are only a small-percent effect even for black holes as small as rS ∼ 103ℓPl, and they decrease with the mass of the black hole, becoming completely negligible for macroscopic black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These investigations also show that the metric passes the stability criterion for tensor and massless scalar field perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, the infrared behavior of the potential is dif- ferent from that in the classical Schwarzschild case, leading to a qualitative difference in the power-law tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These tails play an important role in the mathematical literature but are not astrophysical significant because they occur after the waves are exponentially damped in the quasi-normal ringing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In summary, at present it is not clear whether counter-intuitive features associated with the asymptotic behavior of the effective metric of [29] are indications that it may be inadmissible in the asymptotic region rS/r ≪ 1, or if they are physically harmless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In view of this uncertainty, several investigations are ex- ploring alternate ways of arriving at an effective metric that has the standard asymptotic behavior (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', [33, 38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Another conceptual issue concerns covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' There is a 4-metric in the full quantum extended Kruskal space-time and results on singularity resolution, for example, refer to curvature invariants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' these considerations are all 4-dimensionally covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But to arrive at the effective Einstein’s equations with quantum geometry modifications, one uses sym- metry reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The question is whether there is a covariant action for the full theory without symmetry reduction whose equations of motion reduce to those in Sections II and III in its static, spherically symmetric sector [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is a technically difficult issue that is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Indeed, it took some time to show that the much simpler effective equations of the homogeneous, isotropic sector of LQC [29] can be obtained in this way, but finally the answer turned out to be in the affirmative [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A similar situation arose also in string theory where it seemed for quite some time that the exact 1+1 dimensional stringy black hole [146] did not arise from the symmetry reduction of a covariant action [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the end, it was shown that there is such an action but it requires inclusion of additional fields 41 [148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' There are some concrete indications that the situation is likely to be similar in the LQG black hole sector we discussed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', [38] that introduces a covariant action in the ‘mimetic gravity’ setting that adds a scalar field with a specific potential and uses the same time-like homogeneous slices as in Section III in the symmetry reduced sector, and [84] that uses a reasoning based on the constraint algebra to argue for covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Discussion in Sections II and III was confined to the LQG treatment of the eternal black hole and arrived at the Penrose diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 3 for the quantum extension of the Kruskal space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This entire space-time is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In classical relativity as well as in the discussion of black hole evaporation, Kruskal space-time of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 1 provides useful mathematical tools as well physical intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The same is true of its quantum extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, realistic and more interesting situations involve formation of black holes by gravitational collapse (for which only a part of the full Kruskal space-time is relevant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In Section IV we focused on two complementary issues that have been investigated in dynamical situations featuring gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The first involves the resolution of singularity for collapsing dust models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Here, the emphasis is on quantum geometry effects because the matter is characterized by dust rather than a fundamental quantum field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, non-classical features associated with matter –such as boundedness of the dust density– are induced on matter by the quantum nature of geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' These investigations show that, as in cosmology, the singularity is replaced by a bounce;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' this is a robust result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The second class of investigations focuses on critical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Now the strategy is the opposite in that it is matter that is represented by a quantum scalar field of LQG [100] while geometry is classical to begin with;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' corrections to classical effects on geometry are induced by quantum matter through field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The overall finding is that the quantum corrections to the classical results are small for macroscopic black holes, just as one would hope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, while there is no ‘mass-gap’ in the classical theory –i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' a black hole can be formed with arbitrarily small mass– a mass gap can develop if one uses a quantization scheme that leads to effective equations violating scale invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' While dynamics is at the heart of these investigations, they do not encompass the Hawking process because, in the first set of analyses matter is classical, and the second focuses on critical behavior in gravitational collapse, rather than on the scalar field quanta going out to I +, or the issue of entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' LQG investigations of the evaporation process –including the issue of back reaction on geometry– were discussed in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' They reflect a broad consensus that the arguments that lead to the traditional Penrose diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 4 are flawed in two important respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' First, they assume that a part of classical singularity persists in the quantum theory while it is resolved in LQG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Second, the event horizon plays a key role in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 4 even though it is teleological and can be made to disappear by changing space-time geometry in a Planck scale neighbor- hood of the singularity [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The traditional Penrose diagram is replaced by a new LQG Penrose diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' There is consensus that there is no information loss: The S-matrix from I − to I + is unitary provided, of course, we consider a closed system in which the black hole forms by the gravitational collapse of a quantum field from I − and we use the quantum state of the same field at I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' That the singularity would be resolved by quantum geometry effects is motivated by two considerations: (i) Quantum 42 geometry effects discussed in Section II that provide universal upper bounds to curvature scalars because of a non-zero value of the area gap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' and, (ii) detailed numerical simula- tions in the CGHS case that show that even in the semi-classical theory, the singularity is significantly weakened when back reaction effects are included, which already suffice to make the metric continuous there [149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' There is no EH in the final picture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' what forms classically in the gravitational collapse and evaporates through quantum processes is a DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The evaporation process of LQG can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The Hawking quanta are created in pairs, the outgoing quanta go out to I + as in Hawking’s original paradigm, and their partner quanta fall across the trapping dynamical horizon T-DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the semi-classical regime depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 5, the outgoing quantum state is well approximated by a thermal state at I + (at sufficiently late times), and the partner modes carry a negative energy flux into the trapped region that is bounded by T-DH in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' When the back reaction is included, the geometry in the trapped region changes adiabatically, and space-like sur- faces Σ of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6 get stretched and become long necked surfaces (LNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Suppose that at its formation, the black hole has solar mass M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Although the process of elongation of necks is very slow, it continues for a very long time since the semi-classical phase lasts some 1064 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' At the end of this phase, the necks become astronomically long, stretched to some 1062 light years!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore the modes that have fallen in the trapped region also get enormously stretched (as they do during inflation) and become infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' They can continue to be entangled with their partner modes that went out to I + during this long semi-classical phase –even though the total energy carried by the outgoing modes is al- most M⊙ and that carried by the trapped modes is tiny– precisely because the trapped modes are infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Thus, the quantum state on a surface such as Σ continues to be pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Since the singularity is resolved and replaced by a transition surface T that lies in the shaded (pink) region, these modes can evolve across the transition surface T and emerge on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Then they propagate to the approximately flat region that lies to the future of the anti-trapped dynamical horizon AT-DH, and arrive at I + restoring the cor- relations with the partner modes that reached I + much earlier, during the semi-classical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' (As explained towards the end of section V B, this situation is qualitatively similar to that of burning a piece of coal where correlations are restored at late times when the large wavelength modes emerge from ashes as they cool down, restring correlations with short wavelength modes emitted earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=') What remains largely unexplored so far is the (red) ‘blob’ at the right end to the shaded (pink) region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6 and how it affects the physics at I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As discussed at the end of Section V, the problem is hard because one simultaneously encounters two difficulties: Planck scale curvature and rapid changes that make adiabatic approximation inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But feasible calculations may suffice to reveal whether most, if not all, of the correlations are restored when the infrared modes traverse the anti-trapped region and emerge at I +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' If they are restored, then the fully quantum ‘blob’ would not be that relevant for the issue of information loss and the S-matrix would be unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Although the consensus in LQG favors this possibility, this issue is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' There are arguments involving a fully quantum evolution from past of the blob to its future, but so far they are inconclusive because of the 43 underlying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' At this stage, one cannot, for example, rule out the possibility that the ‘blob’ joins on to a baby universe whose states are inaccessible from I + of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' If this were ti happen, from the perspective of I ± of this figure, information may be lost, although the ‘total’ S-matrix would be unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Showing that this does not happen remains a fascinating challenge in the LQG community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Let us summarize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The LQG community has explored different aspects of the many fascinating properties of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The distinguishing feature of these investigations is their emphasis on quantum geometry that is directly responsible for replacement of the singularity by a transition surface with interesting causal properties, and boundedness of physical observables such as curvature scalars and matter density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As discussed in Section I, these features are not shared by other approaches: Using the AdS/CFT corre- spondence as motivation, it is sometimes argued that singularities should persist also in quantum gravity, and indeed, much of the literature uses the Penrose diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 4 in which a singularity features as part of the future boundary of space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' On the other hand, because quantum geometry effects become important only in the Planck regime, LQG corrections to the classical results are very small near the horizons of astrophysical black holes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' examples we discussed include corrections to the Hawking temperature using the near horizon geometry and the machinery of Euclidean quantum field theory, correc- tions to quasi-normal frequencies of astrophysical black holes, and to results associated with critical collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is also in striking contrast to some other approaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', the ‘firewall scenario’ that emerged from string theory considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' More generally, LQG does not lead to violations of semi-classical expectations of physics near horizons of astrophysical black holes that had been advocated before the LIGO discoveries showed that predictions of classical GR, without such major corrections, are realized in compact binary mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As mentioned in Section V B, there is also a large body of investigations that posit a space-time structure for the entire process and work out its consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' By and large one solves classical Einstein’s equations (with suitable stress-energy tensors) in various patches, and joins them consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Some of these space-time diagrams resem- ble Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' While these investigations do pay close attention to consistency conditions, and often also to energy considerations, the issue of quantum correlations and unitarity received little attention in these works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The LQG line of reasoning of Section V fills this conceptually important gap that is key to the issue of ‘information loss’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' There is also considerable discussion on the issue of young versus old black holes, and long lived remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In LQG, there is indeed an important difference between a young and an old black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' As a concrete example, let us consider two lunar mass black holes – a young one that is freshly formed from gravitational collapse, and an old one what started out as a solar mass black hole and then evaporated down to the lunar mass, as discussed in Section V B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' While their dynamical horizons will have the same radius, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1mm, and mass MT-DH = Mmoon, their external environment as well as internal structure will be very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' In the second case, the evaporation process would have gone on for some 1064 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, there will be a very large number of outgoing Hawking quanta in the exterior region, and an equal number of ingoing quanta in the trapped region, the two being entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Therefore, the small area of T-DH will not be a 44 measure of the entropy of what is in the interior (or exterior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, in LQG, in both cases the area is a measure of the surface degrees of freedom of the horizon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', degrees of freedom that can communicate both the outside and inside regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But it is sometimes argued that there is a potential problem with this scenario: because old black holes can have small energy but an enormous number of modes, it should be easy to produce them in particle accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But these arguments use only the conservation laws normally used in computing scattering amplitudes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' since old black holes have astronomically long necks, it is hard to imagine how such changes in space-time structure can occur on time scales of accelerator physics [150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Finally, let us discuss some of the limitations of the current LQG investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' The analyses we summarized make a strong use of symmetry reduced models and effective equations that capture the leading order quantum corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' However, there have been a number of interesting investigations that aim at arriving at these effective equations start- ing from full LQG (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=', [151, 152]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But they are still in a rather preliminary stage, and further and more detailed investigations are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Another key limitation is that so far the LQG investigations have focused primarily on non-rotating black holes, where the classical singularity is space-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' But for rotating black holes the inner horizons would be unstable and therefore the singularity would be null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' So far quantum geometry consid- erations have not been applied to null singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' This is an outstanding open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Indeed, inclusion of rotating black holes in the discussion of ‘information loss’ during evaporation remains a fascinating problem in all approaches to quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Acknowledgements This work was supported in part by the NSF grant PHY-1806356, PHY-1912274 and PHY-2110207, Penn State research funds associated with the Eberly Chair and Atherton professorship, and by Projects PID2020-118159GB-C43, PID2019-105943GB-I00 (with FEDER contribution), by the Spanish Government, and also by the “Operative Program FEDER2014-2020 Junta de Andaluc´ıa-Consejer´ıa de Econom´ıa y Conocimiento” under project E-FQM-262-UGR18 by Universidad de Granada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' We would like to thank Eugenio Bianchi, Kristina Giesel, Muxin Han, Bao-Fei Li, Guillermo Mena, Sahil Saini and Ed Wilson-Ewing for discussions, and Tommaso De Lorenzo for Figures 4-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Englehardt and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Horowitz, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Mod.' metadata={'source': 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J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Sully, JHEP 09, 018 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Giddings, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' B 738, 92 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [7] D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 23, 5587 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Campiglia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Gambini and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Pullin, AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} 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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='D 78, 044019 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Brannlund, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Kloster and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' DeBenedictis, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Rev.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 31, 095009 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [27] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Dadhich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Joe and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Singh, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 32, 185006 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Ashtekar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Olmedo and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Singh, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Rev.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' D98, 126003 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [30] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Ashtekar and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Olmedo, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' D105, 024069 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [33] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Gambini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Olmedo, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Pullin, Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' 8, 74 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' [34] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Navascu´es, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Garc´ıa-Quismondo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Mena Marug´an, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' D 106, 063516 (2022).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Ni, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Tang, arXiv:1212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content='1265 [60] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Ben Achour, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Lamy, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} +page_content=' Liu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNAzT4oBgHgl3EQfW_wb/content/2301.01309v1.pdf'} 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b/QNE3T4oBgHgl3EQfCwmU/content/tmp_files/2301.04279v1.pdf.txt @@ -0,0 +1,1512 @@ +Collective flows of protons and deuterons in Au + Au collisions at Ebeam = 1.23A GeV +by the IQMD model +Ling-Meng Fang(房灵猛),1, 2 Yu-Gang Ma(马余刚) +ID ,3, 4, ∗ and Song Zhang(张松) +ID 3, 4, † +1Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China +2University of Chinese Academy of Sciences, Beijing 100049, China +3Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), +Institute of Modern Physics, Fudan University, Shanghai 200433, China +4Shanghai Research Center for Theoretical Nuclear Physics，NSFC and Fudan University, Shanghai 200438, China +(Dated: January 12, 2023) +Collective flows of protons and deuterons for Au + Au collisions at beam energy Ebeam = 1.23A +GeV were simulated by an Isospin dependent Quantum Molecular Dynamics (IQMD) model. Two +coalescence models, namely naive coalescence and dynamical coalescence models, for the formation +of deuterons are compared. After reasonable match of rapidity spectra of protons and deuterons to +the High Acceptance DiElectron Spectrometer (HADES) data is reached，we apply an event-plane +method to calculate the first four-order collective flow coefficients as well as the ratios of ⟨v4⟩ / ⟨v2⟩2 +and ⟨v3⟩ /(⟨v1⟩ ⟨v2⟩), and observe the number of constituent nucleon scaling among protons and +deuterons. In addition, the dependence of εn and ⟨vn⟩ as well as the ratio ⟨vn⟩ /εn on the centrality +is obtained. Lastly, we further investigate the Pearson coefficients corr(vn, vm) between the first +four harmonic flows for protons and deuterons as a function of rapidity and centrality. +I. +INTRODUCTION +In heavy-ion collisions, +a highly excited nuclear +medium is created, and its collective expansion produces +the associated particle emission. In a perfectly central +collision, the expansion should be isotropic in the trans- +verse plane, as observed in the transverse mass spectra +of the ejected particles. +The shape of the overlapping +regions becomes more anisotropic in more off-central col- +lisions. In heavy-ion collisions, the collective motion of +final-state particles can be described by the collective +flows, which can be divided into longitudinal flow and +transverse flow according to the motion direction of the +final-state particles. The anisotropic flow is essentially +originated from the asymmetrical azimuthal distribution +of participant nucleons, which can be classified into di- +rected flow, elliptic flow, triangular flow, quadruple flow +and so on according to different terms of the Fourier ex- +pansion of the azimuthal distribution. +The Fourier expansion of the azimuthal distribution of +the final-state emission particles in momentum space can +be expressed as follows [1–3]: +E d3N +d3p = 1 +2π +d2N +ptdptdy +� +1 + +∞ +� +n=1 +2vn cos [n (φ − Ψr)] +� +(1) +where E is the energy of the final particle, pt is the trans- +verse momentum of the particle, y is the rapidity, φ is the +azimuthal angle of the transverse momentum relative to +the fixed plane XZ, Ψr is the azimuthal angle of the re- +action plane relative to the fixed plane XZ. vn at n = +∗ Email: mayugang@fudan.edu.cn +† Email: song zhang@fudan.edu.cn +1, 2, 3, 4 are defined as directed flow, elliptic flow, triangu- +lar flow, and quadruple flow, respectively, as mentioned +before. +In general we know that, the elliptic shape of the trans- +verse momentum distribution of the final particles is lo- +cated in-plane in lower energy below a hundred MeV per +nucleon due to the collective rotation dominated by the +attractive mean-field [4–8]. With the increasing of beam +energy to a few hundred MeV energy, the elliptic shape +could be perpendicular to the reaction plane in the mid- +rapidity region which is mainly because the spectators +have not moved away from the reaction area timely at +such energy range [9–11]. The spectators have a shadow- +ing effect on participants, making particles tend to eject +perpendicular to the direction of the reaction plane. This +phenomenon is called ”squeeze-out effect”, i.e. an ellipti- +cal flow outside of the reaction plane. While in the high +energy region, because of the Lorentz contraction in two +nuclei collisions, the transverse size of the nucleons is neg- +ligible relative to the longitudinal alignment. The time +for the two nuclei to cross is extremely short in compari- +son with the characteristic time of elliptic flow formation, +thus the bystander leaves the reaction area quickly and +almost has no shadowing effect on the reaction zone, so +the final particles tend to exude in the reaction plane, +and the elliptic flow is in the reaction plane [12]. +In 1992, Ollitrault et al. found that the spatial energy +density distribution at the early stage of the collision was +related to the spatial angular distribution of the freeze- +out particles at the later stage of the reaction [13]. In +1996, Voloshin et al. carried out the Fourier expansion +of the particle spectrum of the final state particles, and +proposed a method to express the size of the collective +flows of the final state particles by the coefficients of the +expansion terms [3]. After that, with the continuous in- +depth theoretical researches, people studied the collective +flows of each order in details, and put forward differ- +arXiv:2301.04279v1 [nucl-th] 11 Jan 2023 + +2 +ent calculation methods of collective flows, for example, +event-plane method [14, 15], energy momentum tensor +method [16] and two particle correlation method [2, 16– +18], etc. With the development of the accelerators, high- +energy heavy ion collision experiments can be carried out +under different conditions to study the collective flows of +final state particles at different energies. In 1999, Heisel- +berg and Levy studied the azimuthal asymmetry of the +system reflected by elliptical flow in noncentral collisions +[19]. Stachel concluded that the energy dependence of el- +liptical flow in high-energy heavy ion collisions is related +to QGP phase transition after analyzing the experimental +data of several accelerators [20]. Voloshin and Poskanzer +analyzed Pb + Pb collisions on SPS and found that the +elliptic flow has the centrality and rapidity dependence +[21]. In 2000, Heinz et al. investigated anisotropic flows +and established a deeper connection with QGP phase +transitions [22]. +Recently, the HADES Collaboration made system- +atic measurements on properties of baryon-rich matter +formed in Au + Au collisions at √sNN = 2.4 GeV. +Different probes, including dilepton and virtual photons +[23], identical pion intensity interferometry [24, 25] as +well high-order harmonic flows of light nuclei [26], which +provide an opportunity to investigate the nuclear fireball +properties as well as light nuclei production mechanism +[27–30], and then constrains theoretical model in this re- +action energy region and contributes the understanding +of the ‘ice in the fire’puzzle [31]. +The paper is organized as follows. First, a concise in- +troduction to the IQMD model and coalescence model +as well as the event-plane method for flow analysis are +given in Sec. II. Next, the results of first to fourth order +coefficients of collective flow of protons and deuterons are +presented in Sec. III. The results about the linear corre- +lations between different-order flows and eccentricity are +also given in this section. +Finally, a brief summary is +presented in Sec. IV. +II. +MODELS AND METHODS +In the study of heavy-ion collisions, various models +have been established to simulate the collision processes. +At present, the commonly used heavy-ion reaction mod- +els can be divided into statistical models and transport +models. +In this study, an Isospin dependent Quantum Molecu- +lar Dynamics (IQMD) model, a kind of transport mod- +els, is employed to study the reaction system from initial +state to final stage in medium-high energy heavy-ion col- +lisions. The coalescence model is used to simulate the +generation of light nuclei by using the nucleon phase- +space from IQMD model. And the collective flow of light +nuclei are calculated from the phase-space information at +the freeze-out stage simulated by the IQMD model with +help of the event-plane method. +In the following, the +IQMD model, coalescence model and event plane calcu- +lation methods will be introduced, separately. +A. +The IQMD model +Quantum Molecular Dynamics (QMD) model can pro- +vide the information on both the collision dynamics and +the phase space information [32–36]. The IQMD model +is based on the traditional QMD model, by including the +isospin degree of freedom of nucleons [37]. +In the IQMD model, the normalized wave function of +each nucleon is expressed in the form of a Gaussian wave +packet, +φi(⃗r, t) = +1 +(2πL)3/4 exp(−(⃗r − ⃗ri(t))2 +4L +) exp(i⃗r · ⃗pi(t) +ℏ +), +(2) +here ⃗ri(t) and ⃗pi(t) are time-dependent variables describ- +ing the center of the wave packet in coordinate space and +momentum space, respectively. Given the direction of ⃗ri +and ⃗pi, φi(⃗r, t) is a four-dimensional function. The pa- +rameter L is the width of the wave packet, which is re- +lated to the size of the reaction system and usually fixed +in the simulations. Here the width L is fixed as 2.16fm2 +for Au + Au reactions [38, 39]. +All the nucleons interact with each other through an +effective mean field and two-body scatterings. The inter- +action potential can be expressed as +U = USky + UCoul + UY uk + USym + UMDI, +(3) +where USky, UCoul, UY uk, USym, and UMDI represent +the density-dependent Skyrme potential, Coulomb po- +tential, Yukawa potential, isospin asymmetric potential, +and the momentum-dependent interaction potential, re- +spectively. +The nucleon-nucleon collision cross section +in the medium (σmed +NN ) can be expressed as taken in +Refs. [40–42] +σmed +NN = (1 − η ρ +ρ0 +)σfree +NN , +(4) +where ρ0 is the density of normal nuclear matter, ρ is +the local density, η is the in-medium correction factor, +which is chosen as 0.2 in this paper to better reproduce +the flow data [43], and σfree +NN is the free nucleon-nucleon +cross section. +B. +Coalescence model +There are two types of coalescence models, naive co- +alescence model and dynamical coalescence model. +In +this article, we use both of two coalescence models, and +compare the difference between them. +The naive coalescence model uses the following criteria +to judge the formation of deuterons: +∆p < p0, +∆r < r0, +(5) + +3 +where ∆p = |⃗p1 − ⃗p2|, ∆r = |⃗r1 − ⃗r2|, and p0 = 0.35 +GeV/c, r0 = 3.5 fm are selected in this paper. It is em- +phasized that here the momentum and coordinate should +be at the rest frame of the pair, such as proton and neu- +tron. +The dynamical coalescence model can give the proba- +bility of light nuclei by the overlap of the cluster Wigner +phase-space density with the nucleon phase space distri- +butions at an equal time in the M-nucleon rest frame +at the freeze-out stage [44]. The momentum distribution +of a cluster in a system containing A nucleons can be +expressed by: +d3NM +d3K += G +� A +M +��M +Z +� 1 +AM +� � Z +� +i=1 +fp +� +⃗ri,⃗ki +�� +� +M +� +i=Z+1 +fn +� +⃗ri,⃗ki +�� +× ρW � +⃗ri1,⃗ki1, · · · ,⃗riM−1,⃗kiM−1 +� +× δ +� +⃗K − +� +⃗k1 + · · · + ⃗kM +�� +d⃗r1d⃗k1 · · · d⃗rMd⃗kM, +(6) +where M and Z are the number of the nucleon and pro- +ton of the cluster, respectively; fn and fp are the neutron +and proton phase-space distribution functions at freeze- +out, respectively; ρW is the Wigner density function; +⃗ri1 · · ·⃗riM−1 and ⃗ki1 · · ·⃗kiM−1 are the relative coordinate +and momentum in the M-nucleon rest frame; the spin- +isospin statistical factor G is 3/4 for deuteron in this pa- +per [44–46]. While the neutron and proton phase-space +distribution comes from the transport model simulations, +the multiplicity of a M-nucleon cluster is then given by: +NM = G +� +� +i1>i2>···>iM +d⃗ri1d⃗ki1 · · · d⃗riM−1d⃗kiM−1 +� +ρW +i +� +⃗ri1,⃗ki1, · · · ,⃗riM−1,⃗kiM−1 +�� +, +(7) +where the ⟨· · · ⟩ denotes the event averaging. +C. +The event-plane method for flow analysis +A common method for calculating collective flow is the +event-plane method. The n-th order event-plane angle +Ψ(n) +EP can be defined by the event flow vector Qn,x and +Qn,y as [1, 2, 47–49]: +Ψ(n) +EP = 1 +n tan−1 +�Qn,y +Qn,x +� +, +Qn,x = +� +i +ωi cos(nΦi), +Qn,y = +� +i +ωi sin(nΦi), +(8) +where Φi and ωi are the azimuthal angle of the momen- +tum and the weight for the i-th particle, respectively. ωi +is usually set to unit for theoretical simulation but set as +charges |Z| in this paper, which is suggested in Ref. [26]. +The sums extend over all particles used in the event plane +reconstruction. For systems with finite multiplicity, the +harmonic flow coefficients can be calculated by: +⟨vn⟩ = +� +vobs +n +� +Res {Ψn {EP}}, +� +vobs +n +� += ⟨cos (km (φ − Ψn {EP}))⟩ , +Res {Ψn {EP}} = ⟨cos (km (Ψn {EP} − ΨRP ))⟩ . +(9) +The angular brackets indicate an average over all parti- +cles in all events and km = n in this work. The resolution +of event plane angle Res {Ψn {EP}} owing to the finite +number of particles can be calculated by: +Res {Ψn {EP}} = ⟨cos (km (Ψn {EP} − ΨRP ))⟩ += +√π +2 +√ +2χm exp +� +−χ2 +m/4 +� +× +� +I(k−1)/2 +� +χ2 +m/4 +� ++ I(k+1)/2 +� +χ2 +m/4 +�� +, +(10) +where the χm can be estimated by the sub-event method. +The event used to calculate the event plane angle would +randomly be split into two sub-events, event A and B, +with maximum difference of particle number equal to 1. +χm from sub-event resolution cos(km(ΨA +m − ΨB +m)) mul- +tiplying +√ +2 would be the χm for full event resolution +Res {Ψn {EP}}. +The details for this analysis can be +found in Refs. [2, 47–50] +III. +ANALYSIS AND DISCUSSION +In this paper, we use an IQMD model to simulate Au ++ Au collisions at beam energy Ebeam = 1.23A GeV, +which corresponds to a center of mass energy √sNN = +2.4 GeV. The total number of events included in the sim- +ulation is 1,600,000. The centrality is characterized as +c = (πb2)/(πb2 +max) × 100%, where b is the impact pa- +rameter, and bmax = 1.15(A1/3 +P ++ A1/3 +T ) is the sum of +effective shape radius of projectile and target. With this +definition of centrality, the smaller the c value, the more +central the collisions. +A. +Yield of protons and deuterons +In this paper, we use naive coalescence model and dy- +namical coalescence model to estimate deuterons forma- +tions. Figure 1 shows the rapidity distributions of pro- +tons and deuterons for the 0-10% centralities as well as +the comparison with the HADES results. +It is seen from Fig. 1(b) that the yields of deuterons +from two coalescence models are consistent with each +other and are all in good agreement with the HADES +experimental data. We notice that the yield of protons +from the IQMD model is higher than experimental data +in the most central collisions from Fig. 1(a), but there is +an overtaking in more off-centralities. This behavior re- +produces previous IQMD results or other models as given +in Ref. [52]. + +4 +0 +20 +40 +60 +80 +100 +-2 +-1 +0 +1 +2 +0 +10 +20 +dN/dYcm + 0-10% (HADES) + 10-20% (HADES) + 20-30% (HADES) + 30-40% (HADES) +(a) +p + 0-10% (IQMD) + 10-20% (IQMD) + 20-30% (IQMD) + 30-40% (IQMD) +dN/dYcm +Ycm + HADES + naive-coal + dynamical-coal +(b) +d +0-10% +FIG. 1. Rapidity distribution of protons (a) and deuterons (b) +in Au + Au collisions at Ebeam = 1.23A GeV for 4 central- +ity classes with 10% bin width. The solid and hollow points +in (a) represent the HADES experimental data [51] and the +IQMD simulation results, respectively. The points in (b) are +from the HADES experimental data (black) [52], naive coales- +cence model (red), and dynamical coalescence model (blue), +respectively. +B. +Collective flows of protons and deuterons +To be consistent with the method used by the HADES +experiment in Ref. [26], we use the charges |Z| as the +weight in this paper, and the flow coefficients of all orders +discussed here are defined relative to ΨEP,1 as: +� +vobs +n +� += ⟨cos [n (φ − ΨEP,1)]⟩ +Rn = ⟨cos [n (ΨEP,1 − ΨRP )]⟩ +⟨vn⟩ = +� +vobs +n +� +/Rn. +(11) +Via this method, we obtain the variation of first to +fourth order resolutions versus centrality as shown in +Fig. 2. +As we can see from Fig. 2, the value of reso- +lution decreases significantly as the order increasing, and +0 +10 +20 +30 +40 +0.0 +0.5 +1.0 +ℜn +centrality [%] + ℜ1(HADES) + ℜ2(HADES) + ℜ3(HADES) + ℜ4(HADES) + ℜ1(IQMD) + ℜ2(IQMD) + ℜ3(IQMD) + ℜ4(IQMD) +FIG. 2. The variation of first to fourth order resolutions with +the centrality in Au + Au collisions at Ebeam = 1.23A GeV. +The solid and hollow points represent the HADES experimen- +tal data and the IQMD simulation results, respectively. +the resolution obtained by the event-plane method has +basically the same trend as the HADES experimental +data. In the most central collisions, the emission par- +ticles tend to be more isotropic, so the values of all or- +der resolutions are the smallest. +With the increase of +the centrality value (i.e. more off-central collision), the +anisotropy of the emission particles is gradually obvious, +so the value of resolution tends to increase gradually. As +we can see from Fig. 2, in the most central collisions, the +resolution of IQMD model is higher than that from the +HADES, which corresponds to the higher proton yield +from IQMD model in the most central collision as shown +in Fig. 1. With the increase of centrality value, the proton +yield from the IQMD is gradually lower than that from +the HADES, which explains the overtaking phenomenon +of resolution in more off-central collisions in Fig. 2. +Using the event-plane method as in Eq. (11), we can +calculate the distribution of the collective flow as a func- +tion of rapidity for light nuclei, as shown in Fig. 3(a). +As we can see from Fig. 3(a), ⟨v1⟩ and ⟨v3⟩ are anti- +symmetric with rapidity, while ⟨v2⟩ and ⟨v4⟩ are axis- +symmetric. As for ⟨v2⟩, a negative value in middle ra- +pidity region indicates an out-of-plane emission, which is +caused by the so-called squeeze-out effect, where particles +are blocked from being emitted in the reaction plane by +the spectator nucleons and are therefore emitted mainly +in the out-of-plane-direction. As rapidity increases, the +value of ⟨v2⟩ becomes positive due to the reduced shad- +owing effect of bystanders on the reaction zone. +And +in the middle rapidity region, the ⟨v2⟩ of the protons is +lower than that of deuterons, which indicates that after +the collision, the protons are more likely to eject out of +the plane, while deuterons prefer an in-plane emission. + +5 +-0.5 +0.0 +0.5 +1.0 +-0.2 +0.0 +0.2 +-0.1 +0.0 +0.1 +-0.8 +-0.4 +0.0 +0.4 +0.8 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +-0.5 +0.0 +0.5 +1.0 +-0.2 +0.0 +0.2 +-0.1 +0.0 +0.1 +-0.15 +-0.10 +-0.05 +0.00 +0.05 + + p(HADES) + d(HADES) + + + +ycm + p(IQMD) + d(naive-coal) + d(dynamical-coal) +(a) +-0.3 +-0.2 +-0.1 +0.0 +0.1 +-0.4 +-0.2 +0.0 +-0.1 +0.0 +0.1 +0.2 +0.0 +0.5 +1.0 +1.5 +2.0 +-0.4 +-0.2 +0.0 +0.2 + + p(HADES) + d(HADES) +(b) + p(IQMD) + d(naive-coal) + d(dynamical-coal) + + + +pt [GeV/c] +FIG. 3. (a) Different harmonic flows as a function of rapidity +in Au + Au collisions at Ebeam = 1.23A GeV for 20-30% cen- +trality. The protons and deuterons are selected within trans- +verse momentum of 1 - 1.5 GeV/c. (b) Different harmonic +flows as a function of transverse momentum in Au + Au col- +lisions at Ebeam = 1.23A GeV for 20-30% centrality. For the +odd-order collective flows, the protons and deuterons are se- +lected within rapidity of -0.25 -0.15, while for the even-order +collective flows, the rapidity is selected within -0.05 0.05. In +the figure, the solid symbols with error bars represent the +HADES experimental data, the dotted lines with error bands +represent the IQMD simulation results, respectively. +Also, ⟨vn⟩ has a larger magnitude for lower harmonics +than higher harmonics. Moreover, we can see that the +result of collective flow obtained by the IQMD model is +lower than that from the HADES experiment especially +for the elliptic flow, this phenomenon is consistent with +the results from the UrQMD model in Ref. [53, 54]. +The distributions of different order collective flows as a +function of light nuclei transverse momentum are shown +in Fig. 3(b). The collective flow coefficients of deuterons +follow that of the free protons, and show a similar strong +dependence on the transverse momentum. And the ab- +solute values of collective flow of each order increase with +transverse momentum, which indicates that light nuclei +with higher pt tend to emit more out of plane, as they are +from earlier emission. ⟨v1⟩ of the deuterons have larger +negative values than the protons which can be inferred +from the coalescence mechanism. We can find that the +IQMD model can well describe the experimental results +of ⟨v1⟩, ⟨v3⟩ and ⟨v4⟩, but ⟨v2⟩ obtained by the IQMD +model is slightly lower than that from the HADES ex- +periment. +The scaling of elliptic flow of hadrons with the num- +ber of constituents has been established for more than +a decade with quark recombination [55] or quark coa- +lescence model [56] at RHIC energies, and an empirical +function can also fits the experimental elliptic flow data +[57]. For the coalescence of nucleons into deuterons the +same scaling should be there in terms of the baryon num- +ber. It has been first claimed that nucleon-number scaled +flows should be observed if the coalescence mechanism +is satisfied for light nuclei production in Ref. [6, 58] and +later on the experimental confirmation has been achieved +by the STAR Collaboration [59]. The nucleon-number +scaling of elliptic flow results in the expectation that +� +vd +2 +� � +pd +T +� += 2 ⟨vp +2⟩ +� 1 +2pd +T +� +. +Thus ⟨v2⟩ /A as a function +of pT /A, with A being the baryon number, should yield +the same curves for protons and light nuclei in the coales- +cence picture. Moreover, instead scaling by the baryon +number A for ⟨v2⟩, the measured data ⟨v4⟩ seems to be +scaled by A2 in previous studies [26, 53, 58, 60]. Tak- +ing the data of Fig. 3 we show that the flow of protons +and the scaled deuterons for Au + Au collisions in 20- +30% centrality at a beam energy of 1.23 AGeV in Fig. 4. +From Fig. 4(a) we observe that the simulation predicts +a good scaling among protons and deuterons. Fig. 4(b) +display ⟨v4⟩ /A2 as a function of (pt/A)2, from which we +can see that the ⟨v4⟩ can still be roughly scaled by A2 for +protons and deuterons. +However, the scaling behavior +is not perfect within the present statistics. For example, +the ⟨v2⟩ /A from the IQMD model has a lower magnitude +than that from the HADES, which is probably due to the +underestimate of ⟨v2⟩ as shown in Fig. 3. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-0.2 +-0.1 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +-0.04 +-0.02 +0.00 +0.02 +0.04 +/A +pt/A [GeV/c] + p(IQMD) + d(naive-coal) + d(dynamical-coal) +(a) +(b) +/A2 +(pt/A)2 [(GeV/c)2] + p(HADES) + d(HADES) +FIG. 4. Mass number A scaled ⟨v2⟩(a) and ⟨v4⟩(b) of protons +and deuterons for 1.23A GeV Au + Au collisions in 20-30% +centrality as a function of transverse momentum per nucleon +for |y| < 0.05. +The initial fluctuation can affect the initial geomet- +ric asymmetry of the overlapping region which could be +transferred into the momentum space partially, and then +significantly contribute to higher-order harmonic flows +[61]. In earlier studies in intermediate energy [6, 58] as +well as at ultra-relativistic energies [61], it was found that +triangular and quadrangular flows also roughly present a + +6 +constituent nucleon number scaling in the intermediate- +pT region, similar to the behaviors of elliptic flow. From +those results, a nucleon-number scaling of ⟨vn⟩ /nn/2 for +different light nuclei holds for harmonic flow (⟨vn⟩, n += 2, 3, and 4), which can be related to ⟨vn⟩ scaling. +In ultra-relativistic energies, such extended flow scaling +for high-order harmonic flows has been demonstrated +by the PHENIX Collaboration [62] and STAR Collab- +oration [63, 64]. Fig. 5 show the ratio ⟨v4⟩ / ⟨v2⟩2 and +⟨v3⟩ /(⟨v1⟩ ⟨v2⟩) distributions on rapidity and transverse +momentum [65]. As we can see from Fig. 5(a), for pro- +tons and deuterons, the ⟨v4⟩ / ⟨v2⟩2 value approaches to +the experimental data of 0.5 within the larger error in +mid-rapidity region. However, in the off-middle rapid- +ity interval, the ⟨v4⟩ / ⟨v2⟩2 of protons and deuterons +decreases. Fig. 5(b) demonstrates that the asymptotic +values of ⟨v4⟩ / ⟨v2⟩2 of protons and deuterons (naive or +dynamical) approach 0.42 and 0.41 or 0.78, respectively, +which is overall in agreement with the experimental val- +ues. As for ⟨v3⟩ /(⟨v1⟩ ⟨v2⟩), the results obtained by the +IQMD model are higher than those from the HADES +experiment, and all of them do not show a significant +rapidity correlation. +-4 +-2 +0 +2 +4 +0 +2 +4 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0 +4 +8 +0.0 +0.5 +1.0 +1.5 +2.0-4 +0 +4 +8 +/2 + p(HADES) + d(HADES) +pt:1~1.5GeV/c +pt:1~1.5GeV/c +/2 +|y|<0.05 +/ +ycm +/ +pt [GeV] +-0.1 + p(IQMD) + d(naive-coal) + d(dynamical-coal) +(d) + +(e) + +(f) +/ε2 +centrality [%] +(g) +/ε3 +centrality [%] +(h) +/ε4 +centrality [%] +(i) +FIG. 6. The dependence of the εn (top row) and ⟨vn⟩ (middle +row) as well as the ratio ⟨vn⟩ /εn (bottom row) on the cen- +trality for n = 2 (left column), 3 (middle column), 4 (right +column) in Au + Au collisions at Ebeam = 1.23AGeV for +0-40% centralities from IQMD. For the odd-order collective +flow, the protons and deuterons are selected within rapidity +of -0.5 - 0, while for the even-order collective flow, the rapid- +ity is selected within -0.1 - 0. The transverse momentum is +selected within 1-1.5 GeV/c. +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.10 +-0.05 +0.00 +0.05 +0.10 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.10 +-0.05 +0.00 +0.05 +0.10 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +-0.04 +-0.02 +0.00 +0.02 +0.04 +corr(v1,v2) + proton + d(naive-coal) + d(dynamical-coal) +(a) +corr(v2,v3) +(b) +corr(v3,v4) +(c) +corr(v1,v3) +ycm +(d) +corr(v2,v4) +ycm +(e) +corr(v1,v4) +ycm +(f) +FIG. 7. The Pearson correlation function corr(vn, vm) of pro- +tons and deuterons as a function of rapidity in Au + Au col- +lisions at 1.23A GeV from IQMD. The transverse momentum +of protons and deuterons are selected as 1-1.5 GeV/c. +corr (vn, vm) = ⟨vnvm⟩ − ⟨vn⟩ ⟨vm⟩ +σvnσvm +. +(13) +Here, the standard deviation σvi = +� +⟨v2 +i ⟩ − ⟨vi⟩2 is used +to normalize the covariance. We know that the Pearson +coefficient provides a measure for linear dependence of +two random variables, which equals to 1 implies a perfect +linear dependence, but a vanishing Pearson coefficient +does not rule out any nonlinear correlation. +We show the Pearson correlation function corr(vn, vm) +between the first four flow coefficients of protons and +deuterons in Au+Au collisions at 1.23 AGeV from IQMD +modle as a function of rapidity in Fig. 7, and as a function +of centrality in Fig. 8. +-0.2 +0.0 +0.2 +-0.2 +0.0 +0.2 +-0.2 +0.0 +0.2 +0 +10 +20 +30 +40 +-0.08 +-0.04 +0.00 +0 +10 +20 +30 +40 +-0.08 +-0.04 +0.00 +0 +10 +20 +30 +40 +-0.02 +0.00 +0.02 +corr(v1,v2) +(a) +corr(v2,v3) +(b) +corr(v3,v4) +(c) +corr(v1,v3) +centrality +(d) +-0.5 5.27 GeV , and +an energy difference ∆E = EBtag − Ebeam in the interval [-0.15, 0.1] GeV, where Ebeam is +half of the collision energy, and ⃗pBtag and EBtag are the momentum and energy of the Btag +candidate, all in the center-of-mass frame. The efficiency of the FEI is calibrated with +B → Xℓν decays [5]. +An event-level selection requires more than three tracks, between 2 and 7 GeV of total +energy in the electromagnetic calorimeter, and a ratio of the second to zeroth order Fox +Wolfram moments, R2 [15], to be smaller than 0.4. The remaining B0 or B0 meson (the signal +side Bsig) is reconstructed in its decay of B0 → D∗−ℓ+νℓ ( D∗− → π−D0, D0 → K+π−). +Charged particles are required to originate from the interaction point and have a transverse +momentum greater than 0.2 GeV. To identify electrons and muons, a likelihood-ratio like +quantity for each particle hypothesis is calculated, which combines information from several +detector subsystems. The likelihood performance is calibrated with well-known physics +processes. In addition, electron and muon momenta are required to be greater than 1.0 GeV +in the center-of-mass frame to reject continuum background. Kaon and pion candidates +are combined to reconstruct D0 → K+π− candidates whose invariant mass (m (Kπ)) is +required to be within the interval [1.85, 1.88] GeV. The D0 candidates are combined with +an additional low-momentum pion to reconstruct D∗− → π−D0 candidates restricted to the +D∗−- D0 mass difference ∆m in the range [0.143, 0.149] GeV. Subsequently, Bsig candidates +are reconstructed by combining D∗− candidates with either e+ or µ+ candidates. At least +one combination of Btag and Bsig candidates is required with no remaining tracks. The +missing neutrino mass squared (m2 +miss = (Pbeam − PBtag − PD∗ − Pℓ)2, where P denotes a four +vector) is required to be in the range [-0.5, 0.5] GeV2. If multiple combinations of Btag and +Bsig candidates are found in an event, the candidate with the largest BDT output for the +Btag and the best ∆m for the Bsig is selected. Figure 1 shows the m (Kπ), ∆m, and m2 +miss +distributions. The figures include data points with statistical uncertainties and histograms +for simulated signal and background candidates scaled to the equivalent data luminosity. +The signal yield is estimated by counting the number of selected events on data from which +simulated background is subtracted. Checks based on data in the ∆m sidebands show good +agreement between simulated and experimental background distributions. +5 + +1.8 +1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 +1.9 +D mass (GeV) +0 +20 +40 +60 +80 +100 +120 +Candidates/(0.002 GeV) +Data +Signal +BG +selection +-1 + Ldt=189.3 fb +∫ +Belle II Preliminary + + and cc. +e +ν ++ +e +*- + D +→ + +0 + B +136 +138 +140 +142 +144 +146 +148 +150 +152 +154 +D*-D mass (MeV) +0 +20 +40 +60 +80 +100 +120 +140 +Candidates/(0.5 MeV) +Data +Signal +BG +selection +-1 + Ldt=189.3 fb +∫ +Belle II Preliminary + + and cc. +e +ν ++ +e +*- + D +→ + +0 + B +1 +− +0.5 +− +0 +0.5 +1 +1.5 +2 +2.5 +3 +) +2 + (GeV +miss +2 +m +0 +20 +40 +60 +80 +100 +120 +140 +) +2 +Candidates/(0.10 GeV +Data +Signal +BG +selection +-1 + Ldt=189.3 fb +∫ +Belle II Preliminary + + and cc. +e +ν ++ +e +*- + D +→ + +0 + B +1.8 +1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 +1.9 +D mass (GeV) +0 +20 +40 +60 +80 +100 +120 +140 +Candidates/(0.002 GeV) +Data +Signal +BG +selection +-1 + Ldt=189.3 fb +∫ +Belle II Preliminary + + and cc. +µ +ν ++ +µ +*- + D +→ + +0 + B +136 +138 +140 +142 +144 +146 +148 +150 +152 +154 +D*-D mass (MeV) +0 +20 +40 +60 +80 +100 +120 +140 +160 +Candidates/(0.5 MeV) +Data +Signal +BG +selection +-1 + Ldt=189.3 fb +∫ +Belle II Preliminary + + and cc. +µ +ν ++ +µ +*- + D +→ + +0 + B +1 +− +0.5 +− +0 +0.5 +1 +1.5 +2 +2.5 +3 +) +2 + (GeV +miss +2 +m +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +) +2 +Candidates/(0.10 GeV +Data +Signal +BG +selection +-1 + Ldt=189.3 fb +∫ +Belle II Preliminary + + and cc. +µ +ν ++ +µ +*- + D +→ + +0 + B +FIG. 1. +Distributions of Kπ invariant mass (left), D∗- D0 mass difference (middle), and missing +neutrino mass squared (right) of B0 → D∗−e+ νe (top) and B0 → D∗−µ+νµ (bottom) candidates in +data (points) and simulation (histograms). Vertical lines enclose the regions of the selected events. +4. +MEASUREMENT OF BRANCHING RATIO +The branching ratio for the decay B0 → D∗−ℓ+νℓ is estimated as +B +� +B0 → D∗−ℓ+νℓ +� += +� +N rec − N bg� +ϵ−1 +4NBB (1 + f+0)−1 B +� +D∗− → π−D0� +B +� +D0 → K+π−�, +(1) +where N rec is the number of reconstructed events in data, N bg is the number of reconstructed +background events, ϵ is the signal reconstruction efficiency, NBB is the number of produced +BB pairs, and f+0 is the ratio of the number of produced B+B− and B0B0 pairs. In Eq. 1, the +values corresponding to electron and muon modes are averaged. The values of N bg and ϵ are +estimated from the background and signal simulation. The value of NBB is determined using +the R2 distribution after a subtraction of the continuum background using off-resonance data. +The values for f+0, B(D∗− → π−D0), B(D0 → K+π−), and fraction of mixed and unmixed +B0B0 are taken from Ref. [12]. The input values for the branching fraction measurement are +summarized in Table I. +6 + +TABLE I. Input values for the measurement of branching ratio with the systematic uncertainties, +described in Sec. 6. +Variables +Values +Nrec +545 (data) +Nbg +29.4 ± 11.2 +ϵ +(9.55 ± 0.67) × 10−4 +NBB +(197.17 ± 5.72) × 106 +f+0 +1.058 ± 0.024 +χd +0.1875 ± 0.0017 +B(D∗− → π−D0) +(67.7 ± 0.5)% +B(D0 → K+π−) +(3.950 ± 0.031)% +5. +MEASUREMENT OF |Vcb| +Figure 2 shows distribution of the recoil variable +w = PB · PD∗ +mBmD∗ = m2 +B + m2 +D∗ − q2 +2mBmD∗ +, +(2) +where q2 = (Pℓ + Pνℓ)2 and mB,D∗ are the known masses of the indicated particles. The +B0 → D∗−ℓ+νℓ decay-width differential in w is as follows [4, 16]: +dΓ +dw = η2 +EWG2 +F +48π3 m3 +D∗(mB − mD∗)2g(w)F 2(w)|Vcb|2. +(3) +Here, ηEW is an electroweak correction (calculated to be 1.00662 ± 0.00016 in Ref. [17]). +From lattice QCD, F(1) is calculated as 0.906 ± 0.004 (stat) ± 0.012 (syst) [17]. The product +g(w)F 2(w) describes the phase-space factor and the form factor, which is parameterized with +R1(1), R2(1), ρ2 in the CLN approach [11]. The CKM matrix element |Vcb| is determined by +fitting the ∆Γ/∆w distribution with the form factor parameters. However, here the product +ηEWF(1)|Vcb| is measured instead, in order to separate theory uncertainty of ηEW and F(1). +In this paper, the R1(1) and R2(1) values are taken from external measurements [1], as shown +in Table II. +TABLE II. Input values for R1(1) and R2(1) [1]. +R1(1) +1.270 ± 0.026 +R2(1) +0.852 ± 0.018 +Correlation coefficient of R1(1) and R2(1) +-0.715 +7 + +1 +1.1 +1.2 +1.3 +1.4 +1.5 +w +0 +10 +20 +30 +40 +50 +Candidates/(0.05) +Data +Signal +BG +MC error +-1 + Ldt=189.3 fb +∫ +Belle II Preliminary + + and cc. +e +ν ++ +e +*- + D +→ + +0 + B +1 +1.1 +1.2 +1.3 +1.4 +1.5 +w +0 +10 +20 +30 +40 +50 +60 +70 +Candidates/(0.05) +Data +Signal +BG +MC error +-1 + Ldt=189.3 fb +∫ +Belle II Preliminary + + and cc. +µ +ν ++ +µ +*- + D +→ + +0 + B +FIG. 2. +Distributions of w for B0 → D∗−e+νe (left) and B0 → D∗−µ+νµ (right) candidates in +data (points) and simulation (histogram) after the event selection. The shaded band shows the +systematic uncertainty of the simulation, which is summarized in Sec. 6. +5.1. +Unfolding method +In order to estimate the true w distribution from the observed w values, an iterative +unfolding method is used [18]. The number of signal events populating w bin, Ni, is estimated +from the reconstructed variables as follows: +Ni = +� +j +Uij(N rec +j +− N bg +j ), +(4) +where i is the w bin number. We define 10 bins in the range [1.0, 1.5] each with a width of +0.05. The matrix Uij = P(wtrue +i +|wrec +j ) models the probability that events reconstructed in the +w bin j are in the true-w bin i, which is calculated by Bayes’ theorem according to +Uij = P(wtrue +i +|wrec +j ) += P(wrec +j |wtrue +i +) × P(wtrue +i +)/P(wrec +j ) += P(wrec +j |wtrue +i +) × P(wtrue +i +)/ +� +k +P(wrec +j |wtrue +k +)P(wtrue +k +). +(5) +Here, P(wrec +j |wtrue +i +) is estimated with simulation. To avoid bias from the simulated signal, +P(wtrue) is calculated using the reconstructed w distribution on data as follows. +1. P(wtrue +i +) is assumed uniform (P(wtrue +i +) = 0.1 for all bins). +2. Uij is calculated by using Eq.(5). +3. P(wtrue +i +) is set to � +j Uij(N rec +j +− N bg +j )/ � +ij Uij(N rec +j +− N bg +j ). +4. Steps 2. and 3. are repeated 10 times, until Uij converges. +The unfolding performance is validated with the simulation. +8 + +5.2. +Fitting method +To determine |Vcb|, a binned maximum likelihood fit is performed using +∆χ2 = −2 ln (L) = 2 +� +i +(N exp +i +− Ni + Ni ln (Ni/N exp +i +)) + +� +i +� +j +∆xiW −1 +ij ∆xj, +(6) +where i denotes the w bin, Ni is the number of observed events in the ith bin, xi is a +systematic parameter defined as the normalization uncertainty in the ith reconstructed-w +bin, ∆xi is the deviation of the systematic parameters from the nominal value, and Wij is +the covariance of the systematic parameters, modeled by multivariate Gaussians functions. +Finally, N exp +i +is the expected yield in the ith bin, which is written as follows: +N exp +i +� +B0 → D∗−ℓ+νℓ +� += 4ϵiNBB (1 + f+0)−1 τ +� +B0� +B +� +D∗− → π−D0� +B +� +D0 → K+π−� +� +1 + +� +j +Uij∆xj +� � wmax +i +wmin +i +dw dΓ +dw +� +B0 → D∗−ℓ+νℓ +� +, +(7) +where ϵi is the signal reconstruction efficiency in the ith bin. The differential distribution is +obtained using Eq.(3). In the fit there are two free parameters, ηEWF(1)|Vcb| and ρ, and ten +nuisance parameters ∆xi. The two-dimensional contour of ηEWF(1)|Vcb| and ρ is estimated +by using a marginalized likelihood [19], +Lmarg = 1 +J +J +� +j=1 +exp +� +− +� +i +� +N exp +ij +− Ni + Ni ln +� +Ni/N exp +ij +�� +� +, +(8) +where J = 10000 and N exp +ij +is the expected yield in the ith bin with the jth set of nuisance +parameters, which is generated following the covariance matrix. The fitter performance is +validated with simplified simulated experiments. +6. +SYSTEMATIC UNCERTAINTIES +Systematic uncertainties are evaluated for several sources associated with the detector +response, MC modeling, and physics inputs. For the branching ratio measurement, the +systematic uncertainty of each source is propagated to the result based on Eq.(1) and +summarized in Table III. The Btag reconstruction efficiency with the FEI algorithm is studied +using B → Xℓν decays and a systematic uncertainty of 3.9% is assigned [5]. The tracking +efficiency is studied with τ decays and the maximum data-simulation difference of 0.3% is +taken as systematic uncertainty for each track in the final state. The reconstruction efficiency +of the low momentum π− is studied by using B0 → π+D∗−(D∗− → π−D0) decays. The +data-MC ratio of the π momentum distribution is evaluated relative to the high momentum +distribution; a 3–4% systematic uncertainty is assigned in each momentum bin, which +is dominated by the statistical uncertainty of the control samples. Electron and muon +identification efficiencies and misidentification rates are studied by using e+e− → e+e−ℓ+ℓ−, +e+e− → e+e−(γ), e+e− → µ+µ−γ, decays of J/ψ, D∗, τ, and K0 +s. The lepton identification +9 + +and misidentification uncertainties associated with the size of the control samples, background +contamination, modeling of the fitting function, trigger, and the difference of the results +across samples are evaluated as a functions of each lepton angle and the absolute value +of the lepton momentum. These uncertainties are propagated to the branching fraction +measurement resulting in a total 2.0% systematic error. The potential variations in the +amount of background from B → D∗∗ℓν decays, hadronic B decays and misreconstructed +D∗ mesons are evaluated to propagate the uncertainty of the branching fraction of the +background processes and of beam backgrounds resulting in a 1.2% systematic uncertainty. +The number of produced BB pairs is estimated from the R2 distribution after a subtraction +of the continuum background using off-resonance data. A systematic uncertainty of 2.9% is +assigned to account for the limited statistics of off-resonance data, operation conditions of +the detector and accelerator including beam energy, and selection efficiencies. A systematic +uncertainty for the event-level selection is estimated to be 1.0%, to cover the maximum data- +simulation difference of the total energy in the electromagnetic calorimeter. The uncertainty +from the limited size of simulated samples is estimated to be 1.8%. The following sources of +systematic uncertainty are from external measurements: the ratio of the number of produced +B+B− and B0B0 pairs (1.2%), the ratio of the number of mixed and unmixed B0B0 (0.9%), +the branching fractions of D∗− → π−D0 (0.7%) and D0 → K+π− (0.8%), and form factors +(0.1%) [12]. The uncertainties from the various sources are assumed to be independent +and the quadratic sum is taken as a total systematic uncertainty. For the measurement of +ηEWF(1)|Vcb| and ρ2, the effect of the systematic uncertainty is included in the likelihood +calculation (the second term in Eq.(6)) with the covariance matrix +Wij = +� +k +� +N k +i − µi +� � +N k +j − µj +� +µiµj +. +(9) +Here, k runs over the sources of uncertainties, µi is the mean of the expected yield in the +ith w bin, N k +i is the variation of the expected yield in the ith bin for the kth source of +uncertainties. Figure 3 shows the estimated covariance matrix. +1.00-1.05 +1.05-1.10 +1.10-1.15 +1.15-1.20 +1.20-1.25 +1.25-1.25 +1.30-1.35 +1.35-1.40 +1.40-1.45 +1.45-1.50 + bin +w +1.00-1.05 +1.05-1.10 +1.10-1.15 +1.15-1.20 +1.20-1.25 +1.25-1.25 +1.30-1.35 +1.35-1.40 +1.40-1.45 +1.45-1.50 + bin +w +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +0.014 +0.016 +0.018 +0.02 +Belle II Preliminary + +FIG. 3. +Total covariance matrix for the ηEW F(1)|Vcb| and ρ2 measurement. The axes denote the +w bin intervals. +10 + +TABLE III. Summary of fractional systematic uncertainties on the branching ratio. +Systematic sources +Relative uncertainty (%) +FEI efficiency +3.9 +Low momentum π efficiency +4.1 +Tracking efficiency +0.9 +Lepton particle identification +2.0 +Background +1.2 +NBB +2.9 +f+0 +1.2 +Number of mixed BB +0.9 +B +� +D∗− → π−D0� +0.7 +B +� +D0 → K+π−� +0.8 +ECL energy +1.0 +Form factor +0.1 +MC sample size +1.8 +Total +7.3 +7. +RESULTS AND CONCLUSION +The result for the branching fraction is +B +� +B0 → D∗−ℓ+νℓ +� += (5.27 ± 0.22 (stat) ± 0.38 (syst)) % +(10) +while the results for |Vcb| are +ηEWF(1)|Vcb| × 103 = 34.6 ± 1.8 (stat) ± 1.7 (syst) +(11) +ρ2 = 0.94 ± 0.18 (stat) ± 0.11 (syst) . +(12) +The two-dimensional probability contours for ηEWF(1)|Vcb| and ρ2 are shown in Fig. 4. The +observed ∆Γ/∆w values are shown in Fig. 5 with the best fit function overlaid. The reduced +χ2 of the fit is 1.6 with p-value of 40.7 %, which is estimated by simulation. Under the +assumption that ηEW = 1.00662±0.00016 and F(1) = 0.906±0.004 (stat)±0.012 (syst) [17], +we obtain |Vcb| × 103 = 37.9 ± 2.7. The results are consistent with the world averages of +B(B0 → D∗−ℓ+νℓ) = (5.06±0.12)% and ηEWF(1)|Vcb|×103 = 35.27±0.38 based on exclusive +B → D∗ℓνℓ decays within one standard deviation [1]. +11 + +0 +0.2 0.4 0.6 0.8 +1 +1.2 1.4 1.6 1.8 +2 +2 +ρ +25 +30 +35 +40 +45 +50 +3 + 10 +× +| +cb +F(1)|V +EW +η +-1 + Ldt=189.3 fb +∫ + +Belle II Preliminary + + and cc. +lν ++l +*- + D +→ + +0 + B +Best fit +68% CL +90% CL +FIG. 4. +Two dimensional probability contours for ηEW F(1)|Vcb| and ρ2 at the 68% (solid) and +90% (dashed) confidence level. The best fit point is also shown. +1 +1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 +w +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 + [GeV] / (0.05) +15 + 10 +× +w +∆ +/ +Γ +∆ +-1 + Ldt=189.3 fb +∫ + +Belle II Preliminary + + and cc. +lν ++l +*- + D +→ + +0 + B +Data with total uncertainty +Best fit + uncertainty +σ +1 + uncertainty +σ +2 +FIG. 5. +Observed dΓ(B0 → D∗ℓν)/dw distribution with the best fit function and one and two +standard-deviation bands overlaid. +8. +ACKNOWLEDGEMENT +These acknowledgements are not to be interpreted as an endorsement of any statement +made by any of our institutes, funding agencies, governments, or their representatives. +We thank the SuperKEKB team for delivering high-luminosity collisions; the KEK +cryogenics group for the efficient operation of the detector solenoid magnet; the KEK +computer group and the NII for on-site computing support and SINET6 network support; +and the raw-data centers at BNL, DESY, GridKa, IN2P3, INFN, and the University of +12 + +Victoria for offsite computing support. +[1] Y. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Zhang, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Zhilich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Zhou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Zhou, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Zhukova, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Zhulanov, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' ˇZlebˇc´ık 3 Abstract We present a measurement of the B0 → D∗−ℓ+νℓ (ℓ = e, µ) branching ratio and of the CKM parameter |Vcb| using signal decays accompanied by a fully reconstructed B meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The Belle II data set of electron-positron collisions at the Υ(4S) resonance, corresponding to 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 fb−1 of integrated luminosity, is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' With the Caprini-Lellouch-Neubert form factor parameterization, the parameters ηEWF(1)|Vcb| and ρ2 are extracted, where ηEW is an electroweak correction, F(1) is a normalization factor and ρ2 is a form factor shape parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' We reconstruct 516 signal decays and thereby obtain B(B0 → D∗−ℓ+νℓ) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='22 (stat) ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='38 (syst)) % , ηEW F(1)|Vcb| × 103 = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='8 (stat) ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='7 (syst), and ρ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='18 (stat) ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='11 (syst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' INTRODUCTION A precise understanding of B0 → D∗−ℓ+νℓ decays is important for future measurements of R(D∗) = B(B → D∗τν)/B(B → D∗ℓν) [1, 2] and of the magnitude of the Cabibbo- Kobayashi-Maskawa matrix element Vcb [3, 4], where persistent tensions exist between inclusive B → Xcℓν and exclusive B → D∗ℓν measurements [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' We study e+e− → Υ(4S) → B0B0 events, where the decay of the accompanying B0 or B0 is reconstructed in a hadronic final state using the full event interpretation algorithm (FEI) [5] and the signal bottom meson of opposite flavor is then reconstructed in the D∗±ℓ±νℓ final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' BELLE II EXPERIMENT Belle II [6] is an experiment at the SuperKEKB super B factory [7], an energy-asymmetric e+ (4 GeV) e− (7 GeV) collider in Tsukuba, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Collision data with an integrated luminosity corresponding to 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 fb−1 were collected from March 2019 to July 2021 at a center-of-mass (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=') energy of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='58 GeV, corresponding to the mass of the Υ(4S) resonance, as well as 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='0 fb−1 at 60 MeV below the nominal c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The Belle II detector consists of several nested detector subsystems arranged around the beam pipe in a cylindrical geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The innermost subsystem is the vertex detector, which includes one or two layers of silicon pixels and four outer layers of silicon strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Outside the silicon, the central drift-chamber reconstructs charged-particles trajectories (tracks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Outside the chamber, a Cherenkov light-imaging and time-of-propagation detectors provide charged particle identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Further out is an electromagnatic calorimeter with CsI(Tl) crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' A uniform 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 T magnetic field aligned with the beam axis is provided by a superconducting solenoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Multiple layers of scintillators and resistive plate chambers, located between the magnetic flux-return iron plates, detect K0 L and muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The analysis uses simulated Monte Carlo (MC) samples to determine the signal efficiency and background yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' These samples are generated using EvtGen [8] and consist of e+e− → Υ(4S) → BB (generic) and e+e− → qq processes, where B indicates a B0 or a B+ meson and q indicates an u, d, c, or s quark (continuum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The latter is simulated with KKMC [9] and PYTHIA [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The luminosity of the generic and continuum samples is 1 ab−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The signal is modeled using the CLN form factor parameterization [11], and the time-integrated B0B0-mixing parameter [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' All samples are analyzed with the basf2 framework [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' In this paper, the natural system of units with c = ℏ = 1 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The inclusion of charge-conjugated decay modes is implied unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' EVENT SELECTION The reconstruction begins by fully reconstructing a B0 or B0 (Btag) in hadronic decay modes with the FEI algorithm [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The algorithm starts by selecting candidates for stable particles, which include muons, electrons, pions, protons, kaons, and photons, from tracks and electromagnetic energy deposits in each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Subsequently, the algorithm carries out several stages of reconstruction of intermediate particles such as π0, K0 S, J/ψ, D and D∗ mesons, Σ, Λ, and Λc baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Intermediate particles are reconstructed in specific decay modes from combinations of stable and other intermediate particle candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The final stage of the algorithm reconstructs the B0 mesons in 31 hadronic modes, using boosted decision trees (BDTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The Btag candidates are required to have a BDT classifier output greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='001, a beam constrained mass Mbc = � E2 beam − |⃗pBtag|2 > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='27 GeV , and an energy difference ∆E = EBtag − Ebeam in the interval [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='1] GeV, where Ebeam is half of the collision energy, and ⃗pBtag and EBtag are the momentum and energy of the Btag candidate, all in the center-of-mass frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The efficiency of the FEI is calibrated with B → Xℓν decays [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' An event-level selection requires more than three tracks, between 2 and 7 GeV of total energy in the electromagnetic calorimeter, and a ratio of the second to zeroth order Fox Wolfram moments, R2 [15], to be smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The remaining B0 or B0 meson (the signal side Bsig) is reconstructed in its decay of B0 → D∗−ℓ+νℓ ( D∗− → π−D0, D0 → K+π−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Charged particles are required to originate from the interaction point and have a transverse momentum greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' To identify electrons and muons, a likelihood-ratio like quantity for each particle hypothesis is calculated, which combines information from several detector subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The likelihood performance is calibrated with well-known physics processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' In addition, electron and muon momenta are required to be greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='0 GeV in the center-of-mass frame to reject continuum background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Kaon and pion candidates are combined to reconstruct D0 → K+π− candidates whose invariant mass (m (Kπ)) is required to be within the interval [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='85, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='88] GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The D0 candidates are combined with an additional low-momentum pion to reconstruct D∗− → π−D0 candidates restricted to the D∗−- D0 mass difference ∆m in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='143, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='149] GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Subsequently, Bsig candidates are reconstructed by combining D∗− candidates with either e+ or µ+ candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' At least one combination of Btag and Bsig candidates is required with no remaining tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The missing neutrino mass squared (m2 miss = (Pbeam − PBtag − PD∗ − Pℓ)2, where P denotes a four vector) is required to be in the range [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5] GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' If multiple combinations of Btag and Bsig candidates are found in an event, the candidate with the largest BDT output for the Btag and the best ∆m for the Bsig is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Figure 1 shows the m (Kπ), ∆m, and m2 miss distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The figures include data points with statistical uncertainties and histograms for simulated signal and background candidates scaled to the equivalent data luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The signal yield is estimated by counting the number of selected events on data from which simulated background is subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Checks based on data in the ∆m sidebands show good agreement between simulated and experimental background distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='9 D mass (GeV) 0 20 40 60 80 100 120 Candidates/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='002 GeV) Data Signal BG selection 1 Ldt=189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 fb ∫ Belle II Preliminary and cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' e ν + e *- D → 0 B 136 138 140 142 144 146 148 150 152 154 D*-D mass (MeV) 0 20 40 60 80 100 120 140 Candidates/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 MeV) Data Signal BG selection 1 Ldt=189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 fb ∫ Belle II Preliminary and cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' e ν + e *- D → 0 B 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 3 ) 2 (GeV miss 2 m 0 20 40 60 80 100 120 140 ) 2 Candidates/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='10 GeV Data Signal BG selection 1 Ldt=189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 fb ∫ Belle II Preliminary and cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' e ν + e *- D → 0 B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='82 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='9 D mass (GeV) 0 20 40 60 80 100 120 140 Candidates/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='002 GeV) Data Signal BG selection 1 Ldt=189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 fb ∫ Belle II Preliminary and cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' µ ν + µ *- D → 0 B 136 138 140 142 144 146 148 150 152 154 D*-D mass (MeV) 0 20 40 60 80 100 120 140 160 Candidates/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 MeV) Data Signal BG selection 1 Ldt=189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 fb ∫ Belle II Preliminary and cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' µ ν + µ *- D → 0 B 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 3 ) 2 (GeV miss 2 m 0 20 40 60 80 100 120 140 160 180 ) 2 Candidates/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='10 GeV Data Signal BG selection 1 Ldt=189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 fb ∫ Belle II Preliminary and cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' µ ν + µ *- D → 0 B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Distributions of Kπ invariant mass (left), D∗- D0 mass difference (middle), and missing neutrino mass squared (right) of B0 → D∗−e+ νe (top) and B0 → D∗−µ+νµ (bottom) candidates in data (points) and simulation (histograms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Vertical lines enclose the regions of the selected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' MEASUREMENT OF BRANCHING RATIO The branching ratio for the decay B0 → D∗−ℓ+νℓ is estimated as B � B0 → D∗−ℓ+νℓ � = � N rec − N bg� ϵ−1 4NBB (1 + f+0)−1 B � D∗− → π−D0� B � D0 → K+π−�, (1) where N rec is the number of reconstructed events in data, N bg is the number of reconstructed background events, ϵ is the signal reconstruction efficiency, NBB is the number of produced BB pairs, and f+0 is the ratio of the number of produced B+B− and B0B0 pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 1, the values corresponding to electron and muon modes are averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The values of N bg and ϵ are estimated from the background and signal simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The value of NBB is determined using the R2 distribution after a subtraction of the continuum background using off-resonance data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The values for f+0, B(D∗− → π−D0), B(D0 → K+π−), and fraction of mixed and unmixed B0B0 are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The input values for the branching fraction measurement are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 6 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Input values for the measurement of branching ratio with the systematic uncertainties, described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Variables Values Nrec 545 (data) Nbg 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='4 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='2 ϵ (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='67) × 10−4 NBB (197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='17 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='72) × 106 f+0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='058 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='024 χd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='1875 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='0017 B(D∗− → π−D0) (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5)% B(D0 → K+π−) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='950 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='031)% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' MEASUREMENT OF |Vcb| Figure 2 shows distribution of the recoil variable w = PB · PD∗ mBmD∗ = m2 B + m2 D∗ − q2 2mBmD∗ , (2) where q2 = (Pℓ + Pνℓ)2 and mB,D∗ are the known masses of the indicated particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The B0 → D∗−ℓ+νℓ decay-width differential in w is as follows [4, 16]: dΓ dw = η2 EWG2 F 48π3 m3 D∗(mB − mD∗)2g(w)F 2(w)|Vcb|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' (3) Here, ηEW is an electroweak correction (calculated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='00662 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='00016 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' From lattice QCD, F(1) is calculated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='906 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='004 (stat) ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='012 (syst) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The product g(w)F 2(w) describes the phase-space factor and the form factor, which is parameterized with R1(1), R2(1), ρ2 in the CLN approach [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The CKM matrix element |Vcb| is determined by fitting the ∆Γ/∆w distribution with the form factor parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' However, here the product ηEWF(1)|Vcb| is measured instead, in order to separate theory uncertainty of ηEW and F(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' In this paper, the R1(1) and R2(1) values are taken from external measurements [1], as shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Input values for R1(1) and R2(1) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' R1(1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='270 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='026 R2(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='852 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='018 Correlation coefficient of R1(1) and R2(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='715 7 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 w 0 10 20 30 40 50 Candidates/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='05) Data Signal BG MC error 1 Ldt=189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 fb ∫ Belle II Preliminary and cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' e ν + e *- D → 0 B 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 w 0 10 20 30 40 50 60 70 Candidates/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='05) Data Signal BG MC error 1 Ldt=189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 fb ∫ Belle II Preliminary and cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' µ ν + µ *- D → 0 B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Distributions of w for B0 → D∗−e+νe (left) and B0 → D∗−µ+νµ (right) candidates in data (points) and simulation (histogram) after the event selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The shaded band shows the systematic uncertainty of the simulation, which is summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Unfolding method In order to estimate the true w distribution from the observed w values, an iterative unfolding method is used [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The number of signal events populating w bin, Ni, is estimated from the reconstructed variables as follows: Ni = � j Uij(N rec j − N bg j ), (4) where i is the w bin number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' We define 10 bins in the range [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5] each with a width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The matrix Uij = P(wtrue i |wrec j ) models the probability that events reconstructed in the w bin j are in the true-w bin i, which is calculated by Bayes’ theorem according to Uij = P(wtrue i |wrec j ) = P(wrec j |wtrue i ) × P(wtrue i )/P(wrec j ) = P(wrec j |wtrue i ) × P(wtrue i )/ � k P(wrec j |wtrue k )P(wtrue k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' (5) Here, P(wrec j |wtrue i ) is estimated with simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' To avoid bias from the simulated signal, P(wtrue) is calculated using the reconstructed w distribution on data as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' P(wtrue i ) is assumed uniform (P(wtrue i ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='1 for all bins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Uij is calculated by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' P(wtrue i ) is set to � j Uij(N rec j − N bg j )/ � ij Uij(N rec j − N bg j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Steps 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' are repeated 10 times, until Uij converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The unfolding performance is validated with the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Fitting method To determine |Vcb|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' a binned maximum likelihood fit is performed using ∆χ2 = −2 ln (L) = 2 � i (N exp i − Ni + Ni ln (Ni/N exp i )) + � i � j ∆xiW −1 ij ∆xj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' (6) where i denotes the w bin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Ni is the number of observed events in the ith bin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' xi is a systematic parameter defined as the normalization uncertainty in the ith reconstructed-w bin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' ∆xi is the deviation of the systematic parameters from the nominal value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' and Wij is the covariance of the systematic parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' modeled by multivariate Gaussians functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Finally, N exp i is the expected yield in the ith bin, which is written as follows: N exp i � B0 → D∗−ℓ+νℓ � = 4ϵiNBB (1 + f+0)−1 τ � B0� B � D∗− → π−D0� B � D0 → K+π−� � 1 + � j Uij∆xj � � wmax i wmin i dw dΓ dw � B0 → D∗−ℓ+νℓ � , (7) where ϵi is the signal reconstruction efficiency in the ith bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The differential distribution is obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' In the fit there are two free parameters, ηEWF(1)|Vcb| and ρ, and ten nuisance parameters ∆xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The two-dimensional contour of ηEWF(1)|Vcb| and ρ is estimated by using a marginalized likelihood [19], Lmarg = 1 J J � j=1 exp � − � i � N exp ij − Ni + Ni ln � Ni/N exp ij �� � , (8) where J = 10000 and N exp ij is the expected yield in the ith bin with the jth set of nuisance parameters, which is generated following the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The fitter performance is validated with simplified simulated experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' SYSTEMATIC UNCERTAINTIES Systematic uncertainties are evaluated for several sources associated with the detector response, MC modeling, and physics inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' For the branching ratio measurement, the systematic uncertainty of each source is propagated to the result based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' (1) and summarized in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The Btag reconstruction efficiency with the FEI algorithm is studied using B → Xℓν decays and a systematic uncertainty of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='9% is assigned [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The tracking efficiency is studied with τ decays and the maximum data-simulation difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3% is taken as systematic uncertainty for each track in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The reconstruction efficiency of the low momentum π− is studied by using B0 → π+D∗−(D∗− → π−D0) decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The data-MC ratio of the π momentum distribution is evaluated relative to the high momentum distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' a 3–4% systematic uncertainty is assigned in each momentum bin, which is dominated by the statistical uncertainty of the control samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Electron and muon identification efficiencies and misidentification rates are studied by using e+e− → e+e−ℓ+ℓ−, e+e− → e+e−(γ), e+e− → µ+µ−γ, decays of J/ψ, D∗, τ, and K0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The lepton identification 9 and misidentification uncertainties associated with the size of the control samples, background contamination, modeling of the fitting function, trigger, and the difference of the results across samples are evaluated as a functions of each lepton angle and the absolute value of the lepton momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' These uncertainties are propagated to the branching fraction measurement resulting in a total 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='0% systematic error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The potential variations in the amount of background from B → D∗∗ℓν decays, hadronic B decays and misreconstructed D∗ mesons are evaluated to propagate the uncertainty of the branching fraction of the background processes and of beam backgrounds resulting in a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='2% systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The number of produced BB pairs is estimated from the R2 distribution after a subtraction of the continuum background using off-resonance data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' A systematic uncertainty of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='9% is assigned to account for the limited statistics of off-resonance data, operation conditions of the detector and accelerator including beam energy, and selection efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' A systematic uncertainty for the event-level selection is estimated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='0%, to cover the maximum data- simulation difference of the total energy in the electromagnetic calorimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The uncertainty from the limited size of simulated samples is estimated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The following sources of systematic uncertainty are from external measurements: the ratio of the number of produced B+B− and B0B0 pairs (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='2%), the ratio of the number of mixed and unmixed B0B0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='9%), the branching fractions of D∗− → π−D0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='7%) and D0 → K+π− (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='8%), and form factors (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='1%) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The uncertainties from the various sources are assumed to be independent and the quadratic sum is taken as a total systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' For the measurement of ηEWF(1)|Vcb| and ρ2, the effect of the systematic uncertainty is included in the likelihood calculation (the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' (6)) with the covariance matrix Wij = � k � N k i − µi � � N k j − µj � µiµj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' (9) Here, k runs over the sources of uncertainties, µi is the mean of the expected yield in the ith w bin, N k i is the variation of the expected yield in the ith bin for the kth source of uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Figure 3 shows the estimated covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='00-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='05-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='15-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='20-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='25-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='30-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='35-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='40-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='45-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='50 bin w 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='00-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='05-1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='25-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='30-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='35-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='40-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='45-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='50 bin w 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='02 Belle II Preliminary FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Total covariance matrix for the ηEW F(1)|Vcb| and ρ2 measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The axes denote the w bin intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 10 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Summary of fractional systematic uncertainties on the branching ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Systematic sources Relative uncertainty (%) FEI efficiency 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='9 Low momentum π efficiency 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='1 Tracking efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='9 Lepton particle identification 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='0 Background 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='2 NBB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='9 f+0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='2 Number of mixed BB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='9 B � D∗− → π−D0� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='7 B � D0 → K+π−� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='8 ECL energy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='0 Form factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='1 MC sample size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='8 Total 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' RESULTS AND CONCLUSION The result for the branching fraction is B � B0 → D∗−ℓ+νℓ � = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='22 (stat) ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='38 (syst)) % (10) while the results for |Vcb| are ηEWF(1)|Vcb| × 103 = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='8 (stat) ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='7 (syst) (11) ρ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='18 (stat) ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='11 (syst) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' (12) The two-dimensional probability contours for ηEWF(1)|Vcb| and ρ2 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The observed ∆Γ/∆w values are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 5 with the best fit function overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The reduced χ2 of the fit is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='6 with p-value of 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='7 %, which is estimated by simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Under the assumption that ηEW = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='00662±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='00016 and F(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='906±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='004 (stat)±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='012 (syst) [17], we obtain |Vcb| × 103 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The results are consistent with the world averages of B(B0 → D∗−ℓ+νℓ) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='06±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='12)% and ηEWF(1)|Vcb|×103 = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='38 based on exclusive B → D∗ℓνℓ decays within one standard deviation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 11 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='8 2 2 ρ 25 30 35 40 45 50 3 10 × | cb F(1)|V EW η 1 Ldt=189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 fb ∫ Belle II Preliminary and cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' lν +l *- D → 0 B Best fit 68% CL 90% CL FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Two dimensional probability contours for ηEW F(1)|Vcb| and ρ2 at the 68% (solid) and 90% (dashed) confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' The best fit point is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 w 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='5 5 [GeV] / (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='05) 15 10 × w ∆ / Γ ∆ 1 Ldt=189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content='3 fb ∫ Belle II Preliminary and cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' lν +l *- D → 0 B Data with total uncertainty Best fit uncertainty σ 1 uncertainty σ 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' Observed dΓ(B0 → D∗ℓν)/dw distribution with the best fit function and one and two standard-deviation bands overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' ACKNOWLEDGEMENT These acknowledgements are not to be interpreted as an endorsement of any statement made by any of our institutes, funding agencies, governments, or their representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' We thank the SuperKEKB team for delivering high-luminosity collisions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' the KEK cryogenics group for the efficient operation of the detector solenoid magnet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE3T4oBgHgl3EQfyAt7/content/2301.04716v1.pdf'} +page_content=' the KEK computer group and the NII for on-site computing support and SINET6 network support;' metadata={'source': 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+1,1433 @@ +CONSISTENCY REGULARISATION IN VARYING CONTEXTS AND +FEATURE PERTURBATIONS FOR SEMI-SUPERVISED SEMANTIC +SEGMENTATION OF HISTOLOGY IMAGES +Raja Muhammad Saad Bashir +Tissue Image Analytics Centre +University of Warwick +Coventry, United Kingdom +saad.bashir@warwick.ac.uk +Talha Qaiser +Tissue Image Analytics Centre +University of Warwick +Coventry, United Kingdom +talha.qaiser@warwick.ac.uk +Shan E Ahmed Raza +Tissue Image Analytics Centre +University of Warwick +Coventry, United Kingdom +shan.raza@warwick.ac.uk +Nasir M. Rajpoot +Tissue Image Analytics Centre +University of Warwick +Coventry, United Kingdom +n.m.rajpoot@warwick.ac.uk +ABSTRACT +Semantic segmentation of various tissue and nuclei types in histology images is fundamental to many +downstream tasks in the area of computational pathology (CPath). In recent years, Deep Learning +(DL) methods have been shown to perform well on segmentation tasks but DL methods generally +require a large amount of pixel-wise annotated data. Pixel-wise annotation sometimes requires +expert’s knowledge and time which is laborious and costly to obtain. In this paper, we present a +consistency based semi-supervised learning (SSL) approach that can help mitigate this challenge +by exploiting a large amount of unlabelled data for model training thus alleviating the need for a +large annotated dataset. However, SSL models might also be susceptible to changing context and +features perturbations exhibiting poor generalisation due to the limited training data. We propose +an SSL method that learns robust features from both labelled and unlabelled images by enforcing +consistency against varying contexts and feature perturbations. The proposed method incorporates +context-aware consistency by contrasting pairs of overlapping images in a pixel-wise manner from +changing contexts resulting in robust and context invariant features. We show that cross-consistency +training makes the encoder features invariant to different perturbations and improves the prediction +confidence. Finally, entropy minimisation is employed to further boost the confidence of the final +prediction maps from unlabelled data. We conduct an extensive set of experiments on two publicly +available large datasets (BCSS and MoNuSeg) and show superior performance compared to the +state-of-the-art methods. +Keywords Deep Learning · Semi-Supervised learning · Contrastive Learning · Computational Pathology · Whole Slide +Images +1 +Introduction +Segmentation of fundamental objects and regions in histology images is key to several downstream analysis tasks in +computational pathology (CPath) [1, 2] e.g., cancer type classification [3, 4, 5, 6], tumour and glandular segmentation +[7], and other tasks like mutation prediction [8, 9]. Their utility is not limited to diagnosis, they have also been employed +for prognostic purposes e.g., tumour infiltrating lymphocytes (TILs) have been found to be significant prognostic +biomarker in various types of tumours [10]. Similarly, tumour progression has been linked with interaction between +arXiv:2301.13141v1 [cs.CV] 30 Jan 2023 + +SSCL +the tumour epithelial cells and tumour associate stroma [11]. Hence, it is important to segment different types of +histological objects precisely as their quantification is vital to downstream analysis. +Machine learning based traditional methods accomplished this task using different hand-crafted features e.g., colour +[12], texture [13, 14] and morphological features [15]. Recently, deep learning (DL) algorithms have gained increasing +attention in semantic segmentation due to their superior performance on natural and medical images [16, 17, 18, 19]. +However, DL methods are known to be “data hungry" and require a large amount of annotated data. Precise annotation +of histology images is an expensive and laborious process, requiring up to ∼5-6 hours of an expert histopathologist’s +time to annotate one whole-slide image (WSI) [20]. To alleviate the annotation burden, other modes of training have +been proposed such as patch based segmentation [21, 22], coarse segmentation [23, 24] and interactive segmentation +[1, 25] but these methods still require large-scale weak annotations involving human expert. +Semantic segmentation is a pixel-level classification task of predicting label for each pixel using pixel values. Most +of the early DL methods were based on fully convolutional networks (FCN) [26] where pooling layers aggregate +the information by focusing on “what" rather than “where" resulting in loss of spatial information. The subsequent +studies addressed this shortcoming by using pooling layers with more advanced techniques involving skip connections, +encoders and boundary in formation. As semantic segmentation is more than just assigning labels to pixels, it inevitably +requires some contextual information along with knowledge of colour, edges and resolutions. In this regard, algorithms +like UNet [16], PSPNet [27], HRNet [28] and DeepLab-v3 [17] use techniques like encoder-decoder architecture, wider +receptive fields and dilated/atrous convolutions to improve the segmentation performance. More recently, another +line of work focused on transforming the task of semantic segmentation to sequence-to-sequence prediction, where, +self-attention mechanism is introduced using transformers [29] to encode the global context in each layer [30, 18] for +subsequent decoding. However, a downside of using transformer based technique is their computational complexity. +On the other hand, semi-supervised learning (SSL) can train DL models with a small set of annotated data by leveraging +the unlabelled data for better representation learning hence boosting the performance. SSL methods consist of different +techniques to incorporate unlabelled data for learning including pseudo labelling [31, 32, 33], generative adversarial +modelling [34, 35, 36, 37], consistency training [38, 39, 33, 40] and entropy minimisation [41, 42, 43]. However, SSL +methods have an additional issue related to overfitting of small labelled input data which may lead to poor generalisation. +In this paper, we propose a novel consistency based SSL method for semantic segmentation which leverages unlabelled +data in varying contexts and feature perturbations. Consistency regularisation is enforced by using context-aware +contrastive learning in changing contexts and cross-consistency training is used to handle feature perturbations along +with entropy minimisation for confident predictions. The main purpose of consistency regularisation is to enforce +the model to output consistent predictions for unlabelled data under changing conditions. For consistency to work +effectively, input space must hold the cluster assumption constraint i.e., same label is most likely to be shared among +the nearby samples thus forming a cluster. Therefore, high density regions would correspond to clusters (i.e., samples +with same labels) whereas the low density regions are separation spaces (i.e., object boundaries). As for histology +and natural images, the pixel space might not hold the constraint of cluster assumption as it can be seen in Figure +1. The low density regions (i.e., high average distance) do not align well with the class boundaries in most of the +scenarios e.g., in 1st row we observe low density regions throughout the image, while, in the last row there exist a +cluster of high density regions for foreground object i.e., road. However, the cluster assumption holds in encoders +latent feature space [39], as we show and discuss later in this paper’s Figure 10. Therefore, we applied the feature +perturbations to encoders output rather than the input images. Also, due to the limited labelled data, the model may +become overly dependent on just context overlooking the objects themselves, losing self-awareness [40]. Therefore, +to enforce consistency against changing contexts, we propose context-aware contrastive learning which helps the +model learn high-level semantic features by contrasting the positive and negative pairs of images in different contexts. +As shown in Figure 2, under varying contexts the model trained in a fully supervised manner is unable to produce +consistent feature distributions as compared to our proposed method consistency regularisation in varying contexts and +feature perturbations for semi-supervised semantic segmentation of Histology Images (CRCFP) with consistent feature +distributions. While context-aware consistency brings robustness to changing contexts, cross-consistency training can +help the model learn invariant feature representations that is robust to small perturbations. While context-aware and +cross-consistency training regularisation can bring consistency in encoder’s features representations, it often fails to +optimise the pixel classifier leading to less confident prediction maps. Finally, entropy minimisation coupled with +aforementioned techniques helps the model acquire high quality and confident predictions. We extensively evaluated our +proposed CRCFP on two publicly available histology image datasets BCSS [44] and MoNuSeg [45] for two different +semantic segmentation tasks i.e., tissue region segmentation and nuclei segmentation. In summary, our contributions +are as follows: +• We propose a consistency regularisation based SSL method against varying contexts and perturbations using a +novel combination of context-aware consistency loss and cross-consistent training for feature generalisability. +2 + +SSCL +Figure 1: (1st column) Example images from histological (BCSS) and natural (PASCAL VOC 2012) datasets; (2nd +column) Respective masks showing the foreground and background objects with boundaries; (3rd column) Average +Euclidean distance L2 between the central patch of size 21 × 21 with four overlapping patches in the immediate +neighbours in RGB colour space. Note that the darker blue colour represents the low density regions corresponding to +high average distance. +• To improve the confidence of final predictions for pseudo labelling, entropy minimisation is employed on top +of context-aware and cross-consistent regularisation. +• We demonstrate our method on two different semantic segmentation tasks i.e., cancer region and nuclei +segmentation on two publicly available large histology datasets. +• Extensive experiments showed superior performance of our method outperforming the state-of-the-art (SOTA) +SSL methods with extensive ablation studies. +3 + +Image +L2 +maskSSCL +Figure 2: (a) Images from the BCSS dataset with overlapping regions cropped sequentially from the same image +to mimic changing contexts; (b) UMAP visualisations of features embedding distributions extracted from a fully +supervised model; (c) UMAP visualisations of feature embedding distributions extracted from our semi-supervised +model. Note that the feature embeddings are represented in the same UMAP space where dots with same colour +represents feature embedding from the same class. +2 +Literature Review +2.1 +Semantic Segmentation +The transformation of pixel values of an image to class labels using high level features is known as semantic segmentation +and is fundamentally a challenging task. FCN extracts meaningful visual hierarchical features for various computer +vision tasks e.g., classification, segmentation and object detection. However, due to the pooling layers spatial information +is lost in aggregation which is vital in segmentation tasks and results in smaller output [26]. Encoder-decoder based +architectures solve this issue by recovering and refining the output spatially in a step wise fashion [46, 47, 48]. Further +improvements can be made possible with the help of skip connections which results in more refined boundaries and +confident predictions [16]. However, the downside of the encoder-decoder architectures is a limited receptive field +resulting in missing long-range dependencies. Dilated/atrous convolutions [17, 49, 50, 24], spatial pyramid pooling +[51, 27, 52, 7] and attention based algorithms [53, 54, 55, 18] can enable aggregation of context by using larger receptive +fields or maintaining spatial information. More recently attention mechanism [29] has been used to replace limited +local receptive field of convolutions with global contexts using transformers. Images are transformed into sequence of +patches for transformer [56] to process as transformers capture more consistent global contexts due to their self-attention +mechanisms [30, 18, 57]. Despite the advancements and improvements in semantic segmentation the bottleneck for +high accuracy still remains to be dependent on pixel-wise annotations. +4 + +a) +b) +c)SSCL +Figure 3: Overview of the proposed framework (CRCFP). The encoder and decoder are trained in a supervised manner +with the cross-entropy (CE) loss for the labelled instances. For unlabelled instances, two cropped patches with partial +overlap together with the input image were passed through the encoder, where the input image is used for contrastive +and cross-consistency learning. +2.2 +Semi-Supervised Learning +Semi-supervised learning (SSL) exploits the unlabelled data on top of limited labelled data for improving the model +performance and internal feature representation. Recently SSL based methods have been widely adopted in the computer +vision domain [58]. Popular SSL techniques include pseudo labelling [31, 32, 33] where the model trained on limited +data is used to predict the labels for unlabelled data known as pseudo labels. Generative adversarial based methods +improve the generalisability of the trained model using various perturbations in the direction of maximum vulnerability, +resulting in aligning the distributions of labelled and unlabelled input in latent space [35, 36, 59, 37]. Data interpolation +based methods aim to augment input space to create perturbed linear inputs for training models [60, 61, 62], Temporal +ensembling based methods aim to ensemble predictions over the epochs using momentum/moving average to enforce +consistency between the predictions [63, 64]. Self-supervised learning based consistency training aims to contrast the +unlabelled input using pre-text tasks for learning important representations [65, 66, 67, 33, 40] and entropy minimisation +based method aims to maximise label assignment to either of the labels [41, 42, 43]. +2.3 +Contrastive Learning +Learning by contrasting pairs of similar (positive) and dissimilar (negative) images for improved representation learning +is known as contrastive learning [68, 65, 69]. Several loss functions have been proposed from maximum margin loss +[70], triplet loss [71], N-pair loss [72] to contrastive predicting coding (CPC) [73] proposing mutual information based +InfoNCE loss to improve contrastive learning. Contrastive learning has been used in both supervised and unsupervised +learning tasks in conjunction with self supervision [66, 65, 74]. Recently, it has been established that using more +accurate positive and negative pairs along with larger batch sizes improves the quality of learned representations with +heavy augmentations. Memory banks are adopted when large batches are not computationally feasible (i.e., doesn’t fit +the GPU memory) for contrastive loss using a large set of negative samples. +2.4 +Semi-Supervised Semantic Segmentation +SSL based semantic segmentation approaches utilise the aforementioned techniques to extract knowledge from +unlabelled data. Recently, CutMix, MixUp, and CutOut based augmentation techniques were used togather with the +5 + +Supervised branch +CE Loss Lsup +Pix. Classifier +Encoder +Decoder +() +(h) +(g) +Entropy Lent +Yi +xi +Projector +Yu +(Φ) +Unsupervised branch +Xul +βu1 +u2 +-ve +Aux. Pix ++ve +Classifier +(Cj) +Perturbation +. +Aux. Pix. +Classifier +Xu2 +(CK) +ul +Yu2SSCL +Figure 4: Directional contrastive loss working for context-aware consistency, where from ϕu1, ϕu2 overlapping area +(yellow overlay) positive pixels with higher confidence pull each other closer (orange arrows) while negative pixels +from ϕu2 as well as from memory bank push each other apart (red arrows). Where class masks ˆyu1, ˆyu2 (dashed green +arrows) were applied to get the negative samples from ϕu2 and from the memory bank illustrated in the grey overlay. +student-teacher model where consistency was enforced between the mixed predictions [75]. Guided collaborative +training (GCT) by [76] performed network perturbations with the help of different network initialisation and enforced +the dynamic consistency constraint between the predictions. Cross-consistency training (CCT) by [39] performed +perturbations on the main encoder’s features and enforced consistency over the multiple decoders output making it +robust to various perturbation types. Context-aware consistency by [40] proposed directional consistency loss for +contrasting different contexts by cropping two overlapping patches of the same input to improve the representation +learning. Recently, in the field of computational pathology, a few methods for semi-supervised semantic segmentation +have been proposed. [77] proposed a semi-supervised method for signet detection using with the help of self-supervised +learning for label generation. [78] proposed a two stage SIM-FixMatch approach utilising self-supervised learning +in the first stage and then using FixMatch for pseudo label generation along with consistency regularisation. [79] +proposed an exponential moving average (EMA) student-teacher framework where the model is trained using the noisy +labels to enforce the consistency over similar and dissimilar patch pairs. Cross-patch dense contrastive learning by +[43] proposed a student-teacher based method to enforce EMA based consistency over predictions and to improve the +internal representations. Pixel-wise contrastive loss was applied to background and foreground patches for improving +the internal feature representations. +In this work, we show that (a) by enforcing consistency over varying contexts and feature perturbations in encoder’s +latent space, models can generalise better and (b) minimising entropy in output prediction maps can boost the confidence +of the final predictions resulting in improved performance. +3 +The Proposed Method +Figure 3 shows an overview of the proposed framework (CRCFP), where L = {(x1 +l , y1 +l ), ..., (xn +l , yn +l ) : n ∈ [1, ..., N]} +represents the N labelled images while U = {(x1 +u), ..., (xm +u ) : m ∈ [1, ..., M]} represents the M unlabelled images. +Labelled and unlabelled images xl and xu were sampled from L and U respectively in batches. Both images xl, xu are +of H × W × D spatial dimensions with corresponding pixel-wise mask yl = RC×H×W only for labelled image where +C is the number of classes. Each labelled image xl is passed through the supervised pathway of the CRCFP framework +(blue arrows in Figure 3) whereas the unlabelled images xu pass through the unsupervised pathways of the framework +(brown arrows in Figure 3) along with two overlapping patches extracted randomly from xu denoted as xu1, xu2 (green +arrows in Figure 3). Feature maps are extracted from the input image using the shared encoder h(·; θh) and decoder +g(·; θg) as f(·; θf) = h(·; θh) ◦ g(·; θg) resulting in feature maps for each input as fl = f(xl; θf), fu = f(xu; θf), +fu1 = f(xu1; θf) and fu2 = f(xu2; θf). Further, fl and fu are processed by a pixel classifier Cf for final prediction +as ˆyl = Cf(fl; θp) and ˆyu = Cf(fu; θp) where ˆyl is optimised using the cross-entropy loss over yl as Lsup shown in +equation 1. +Lsup = − 1 +N +C +� +c=1 +N +� +i=1 +yi,c log(ˆyi,c) +(1) +6 + +Pu1 +Pu2 +-ve +e ++ve +Push Away +Pull Closer +Overlap +Mask +Ju1 +Ju2 +Memory Bank +Contrastive LossSSCL +3.1 +Context-Aware Consistency +With only the supervised loss Lsup, the model may start relying excessively on contexts due to limited labelled data. +Context-aware consistency can alleviate this issue by aligning the two different contexts of the same patch with the help +of contrastive learning. For this purpose, encoded feature maps fu1 and fu2 are projected to a low-dimensional space +using a non-linear projector φ to preserve important contextual information. The choice of non-linear projection head +as compared to linear and identity projection head is due to its superior performance [65]. The projection head φ(·; θz) +outputs projection maps as ϕu1 = φ(fu1; θz) and ϕu2 = φ(fu2; θz). Similar to [40], context-aware consistency is +maintained between the overlapping regions of ϕu1 and ϕu2 using the directional contrastive loss Lcont to keep the +feature representation consistent under different contexts as shown in 4. For computing directional consistency loss, +first class maps ˆyui were extracted using pixel classifier Cf and then maximum probability among all classes C is +maintained using max probability as it is linked with higher confidence as shown in equation 2. +ˆyui = arg max +x ∈ U +Cf(fui; θp) +(2) +where i ∈ {1, 2} and higher probability features are used to align less confident features towards the more confident +features [76, 40, 43] which can improve learning by avoiding the exchange of unreliable knowledge from the less +confident feature as shown in Figure 4. In order to extract negative samples (i.e., negative pairs), class maps as +ˆyu1 = Cf(fu1; θp) and ˆyu2 = Cf(fu2; θp) were used. A positive feature projection ϕu1+ with class map ˆyu1+ (i.e., in +case of u1 → u2), the negative samples η should have (ˆyu1+ ̸= ˆyu−) as shown in 5. Further, to avoid less confident +features from contributing towards the loss, a threshold λ is applied to avoid an exchange of knowledge between less +confident features. The ℓcont(ϕu1,ϕu2) loss for one pair is calculated as shown below, +ℓcont(ϕu1,ϕu2) = − 1 +M +� +h,w +M+. log +sim(ϕu1, ϕu2) +sim(ϕu1, ϕu2) + +� +ϕu−∈η +Mu− · sim(ϕu1, ϕu−) +(3) +sim(ϕu1, ϕu2) = exp( +ϕT +u1ϕu2 +∥ϕu2∥ ∥ϕu2∥ τ ) +(4) +M− = +� +1 +if ˆyu1+ ̸= ˆyu−, +0 +otherwise. +(5) +M+ = +� +Mc+ +if max Cf(fu1; θp) > λ, +0 +otherwise. +(6) +Mc+ = +� +1 +max Cf(fu1; θp) < max Cf(fu2; θp), +0 +otherwise. +(7) +where sim(.) is the cosine similarity measure with temperature τ, Mc+ represents the binary mask for confident +features corresponding to ϕu1+. M+ is the binary mask for positive confident samples above threshold λ. M− is the +binary mask for negative samples indicating different pseudo labels between ϕu+ and ϕu−. To increase the negative +samples, we have used the memory bank which stores features from recent batches to further increase the negative +samples for better contrastive performance [65, 40, 43]. Finally, the directional contrastive loss Lcont is calculated as +below: +Lcont = ℓcont(ϕu1,ϕu2) + ℓcont(ϕu2,ϕu1) +(8) +3.2 +Cross-Consistency Training +As context-aware consistency improves the model’s robustness towards changing contexts without losing self-awareness, +the model is still susceptible to small perturbations in the input due to limited labelled data. Therefore, in order to +leverage unlabelled data and make the model invariant to small perturbations, we utilise the cross-consistency training +[39] where fu is perturbed K times for each perturbation type and consistency is maintained between the output of +pixel classifier and auxiliary classifiers. This not only improves the model’s robustness but also regularises the main +pixel classifier towards correct predictions. We use ˆyu to regularise the pixel classifier over the mean square error +(MSE) loss by measuring the distance between the output of the main pixel classifier Cf and the output of auxiliary +7 + +SSCL +classifiers Ck +f . Formally, a perturbation function pk with k ∈ {1, K} perturbations outputs a perturbed version of the +fu as ˆf k +u = pk( ˆfu) for a perturbation type and the cross-consistency training loss Lcross can be defined as below, +Lcross = 1 +M +1 +K +� +xu∈U +K +� +k=1 +d(ˆyu, Ck +f ( ˆf k +u)) +(9) +where d measures the squared distance between the output probabilities of the main pixel classifier and perturbed pixel +classifier output. Following perturbations are applied to enforce the consistency: +Feature Noise: A uniformly sampled noise from the interval [α, β] is added to the features map fu in two steps. First +sampled noise is multiplied with fu to scale the noise relative to feature activations. Second, the scaled sample noise is +then added to the feature map fu. This makes the noise proportional to each feature activation as shown below. +Ω ∼ U(α, β) +(10) +f noise +u += (fu ⊙ Ω) + fu +(11) +Feature Dropout: A uniform sample threshold γ is used to prune the less confident activations to stop the model from +relying on those activations. This is done by first summing the fu over different channels and then normalising it using +min-max normalisation resulting in f ′ +u. Anything below γ is dropped as seen below: +γ ∼ U(α, β) +(12) +Mdrop = +� +1 +if f ′ +u < γ, +0 +otherwise. +(13) +f drop +u += Mdrop ⊙ f ′ +u +(14) +where Mdrop is the binary mask containing threshold values for pruning the activations. +DropOut: A fraction of activations are dropped out spatially where the fraction is decided using the Bernoulli +distribution with probability δ. +Mdropout ∼ Bernoulli(δ) +(15) +f dropout +u += Mdropout ⊙ fu +(16) +3.3 +Entropy Minimisation +Context-aware contrastive learning and cross-consistency training improves the encoder’s features but it often fails +to improve the final pixel classifier leading to less reliable pseudo labels corrupting the training from unlabelled data. +As higher confidence means better prediction maps resulting in more refined pseudo labels which can help train both +context-ware and cross-training with improved positive/negative pairs and pseudo labels. Hence, in order to improve +the confidence of predictions, we employ entropy regularisation following its applications in semi-supervised learning +[41, 80, 69, 43] as shown in 17 where it penalises the uncertain prediction in the unlabelled data and improves the +overall confidence of the prediction maps. +Lent = − 1 +M +M +� +m=1 +C +� +c=1 +ˆyu log ˆyu +(17) +3.4 +Training +Finally, the entire framework is trained in an end-to-end fashion using a weighted combination of the above mentioned +losses as shown below, +L = wsupLsup + wcontLcont + wcrossLcross + wentLent +(18) +where wsup, wcont, wcross and went correspond to the weights for each loss component respectively. +8 + +SSCL +Table 1: Comparison of the state-of-the-art methods with mIoU, dice score and accuracy aggregated +for 3 different random seeds as mean (standard deviation). The first column represents the fraction of +data used for training the model. +BCSS +Fraction +Method +mIoU +Dice +Accuracy +1/8 +DeepLab-v3 [17] +40.99 (7.96) +55.96 (9.1) +66.53 (6.07) +1/8 +CCT [39] +22.84 (0.54) +32.01 (0.69) +56.14 (0.70) +1/8 +CAC [40] +44.67 (6.32) +58.97 (7.51) +72.43 (3.40) +1/8 +CDCL [43] +- +- +- +1/8 +CRCFP +47.09 (6.18) +61.84 (6.59) +73.20 (3.31) +1/4 +DeepLab-v3 [17] +53.03 (0.88) +68.52 (0.94) +75.70 (0.65) +1/4 +CCT [39] +30.63 (2.19) +43.24 (3.33) +62.43 (0.89) +1/4 +CAC [40] +58.65 (0.65) +73.33 (0.55) +78.60 (0.42) +1/4 +CDCL [43] +- +- +- +1/4 +CRCFP +61.06 (0.98) +75.21 (0.74) +80.87 (0.89) +1/2 +DeepLab-v3 [17] +56.26 (1.19) +71.33 (0.98) +78.07 (1.031) +1/2 +CCT [39] +29.78 (2.56) +41.63 (4.28) +61.94 (0.17) +1/2 +CAC [40] +60.44 (1.48) +74.67 (1.16) +80.50 (0.92) +1/2 +CDCL [43] +- +- +- +1/2 +CRCFP +61.86 (0.63) +75.73 (0.59) +81.18 (0.27) +1/1 +DeepLab-v3 [17] +61.29 (0.26) +75.49 (0.12) +81.10 (0.09) +4 +Experiments +4.1 +Datasets +We evaluated the proposed framework on two publicly available datasets, the Breast Cancer Semantic Segmentation +(BCSS) [44] and Multi-organ Nucleus Segmentation Challenge (MoNuSeg) [45] dataset for semantic segmentation. +The data was obtained from the respective challenge pages hosted on Grand Challenge for Medical Image Analysis +website (https://grand-challenge.org/). +MoNuSeg. The MoNuSeg challenge was organised as a MICCAI 2018 satellite event and contains 21,623 annotated +nuclei from 30 H&E stained images for training and contains 7223 annotated nuclei from 14 H&E stained images +for testing purposes. Annotations were done by engineering students and then an expert pathologist served as quality +control for the annotated nuclei. Each image is of size 1000 × 1000 extracted from a WSI scanned at 40× resolution +of an individual patient obtained from The Cancer Genome Atlas Program (TCGA) [81]. WSIs are sampled from 18 +different centres and 7 different organs including breast, liver, kidney, prostate, bladder, colon and stomach with various +tumour stages. +BCSS. The BCSS challenge was conducted in 2021 and contains over 20,000 annotated regions of interest (ROI) from +151 H&E stained WSIs with the same number of patients from TCGA [81]. 25 annotators including pathologists, +residents, and medical students helped annotate this large scale data into 25 refined categories which are later merged +into 5 broad categories as tumour, stroma, inflammatory, necrosis, and others. For this work, we have used the same 5 +broad categories by relabelling the regions and then split them into training and test centres according to the [44] where +there were 14 centres for training and 7 centres for testing. +4.2 +Implementation Details +4.3 +Data Preparation +In order to validate the CRCFP framework, we evaluated it against different label proportions of each dataset. Where +for BCSS different label proportions were collected from different centres (hospitals) to make training more susceptible +to variation in colours enabling domain shift. DL methods often fail to perform well on samples from a different +domain (centres), mainly due to domain shift, this also makes it a domain generalisation problem. Therefore, the +training set was divided into portions by diving the total training centres as 1/1 (full), 1/2 (half), 1/4 (quarter), and 1/8 +(one-eighth) centres where 1/8 results in training images coming from only 1 centre, while the test set remains intact +as it is. Similarly, for 1/4 (quarter) training images comes from 4 centres and 7 centres for 1/2 (half). For MoNuSeg, +different label proportions were based on training images themselves and are then divided into 1/1, 1/8, 1/16, and 1/32 +9 + +SSCL +proportions to make it comparable to the work of [43]. Further, this whole process is repeated using 3 different random +seeds and then the results are reported using mean aggregation with standard deviation. +4.4 +Evaluation metrics +In order to compare our proposed method quantitatively with other state-of-the-art methods (SOTA), we have used +different quantitative measures including accuracy, F1-score (Dice) and mean intersection over union (mIoU) for both +the datasets. +4.5 +Network Architecture +We used DeepLab-v3 [17] as base segmentation network with ResNet-50 [82] encoder pretrained on ImageNet [83]. +Where the projector consists of two fully connected (FC) layers of size 128 with ReLU as an intermediate activation +layer, FC → ReLU → FC. Pixel classifiers consist of convolutional layers with a kernel of size 1 × 1 to reduce the +number of channels to total classes with non-linear ReLU activation. The final layers upsamples the output using +bi-linear interpolation to match the input size as H × W × C. +4.6 +Experimental Settings +The input size for the proposed framework for both labelled and unlabelled images was 320 × 320. For contrastive +learning, two patches xu1 and xu2 were randomly cropped from the unlabelled image with an overlap in the range of +[0.1, 1.0] and are resized to match the input dimensions. For positive filtering mask λ was set to 0.75 and τ = 0.1 +as temperature for cosine similarity. For cross-consistency training, number of auxiliary pixel classifiers were set to +K = 4 for each perturbation type and for feature noise perturbation the parameters α = −0.3, β = 0.3 were used. +For feature dropout perturbation, α = 0.75, β = 0.9 were used as they can help remove approximately 10% to 30% +of active regions from the feature map. Also, for simple Dropout the probability for Bernoulli distribution was set to +δ = 0.5. During training, a set of standard augmentations were applied to the input images including horizontal and +vertical flipping, gaussian blur, colour and grey scaling. PyTorch was used for implementing this framework where +for optimisation we train the whole framework for 80 epochs. For the initial 5 epochs, only supervised loss Lsup was +used to train the whole framework as this provides a stable head start for the semi-supervised learning. The batch size +of 8 was used for labelled and unlabelled images with stochastic gradient descent (SGD) optimiser and a learning +rate of 0.001. As a common practice, poly learning rate decay policy was used where the learning rate is scaled using +1 − ( +iter +max _iter)power at each iteration with power = 0.9. Weights with respect to different losses Lsup, Lcont, Lcross +and Lend were set to fixed values as wsup = 1, wcont = 0.1, wcross = 0.01 and went = 0.01 respectively. All models +were trained with the same configurations for both datasets where two Nvidia GeForce 1080Ti GPUs are used for +training. +5 +Results +The performance of our proposed method (CRCFP) compared to recent SOTA semi-supervised semantic segmentation +methods including DeepLab [17], CCT [39], CAC [40] and CDCL [43] 1 is shown in Table 1 and Table 2. As these +methods are implemented using different configurations and baseline segmentation models. For a fair comparison, +we have implemented these methods within a unified framework with the same segmentation baseline, experimental +settings and data augmentations. +Table 1 shows the performance of our CRCFP model compared to supervised and semi-supervised methods on all +matrices for the BCSS dataset. Particularly, when 1/8 proportion of the training centres was used, it can be seen that +in terms of mIoU our method performs ∼6% better than the supervised method and ∼3% better than the recent CAC +[40]. Similarly, its worth noting that with 1/4 of the total centres, the CRCFP performance is almost similar to fully +supervised method with all data. On the other hand, the poor performance of CCT [39] can be attributed towards heavy +perturbations applied directly to the features where it brings perturbed features from different contexts closer without +pushing dissimilar apart whereas CAC [40] not only bring them closes but also pushes away the features from different +classes. However, it focuses more on encoder feature generalisation leaving pixel-classifier with less confident features. +Figure 5 shows visual comparison of CRCFP with the SOTA algorithms, where, it can be observed that prediction maps +of CRCFP are better as compared to the rest, specially highligted in the dashed red boxes. +1CDCL [43] cannot be applied to the multi-class problem as it divides the patches into foreground and background only for +contrastive learning. +10 + +SSCL +Figure 5: Visual comparison of the CRCFP with different state-of-the-art methods for tissue region +segmentation with 1/2 training data only. Dashed red box highlights superior performance of our +method as compared to SOTA methods. +Table 2 shows the performance of CRCFP surpassing other SOTA methods in all data proportions and metrics, especially +in 1/32 proportion of the MoNuSeg dataset. It can be seen that our CRCFP outperforms the CAC [40] by 4.32% in mIoU +with a smaller standard deviation of 0.22. It can also be observed that fully supervised models are more susceptible +to domain generalisation problem from the table as in 1/32 proportion of the training images the performance of +DeepLab-v3 [17] is 4% better than the 1/16 proportion of the training images whereas there is more data available in +the latter. This is due to the fact that in a random sampling of training images some training images are better indicators +of the testing distribution due to similarities in the same stain, organ and tumour stage. However, most of the SOTA +semi-supervised algorithms solve this issue with the help of unlabelled data as it can be seen that the performance +increase with the increase in data for all these methods. Figure 6 shows a visual comparison of CRCFP with SOTA +methods where it can be seen that our approach predicts fewer false positives as compared to CDCL [43]. +Further, in order to validate the contribution of each component (i.e., context-aware consistency, cross-consistency +training and entropy minimisation) we conducted an extensive ablation study. The ablation study is performed on the +BCSS dataset due to its complexity and multi-class nature, where we studied the effect of using all data proportions for +the different encoders and in stripping the framework. While studying the effect of negative samples and the number of +auxiliary pixel classifiers we used 1/8 data proportion. +5.1 +Encoder +To verify the performance boost by plugging in a bigger encoder in the base segmentation network, we replaced ResNet- +50 with ResNet-101 for all data proportions. Table 3 shows the performance of the proposed CRCFP framework with a +11 + +Input +GT +SupOnly +CCT +CAC +Ours +Tumor | Stroma / Inflammatory/ Necrosis / OthersSSCL +Table 2: Comparison of the state-of-the-art methods with mIoU, dice score and accuracy aggregated +for 3 different random seeds as mean (standard deviation). The first column represents the fraction of +data used for training the model. +MoNuSeg +Fraction +Method +mIoU +Dice +Accuracy +1/32 +DeepLab-v3 [17] +60.09 (2.07) +73.89 (1.77) +79.45 (1.40) +1/32 +CCT [39] +41.13 (0.06) +50.31 (0.29) +74.33 (0.35) +1/32 +CAC [40] +67.40 (1.12) +79.33 (0.90) +86.14 (0.62) +1/32 +CDCL [43] +62.72 (1.83) +75.95 (75.95) +81.66 (1.57) +1/32 +CRCFP +71.72 (0.22) +82.60 (0.24) +88.86 (0.23) +1/16 +DeepLab-v3 [17] +56.20 (5.76) +70.80 (4.58) +75.27 (75.27) +1/16 +CCT [39] +40.99 (0.08) +49.56 (0.29) +75.17 (0.36) +1/16 +CAC [40] +71.44 (1.11) +82.47 (0.92) +88.27 (0.11) +1/16 +CDCL [43] +60.63 (1.15) +74.40 (0.90) +79.99 (1.37) +1/16 +CRCFP +72.08 (2.07) +82.91 (1.52) +88.58 (1.16) +1/8 +DeepLab-v3 [17] +59.67 (2.99) +73.59 (2.32) +78.98 (2.89) +1/8 +CCT [39] +40.9 (0.13) +50.00 (0.54) +74.63 (0.42) +1/8 +CAC [40] +74.56 (0.42) +84.73 (0.30) +89.91 (0.23) +1/8 +CDCL [43] +57.07 (1.45) +71.63 (1.14) +76.29 (1.48) +1/8 +CRCFP +75.57 (0.85) +85.19 (0.54) +90.28 (0.42) +1/1 +DeepLab-v3 [17] +71.29 (0.16) +82.49 (0.11) +87.52 (0.09) +Figure 6: Visual comparison of the CRCFP with different state-of-the-art techniques in nuclei image +segmentation with 1/8 training data only. GT represents the ground truth nuclei masks, and SupOnly +shows the models trained with labelled training data only. Red pixels correspond to the ground truth +while green shows the prediction. Yellow pixels represent the overlap regions between the prediction +and ground truth. +bigger encoder and it can be seen that there is a performance boost overall for most of the methods, especially for CCT +12 + +Input +GT +SupOnly +CCT +CAC +CDCL +OursSSCL +Table 3: Comparison of the state-of-the-art methods on mean (standard deviation) of mean intersection +of union (mIoU), dice score and accuracy with baseline encoder as ResNet-101. The first column +represents the fraction of data used for training the model. +BCSS +Fraction +Method +mIoU +Dice +Accuracy +1/8 +DeepLab-v3 [17] +37.50 (6.61) +51.73 (7.51) +64.89 (5.92) +1/8 +CCT [39] +31.71 (4.64) +45.66 (5.96) +59.42 (3.46) +1/8 +CAC [40] +46.91 (6.79) +61.92 (6.74) +72.01 (3.85) +1/8 +CRCFP +47.15 (6.76) +61.27 (7.72) +72.57 (2.82) +1/4 +DeepLab-v3 [17] +55.18 (1.88) +70.30 (1.70) +77.37 (1.25) +1/4 +CCT [39] +42.63 (0.98) +58.35 (1.26) +66.94 (0.73) +1/4 +CAC [40] +61.48 (0.73) +75.52 (0.47) +80.78 (0.84) +1/4 +CRCFP +62.01 (0.40) +75.94 (0.29) +81.18 (0.49) +1/2 +DeepLab-v3 [17] +60.37 (1.89) +74.5 (1.58) +80.57 (0.86) +1/2 +CCT [39] +44.01 (0.65) +59.64 (0.55) +67.66 (1.33) +1/2 +CAC [40] +61.95 (0.72) +75.77 (0.67) +81.09 (0.27) +1/2 +CRCFP +63.01 (0.09) +76.57 (0.09) +81.67 (0.12) +1/1 +DeepLab-v3 [17] +62.33 (1.04) +76.22 (0.73) +81.68 (0.58) +[39]. However, it can be observed that CRCFP with a smaller encoder (i.e., ResNet-50) still performs comparable/better +than other SOTA techniques with a bigger encoder e.g., in 1/8 proportion CAC [40] with ResNet-101 achieves mIoU of +46.91 where CRCFP with ResNet-50 achieves mIoU of 47.09 showing superiority of our proposed method. Also, it is +worth mentioning that with ResNet-101 the standard deviation we observed with ResNet-50 was reduced, owing to the +fact that bigger encoders are more stable for semi-supervised learning frameworks. Overall the CRCFP framework +provides improved and stable performance with bigger encoders as compared to the other methods. +5.2 +Network Schemes +We validated the contribution of each component by breaking down the whole framework with respect to different losses +and called them network schemes. We started with a baseline segmentation network i.e., DeepLab-v3 with ResNet-50 +as SupOnly, Scheme.1 consists of using context-aware consistency loss, Scheme.2 consists of using context-aware +consistency loss with entropy minimisation and finally Scheme.3 is our proposed framework with context-aware +consistency loss with cross-consistency training and entropy minimisation. Table 4 shows the schemes with respect +to their respective losses being used, it can be seen that with each component’s addition we can see improvement +in overall performance. E.g., in 1/8 data proportion, the addition of context-aware consistency brings about 4% of +improvement while entropy minimisation further bumps it up by 1% and finally cross-consistent training beings about +2% of improvement accumulating the overall performance to ∼7% from baseline supervised model. Also, for other +data proportions the performance boost is not that much significant with the addition of these Scheme.2 and Scheme.3 +as compared to Scheme.1. However, its worth mentioning that the standard deviation of Scheme.2 and Scheme.3 as +compared to Scheme.1 is smaller which is due to the fact that these schemes brings confidence in prediction maps thus +improving the overall performance with stability. +5.3 +Negative Samples +As increasing the negative samples in training contrastive learning framework boosts the performance of the underlying +model. This is done mostly by increasing the batch size to 2048 or 4096 where possible as the bigger the batch size +the more samples you get for comparisons [65, 84]. However, where it is not possible, another workaround is to use a +memory bank where negative samples from previous batches were stored for later use. Therefore, in order to get the +upper bound of performance in our framework with respect to negative samples, we have experimented with different +number of negative samples as seen in Table 5. It can be noticed that with increasing negative samples, the performance +increases for a while and then it reaches the plateau and then increases with very little gain as it can also be observed +visually in Figure 7. This can be due to the fact that there might not be many variations to cover in the training set +with more negative samples, thus reaching stable performance or very little performance gain. Also, due to gradient +checkpoint functionality in PyTorch adding more negative samples does not effect the training efficiency drastically but +does consume more compute time and memory. Hence, based on these observations, for this study, we set the number +of negative samples to 1200 for its memory vs accuracy trade-off. +13 + +SSCL +Table 4: CRCFP breakdown in different Schemes with respect to their loss functions. SupOnly correspond to baseline +segmentation model with Lsup loss only. Scheme.1 corresponds to addition of Lcont loss on top of SupOnly. Scheme.2 +corresponds to addition of Lent on top of Scheme.1 and finally Scheme.3 is addition of Lcons on top of Scheme.2. +Method +Split +Lsup +Lcont +Lent +Lcons +mIoU +SupOnly +1/8 +✓ +× +× +× +40.99 (7.96) +Scheme.1 +1/8 +✓ +✓ +× +× +44.67 (6.32) +Scheme.2 +1/8 +✓ +✓ +✓ +× +45.76 (6.12) +Scheme.3 +1/8 +✓ +✓ +✓ +✓ +47.09 (6.18) +SupOnly +1/4 +✓ +× +× +× +53.03 (0.88) +Scheme.1 +1/4 +✓ +✓ +× +× +58.65 (0.65) +Scheme.2 +1/4 +✓ +✓ +✓ +× +59.97 (1.47) +Scheme.3 +1/4 +✓ +✓ +✓ +✓ +61.06 (0.98) +SupOnly +1/2 +✓ +× +× +× +56.26 (1.19) +Scheme.1 +1/2 +✓ +✓ +× +× +60.44 (1.48) +Scheme.2 +1/2 +✓ +✓ +✓ +× +60.87 (1.39) +Scheme.3 +1/2 +✓ +✓ +✓ +✓ +61.86 (0.63) +Table 5: Performance of CRCFP with respect different number of negatives samples used while training Lcont loss with +BCSS data split of 1/8 +# +mIoU +Dice +Accuracy +100 +45.62 (8.10) +60.46 (8.87) +67.38 (8.14) +500 +45.81 (7.78) +59.86 (8.88) +70.05 (5.30) +1200 +47.09 (6.18) +61.84 (6.59) +73.20 (3.31) +1600 +47.16 (6.70) +61.81 (7.05) +72.68 (3.07) +2400 +47.60 (6.09) +62.14 (6.80) +73.58 (3.43) +3200 +48.34 (5.25) +63.83 (5.01) +73.06 (3.59) +5.4 +Auxiliary Pixel Classifier +To see the effect of a varying number of auxiliary pixel classifiers with respect to different perturbations we conducted +experiments with K ∈ {1, 2, 4, 6, 8, 10} as seen in Table 6. It can be seen that increasing the number of pixel classifiers +per perturbation increases the performance but the upper bound is achieved soon after it reaches K = 4, from where the +performance drops slightly as can be observed in the Figure 8. Increasing the number of perturbations can result in +more aggressive penalisation of the model overall as it accumulates to K × 3 losses which can deviate the model from +learning meaningful representations. Based on this observation we set the number K = 4 for our study for the rest of +the comparisons for both datasets. +6 +Discussion +Interpretable features from histology slides can be extracted by segmenting objects/structures from ROIs e.g., nuclei, +glands, stroma, tumours etc. Intrepretable features can enable discovery of novel digital bio-markers with explanations +for histology images for hard tasks like survival analysi [85, 10] and mutation prediction [86, 87, 8]. Therefore, it is +vital for the downstream tasks to have good quality and precise segmentation of region of interests. For this purpose, +utilising unlabelled data for representation learning not only improves performance but also improves the internal +Table 6: Performance of CRCFP with respect different number of K auxiliary classifiers used while training Lcons loss +with BCSS data split of 1/8 +# +mIoU +Dice +Accuracy +1 +43.94 (7.95) +58.9 (8.27) +69.14 (5.54) +2 +45.76 (7.51) +60.44 (7.88) +71.23 (4.23) +4 +47.09 (6.18) +61.84 (6.59) +73.20 (3.31) +6 +46.48 (6.26) +61.01 (6.73) +72.60 (3.68) +8 +46.72 (6.88) +61.38 (7.29) +72.25 (3.89) +10 +45.68 (6.79) +60.64 (7.20) +71.84 (3.99) +14 + +SSCL +Figure 7: Performance graph with respect varying number of negatives samples used while training Lcont loss with +BCSS data split of 1/8 +representations for better learning. The qualitative and quantitative results along with the ablation study has shown +superior performance of our proposed CRCFP with respect to other SOTA methods. However, it’s worth exploring +internal representations of the learned models (i.e.,feature embeddings) to account for (1) Consistency in feature space +(2) Cluster assumption, for the sake of validation of aforementioned claims in the introduction section. +6.1 +Feature Space Visualisation +In order to observe the consistency in feature space, feature embeddings were extracted from both our SSL based +CRCFP trained on 1/2 proportion of the training data vs DeepLab-v3 trained on all data (i.e., fully supervised), since +they achieved same performance. Extracted feature maps were upsampled to match the size of the input image (i.e., 320 +× 320) and are then mapped to lower dimensions using UMAP [88] for visualisation purposes. It can be seen in Figure +9 that the feature embedding distributions are consistent with varying contexts specially in the 1st and 2nd column for +our CRCFP model as compared to fully supervised ones. Similarly, it can be observed in the other examples where the +varying context is inherent due to the sequential overlap in patch tessellation process. Whereas, the fully supervised +model is susceptible to perturbations in contextual cues as can be observed. It is worth noting the last two columns +where the shape of feature embedding distribution changes along with the orientation of same samples points from the +same class. Specially, the ones shown in yellow dots as compared to our proposed framework where the distributions +are almost consistent under these perturbations. +6.2 +Cluster Assumption +Consistency regularisation based methods work on the basis of cluster assumption and have achieved SOTA results +in semi-supervised classification and segmentation. The main idea behind consistency regularisation is to have high +and low density regions where samples closer to each other are likely to share the same label forming a high density +region with a low average distance. While the class boundaries are likely to be aligned with the low density regions +15 + +→mloU --Dice -→Accuracy +80.0 +75.0 +70.0 +65.0 +PERFORMANC +60.0 +55.0 +50.0 +45.0 +40.0 +100 +500 +1200 +1600 +2400 +3200 +# OF NEGATIVE SAMPLESSSCL +Figure 8: Performance graph with respect varying number of pixel classifiers used while training Lcons loss with BCSS +data split of 1/8 +Figure 9: (a) BCSS dataset images with overlapping regions cropped sequentially from the same +image to mimic changing contexts. (b) UMAP visualisations of features embedding distributions +extracted from a fully supervised model. (c) UMAP visualisations of feature embedding distributions +extracted from a semi-supervised model. Note that the feature embeddings are represented in the +same UMAP space where dots with same colour represents feature embedding from the same class. +16 + +→mloU -Dice →-Accuracy +75.0 +70.0 +65.0 +PERFORMANCE +60.0 +55.0 +50.0 +45.0 +40.0 +1 +2 +4 +6 +8 +10 +# OF AUX. CLASSIFIERSb) +CSSCL +Figure 10: (a) Example images from BCSS test dataset. (b) Respective masks showing the foreground +and background pixels. (c,d) Average euclidean distance L2 between the central patch of size 21 × 21 +with four overlapping patches in the immediate neighbours in RGB colour space and feature space +respectively. Note that for feature space visualisation encoder embeddings were upsampled to map +input size. The darker blue colour represents the low density regions corresponding to high average +distance. +i.e., high average distance. In order to observe cluster assumption, feature embeddings were extracted from CRCFP +and were compared against RGB colour space as shown in Figure 10. Extracted feature maps were upsampled to +match the size of the input image and then the average euclidean distance between each patch of size 21 × 21 centred +around its 4 immediate spatial neighbours (left, right, top and bottom) was calculated. It can be seen in Figure 10(d) +that the class boundaries are much more aligned and apparent in the feature space as compared to the colour space +where the boundaries doesn’t align well thus violating cluster assumption. This can be due to the fact that the CNNs +at higher layers tends to learn more semantic based features from the basic low-level features. Also, interestingly the +background/fat represented in white colour in input images somewhat holds the high density regions because there is +not much change in colour values for that region. While the rest of the tissue area is not very homogeneous in pixel +values due to the presence of cells of various shapes and sizes. +7 +Conclusions +In this work, we haved presented a novel consistency based semi-supervised learning based semantic segmentation +framework for region and nuclei segmentation in histology images. The proposed method is invariant to varying +contexts and perturbations making it efficient and robust for semantic segmentation tasks. We have shown that context- +aware consistency learning can exploit unlabelled images efficiently with the help of cross-consistency training and +entropy minimisation. Extensive experiments on two publicly available large histopathological datasets have shown +the superiority of the CRCFP framework by achieving new SOTA results for semi-supervised semantic segmentation. +Also, detailed ablation studies for different network parameters and components show the contribution of each network +component, demonstrating the effectiveness of our method. Future directions include improvements to the proposed +method with respect to improving the context-aware loss for minor classes and finding histology specific perturbation +e.g., targeting stain variations, on a large multi-centric histopathological dataset. Large multi-centric data is vital for the +validation of the study as the quality of downstream analysis is highly dependent on the segmented histology primitives. +17 + +a) +b) +c) +d)SSCL +8 +Declaration of competing interest +The authors declare that they have no known competing financial interests or personal relationships that could have +appeared to influence the work reported in this paper. +9 +Acknowledgements +RMSB is funded by the Chancellor Scholarship from University of Warwick. SEAR and NMR are part of the PathLAKE +digital pathology consortium, which is funded by the Data to Early Diagnosis and Precision Medicine strand of the +governments Industrial Strategy Challenge Fund, managed and delivered by UK Research and Innovation (UKRI). +NMR was also supported by the UK Medical Research Council (grant award MR/P015476/1) and the Alan Turing +Institute. +References +[1] Navid Alemi Koohbanani, Mostafa Jahanifar, Neda Zamani Tajadin, and Nasir Rajpoot. 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Umap: Uniform manifold approximation and projection for +dimension reduction. arXiv preprint arXiv:1802.03426, 2018. +22 + diff --git a/R9FPT4oBgHgl3EQfqTWc/content/tmp_files/load_file.txt b/R9FPT4oBgHgl3EQfqTWc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1798fde0fa37424e821397f66d503bfd287fa84 --- /dev/null +++ b/R9FPT4oBgHgl3EQfqTWc/content/tmp_files/load_file.txt @@ -0,0 +1,1111 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf,len=1110 +page_content='CONSISTENCY REGULARISATION IN VARYING CONTEXTS AND FEATURE PERTURBATIONS FOR SEMI-SUPERVISED SEMANTIC SEGMENTATION OF HISTOLOGY IMAGES Raja Muhammad Saad Bashir Tissue Image Analytics Centre University of Warwick Coventry, United Kingdom saad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='bashir@warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='uk Talha Qaiser Tissue Image Analytics Centre University of Warwick Coventry, United Kingdom talha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='qaiser@warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='uk Shan E Ahmed Raza Tissue Image Analytics Centre University of Warwick Coventry, United Kingdom shan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='raza@warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='uk Nasir M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Rajpoot Tissue Image Analytics Centre University of Warwick Coventry, United Kingdom n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='rajpoot@warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='uk ABSTRACT Semantic segmentation of various tissue and nuclei types in histology images is fundamental to many downstream tasks in the area of computational pathology (CPath).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' In recent years, Deep Learning (DL) methods have been shown to perform well on segmentation tasks but DL methods generally require a large amount of pixel-wise annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Pixel-wise annotation sometimes requires expert’s knowledge and time which is laborious and costly to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' In this paper, we present a consistency based semi-supervised learning (SSL) approach that can help mitigate this challenge by exploiting a large amount of unlabelled data for model training thus alleviating the need for a large annotated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, SSL models might also be susceptible to changing context and features perturbations exhibiting poor generalisation due to the limited training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' We propose an SSL method that learns robust features from both labelled and unlabelled images by enforcing consistency against varying contexts and feature perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The proposed method incorporates context-aware consistency by contrasting pairs of overlapping images in a pixel-wise manner from changing contexts resulting in robust and context invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' We show that cross-consistency training makes the encoder features invariant to different perturbations and improves the prediction confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Finally, entropy minimisation is employed to further boost the confidence of the final prediction maps from unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' We conduct an extensive set of experiments on two publicly available large datasets (BCSS and MoNuSeg) and show superior performance compared to the state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Keywords Deep Learning · Semi-Supervised learning · Contrastive Learning · Computational Pathology · Whole Slide Images 1 Introduction Segmentation of fundamental objects and regions in histology images is key to several downstream analysis tasks in computational pathology (CPath) [1, 2] e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', cancer type classification [3, 4, 5, 6], tumour and glandular segmentation [7], and other tasks like mutation prediction [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Their utility is not limited to diagnosis, they have also been employed for prognostic purposes e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', tumour infiltrating lymphocytes (TILs) have been found to be significant prognostic biomarker in various types of tumours [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Similarly, tumour progression has been linked with interaction between arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='13141v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='CV] 30 Jan 2023 SSCL the tumour epithelial cells and tumour associate stroma [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Hence, it is important to segment different types of histological objects precisely as their quantification is vital to downstream analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Machine learning based traditional methods accomplished this task using different hand-crafted features e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', colour [12], texture [13, 14] and morphological features [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Recently, deep learning (DL) algorithms have gained increasing attention in semantic segmentation due to their superior performance on natural and medical images [16, 17, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, DL methods are known to be “data hungry" and require a large amount of annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Precise annotation of histology images is an expensive and laborious process, requiring up to ∼5-6 hours of an expert histopathologist’s time to annotate one whole-slide image (WSI) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' To alleviate the annotation burden, other modes of training have been proposed such as patch based segmentation [21, 22], coarse segmentation [23, 24] and interactive segmentation [1, 25] but these methods still require large-scale weak annotations involving human expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Semantic segmentation is a pixel-level classification task of predicting label for each pixel using pixel values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Most of the early DL methods were based on fully convolutional networks (FCN) [26] where pooling layers aggregate the information by focusing on “what" rather than “where" resulting in loss of spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The subsequent studies addressed this shortcoming by using pooling layers with more advanced techniques involving skip connections, encoders and boundary in formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' As semantic segmentation is more than just assigning labels to pixels, it inevitably requires some contextual information along with knowledge of colour, edges and resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' In this regard, algorithms like UNet [16], PSPNet [27], HRNet [28] and DeepLab-v3 [17] use techniques like encoder-decoder architecture, wider receptive fields and dilated/atrous convolutions to improve the segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' More recently, another line of work focused on transforming the task of semantic segmentation to sequence-to-sequence prediction, where, self-attention mechanism is introduced using transformers [29] to encode the global context in each layer [30, 18] for subsequent decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, a downside of using transformer based technique is their computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' On the other hand, semi-supervised learning (SSL) can train DL models with a small set of annotated data by leveraging the unlabelled data for better representation learning hence boosting the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' SSL methods consist of different techniques to incorporate unlabelled data for learning including pseudo labelling [31, 32, 33], generative adversarial modelling [34, 35, 36, 37], consistency training [38, 39, 33, 40] and entropy minimisation [41, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, SSL methods have an additional issue related to overfitting of small labelled input data which may lead to poor generalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' In this paper, we propose a novel consistency based SSL method for semantic segmentation which leverages unlabelled data in varying contexts and feature perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Consistency regularisation is enforced by using context-aware contrastive learning in changing contexts and cross-consistency training is used to handle feature perturbations along with entropy minimisation for confident predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The main purpose of consistency regularisation is to enforce the model to output consistent predictions for unlabelled data under changing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For consistency to work effectively, input space must hold the cluster assumption constraint i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', same label is most likely to be shared among the nearby samples thus forming a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Therefore, high density regions would correspond to clusters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', samples with same labels) whereas the low density regions are separation spaces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', object boundaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' As for histology and natural images, the pixel space might not hold the constraint of cluster assumption as it can be seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The low density regions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', high average distance) do not align well with the class boundaries in most of the scenarios e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', in 1st row we observe low density regions throughout the image, while, in the last row there exist a cluster of high density regions for foreground object i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, the cluster assumption holds in encoders latent feature space [39], as we show and discuss later in this paper’s Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Therefore, we applied the feature perturbations to encoders output rather than the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Also, due to the limited labelled data, the model may become overly dependent on just context overlooking the objects themselves, losing self-awareness [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Therefore, to enforce consistency against changing contexts, we propose context-aware contrastive learning which helps the model learn high-level semantic features by contrasting the positive and negative pairs of images in different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' As shown in Figure 2, under varying contexts the model trained in a fully supervised manner is unable to produce consistent feature distributions as compared to our proposed method consistency regularisation in varying contexts and feature perturbations for semi-supervised semantic segmentation of Histology Images (CRCFP) with consistent feature distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' While context-aware consistency brings robustness to changing contexts, cross-consistency training can help the model learn invariant feature representations that is robust to small perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' While context-aware and cross-consistency training regularisation can bring consistency in encoder’s features representations, it often fails to optimise the pixel classifier leading to less confident prediction maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Finally, entropy minimisation coupled with aforementioned techniques helps the model acquire high quality and confident predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' We extensively evaluated our proposed CRCFP on two publicly available histology image datasets BCSS [44] and MoNuSeg [45] for two different semantic segmentation tasks i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', tissue region segmentation and nuclei segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' In summary, our contributions are as follows: We propose a consistency regularisation based SSL method against varying contexts and perturbations using a novel combination of context-aware consistency loss and cross-consistent training for feature generalisability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 2 SSCL Figure 1: (1st column) Example images from histological (BCSS) and natural (PASCAL VOC 2012) datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' (2nd column) Respective masks showing the foreground and background objects with boundaries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' (3rd column) Average Euclidean distance L2 between the central patch of size 21 × 21 with four overlapping patches in the immediate neighbours in RGB colour space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Note that the darker blue colour represents the low density regions corresponding to high average distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' To improve the confidence of final predictions for pseudo labelling, entropy minimisation is employed on top of context-aware and cross-consistent regularisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' We demonstrate our method on two different semantic segmentation tasks i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', cancer region and nuclei segmentation on two publicly available large histology datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Extensive experiments showed superior performance of our method outperforming the state-of-the-art (SOTA) SSL methods with extensive ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 3 Image L2 maskSSCL Figure 2: (a) Images from the BCSS dataset with overlapping regions cropped sequentially from the same image to mimic changing contexts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' (b) UMAP visualisations of features embedding distributions extracted from a fully supervised model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' (c) UMAP visualisations of feature embedding distributions extracted from our semi-supervised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Note that the feature embeddings are represented in the same UMAP space where dots with same colour represents feature embedding from the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 2 Literature Review 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 Semantic Segmentation The transformation of pixel values of an image to class labels using high level features is known as semantic segmentation and is fundamentally a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' FCN extracts meaningful visual hierarchical features for various computer vision tasks e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', classification, segmentation and object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, due to the pooling layers spatial information is lost in aggregation which is vital in segmentation tasks and results in smaller output [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Encoder-decoder based architectures solve this issue by recovering and refining the output spatially in a step wise fashion [46, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Further improvements can be made possible with the help of skip connections which results in more refined boundaries and confident predictions [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, the downside of the encoder-decoder architectures is a limited receptive field resulting in missing long-range dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Dilated/atrous convolutions [17, 49, 50, 24], spatial pyramid pooling [51, 27, 52, 7] and attention based algorithms [53, 54, 55, 18] can enable aggregation of context by using larger receptive fields or maintaining spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' More recently attention mechanism [29] has been used to replace limited local receptive field of convolutions with global contexts using transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Images are transformed into sequence of patches for transformer [56] to process as transformers capture more consistent global contexts due to their self-attention mechanisms [30, 18, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Despite the advancements and improvements in semantic segmentation the bottleneck for high accuracy still remains to be dependent on pixel-wise annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 4 a) b) c)SSCL Figure 3: Overview of the proposed framework (CRCFP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The encoder and decoder are trained in a supervised manner with the cross-entropy (CE) loss for the labelled instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For unlabelled instances, two cropped patches with partial overlap together with the input image were passed through the encoder, where the input image is used for contrastive and cross-consistency learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2 Semi-Supervised Learning Semi-supervised learning (SSL) exploits the unlabelled data on top of limited labelled data for improving the model performance and internal feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Recently SSL based methods have been widely adopted in the computer vision domain [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Popular SSL techniques include pseudo labelling [31, 32, 33] where the model trained on limited data is used to predict the labels for unlabelled data known as pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Generative adversarial based methods improve the generalisability of the trained model using various perturbations in the direction of maximum vulnerability, resulting in aligning the distributions of labelled and unlabelled input in latent space [35, 36, 59, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Data interpolation based methods aim to augment input space to create perturbed linear inputs for training models [60, 61, 62], Temporal ensembling based methods aim to ensemble predictions over the epochs using momentum/moving average to enforce consistency between the predictions [63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Self-supervised learning based consistency training aims to contrast the unlabelled input using pre-text tasks for learning important representations [65, 66, 67, 33, 40] and entropy minimisation based method aims to maximise label assignment to either of the labels [41, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3 Contrastive Learning Learning by contrasting pairs of similar (positive) and dissimilar (negative) images for improved representation learning is known as contrastive learning [68, 65, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Several loss functions have been proposed from maximum margin loss [70], triplet loss [71], N-pair loss [72] to contrastive predicting coding (CPC) [73] proposing mutual information based InfoNCE loss to improve contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Contrastive learning has been used in both supervised and unsupervised learning tasks in conjunction with self supervision [66, 65, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Recently, it has been established that using more accurate positive and negative pairs along with larger batch sizes improves the quality of learned representations with heavy augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Memory banks are adopted when large batches are not computationally feasible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', doesn’t fit the GPU memory) for contrastive loss using a large set of negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='4 Semi-Supervised Semantic Segmentation SSL based semantic segmentation approaches utilise the aforementioned techniques to extract knowledge from unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Recently, CutMix, MixUp, and CutOut based augmentation techniques were used togather with the 5 Supervised branch CE Loss Lsup Pix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Classifier Encoder Decoder () (h) (g) Entropy Lent Yi xi Projector Yu (Φ) Unsupervised branch Xul βu1 u2 ve Aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Pix +ve Classifier (Cj) Perturbation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Pix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Classifier Xu2 (CK) ul Yu2SSCL Figure 4: Directional contrastive loss working for context-aware consistency, where from ϕu1, ϕu2 overlapping area (yellow overlay) positive pixels with higher confidence pull each other closer (orange arrows) while negative pixels from ϕu2 as well as from memory bank push each other apart (red arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Where class masks ˆyu1, ˆyu2 (dashed green arrows) were applied to get the negative samples from ϕu2 and from the memory bank illustrated in the grey overlay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' student-teacher model where consistency was enforced between the mixed predictions [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Guided collaborative training (GCT) by [76] performed network perturbations with the help of different network initialisation and enforced the dynamic consistency constraint between the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Cross-consistency training (CCT) by [39] performed perturbations on the main encoder’s features and enforced consistency over the multiple decoders output making it robust to various perturbation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Context-aware consistency by [40] proposed directional consistency loss for contrasting different contexts by cropping two overlapping patches of the same input to improve the representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Recently, in the field of computational pathology, a few methods for semi-supervised semantic segmentation have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' [77] proposed a semi-supervised method for signet detection using with the help of self-supervised learning for label generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' [78] proposed a two stage SIM-FixMatch approach utilising self-supervised learning in the first stage and then using FixMatch for pseudo label generation along with consistency regularisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' [79] proposed an exponential moving average (EMA) student-teacher framework where the model is trained using the noisy labels to enforce the consistency over similar and dissimilar patch pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Cross-patch dense contrastive learning by [43] proposed a student-teacher based method to enforce EMA based consistency over predictions and to improve the internal representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Pixel-wise contrastive loss was applied to background and foreground patches for improving the internal feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' In this work, we show that (a) by enforcing consistency over varying contexts and feature perturbations in encoder’s latent space, models can generalise better and (b) minimising entropy in output prediction maps can boost the confidence of the final predictions resulting in improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 3 The Proposed Method Figure 3 shows an overview of the proposed framework (CRCFP), where L = {(x1 l , y1 l ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', (xn l , yn l ) : n ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', N]} represents the N labelled images while U = {(x1 u), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', (xm u ) : m ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', M]} represents the M unlabelled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Labelled and unlabelled images xl and xu were sampled from L and U respectively in batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Both images xl, xu are of H × W × D spatial dimensions with corresponding pixel-wise mask yl = RC×H×W only for labelled image where C is the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Each labelled image xl is passed through the supervised pathway of the CRCFP framework (blue arrows in Figure 3) whereas the unlabelled images xu pass through the unsupervised pathways of the framework (brown arrows in Figure 3) along with two overlapping patches extracted randomly from xu denoted as xu1, xu2 (green arrows in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Feature maps are extracted from the input image using the shared encoder h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θh) and decoder g(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θg) as f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θf) = h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θh) ◦ g(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θg) resulting in feature maps for each input as fl = f(xl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θf), fu = f(xu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θf), fu1 = f(xu1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θf) and fu2 = f(xu2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Further, fl and fu are processed by a pixel classifier Cf for final prediction as ˆyl = Cf(fl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θp) and ˆyu = Cf(fu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θp) where ˆyl is optimised using the cross-entropy loss over yl as Lsup shown in equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Lsup = − 1 N C � c=1 N � i=1 yi,c log(ˆyi,c) (1) 6 Pu1 Pu2 ve e +ve Push Away Pull Closer Overlap Mask Ju1 Ju2 Memory Bank Contrastive LossSSCL 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 Context-Aware Consistency With only the supervised loss Lsup, the model may start relying excessively on contexts due to limited labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Context-aware consistency can alleviate this issue by aligning the two different contexts of the same patch with the help of contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For this purpose, encoded feature maps fu1 and fu2 are projected to a low-dimensional space using a non-linear projector φ to preserve important contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The choice of non-linear projection head as compared to linear and identity projection head is due to its superior performance [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The projection head φ(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θz) outputs projection maps as ϕu1 = φ(fu1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θz) and ϕu2 = φ(fu2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Similar to [40], context-aware consistency is maintained between the overlapping regions of ϕu1 and ϕu2 using the directional contrastive loss Lcont to keep the feature representation consistent under different contexts as shown in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For computing directional consistency loss, first class maps ˆyui were extracted using pixel classifier Cf and then maximum probability among all classes C is maintained using max probability as it is linked with higher confidence as shown in equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' ˆyui = arg max x ∈ U Cf(fui;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θp) (2) where i ∈ {1, 2} and higher probability features are used to align less confident features towards the more confident features [76, 40, 43] which can improve learning by avoiding the exchange of unreliable knowledge from the less confident feature as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' In order to extract negative samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', negative pairs), class maps as ˆyu1 = Cf(fu1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θp) and ˆyu2 = Cf(fu2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θp) were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' A positive feature projection ϕu1+ with class map ˆyu1+ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', in case of u1 → u2), the negative samples η should have (ˆyu1+ ̸= ˆyu−) as shown in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Further, to avoid less confident features from contributing towards the loss, a threshold λ is applied to avoid an exchange of knowledge between less confident features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The ℓcont(ϕu1,ϕu2) loss for one pair is calculated as shown below, ℓcont(ϕu1,ϕu2) = − 1 M � h,w M+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' log sim(ϕu1, ϕu2) sim(ϕu1, ϕu2) + � ϕu−∈η Mu− · sim(ϕu1, ϕu−) (3) sim(ϕu1, ϕu2) = exp( ϕT u1ϕu2 ∥ϕu2∥ ∥ϕu2∥ τ ) (4) M− = � 1 if ˆyu1+ ̸= ˆyu−, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' (5) M+ = � Mc+ if max Cf(fu1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θp) > λ, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' (6) Mc+ = � 1 max Cf(fu1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θp) < max Cf(fu2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' θp), 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' (7) where sim(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=') is the cosine similarity measure with temperature τ, Mc+ represents the binary mask for confident features corresponding to ϕu1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' M+ is the binary mask for positive confident samples above threshold λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' M− is the binary mask for negative samples indicating different pseudo labels between ϕu+ and ϕu−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' To increase the negative samples, we have used the memory bank which stores features from recent batches to further increase the negative samples for better contrastive performance [65, 40, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Finally, the directional contrastive loss Lcont is calculated as below: Lcont = ℓcont(ϕu1,ϕu2) + ℓcont(ϕu2,ϕu1) (8) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2 Cross-Consistency Training As context-aware consistency improves the model’s robustness towards changing contexts without losing self-awareness, the model is still susceptible to small perturbations in the input due to limited labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Therefore, in order to leverage unlabelled data and make the model invariant to small perturbations, we utilise the cross-consistency training [39] where fu is perturbed K times for each perturbation type and consistency is maintained between the output of pixel classifier and auxiliary classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' This not only improves the model’s robustness but also regularises the main pixel classifier towards correct predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' We use ˆyu to regularise the pixel classifier over the mean square error (MSE) loss by measuring the distance between the output of the main pixel classifier Cf and the output of auxiliary 7 SSCL classifiers Ck f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Formally, a perturbation function pk with k ∈ {1, K} perturbations outputs a perturbed version of the fu as ˆf k u = pk( ˆfu) for a perturbation type and the cross-consistency training loss Lcross can be defined as below, Lcross = 1 M 1 K � xu∈U K � k=1 d(ˆyu, Ck f ( ˆf k u)) (9) where d measures the squared distance between the output probabilities of the main pixel classifier and perturbed pixel classifier output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Following perturbations are applied to enforce the consistency: Feature Noise: A uniformly sampled noise from the interval [α, β] is added to the features map fu in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' First sampled noise is multiplied with fu to scale the noise relative to feature activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Second, the scaled sample noise is then added to the feature map fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' This makes the noise proportional to each feature activation as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Ω ∼ U(α, β) (10) f noise u = (fu ⊙ Ω) + fu (11) Feature Dropout: A uniform sample threshold γ is used to prune the less confident activations to stop the model from relying on those activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' This is done by first summing the fu over different channels and then normalising it using min-max normalisation resulting in f ′ u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Anything below γ is dropped as seen below: γ ∼ U(α, β) (12) Mdrop = � 1 if f ′ u < γ, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' (13) f drop u = Mdrop ⊙ f ′ u (14) where Mdrop is the binary mask containing threshold values for pruning the activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' DropOut: A fraction of activations are dropped out spatially where the fraction is decided using the Bernoulli distribution with probability δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Mdropout ∼ Bernoulli(δ) (15) f dropout u = Mdropout ⊙ fu (16) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3 Entropy Minimisation Context-aware contrastive learning and cross-consistency training improves the encoder’s features but it often fails to improve the final pixel classifier leading to less reliable pseudo labels corrupting the training from unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' As higher confidence means better prediction maps resulting in more refined pseudo labels which can help train both context-ware and cross-training with improved positive/negative pairs and pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Hence, in order to improve the confidence of predictions, we employ entropy regularisation following its applications in semi-supervised learning [41, 80, 69, 43] as shown in 17 where it penalises the uncertain prediction in the unlabelled data and improves the overall confidence of the prediction maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Lent = − 1 M M � m=1 C � c=1 ˆyu log ˆyu (17) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='4 Training Finally, the entire framework is trained in an end-to-end fashion using a weighted combination of the above mentioned losses as shown below, L = wsupLsup + wcontLcont + wcrossLcross + wentLent (18) where wsup, wcont, wcross and went correspond to the weights for each loss component respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 8 SSCL Table 1: Comparison of the state-of-the-art methods with mIoU, dice score and accuracy aggregated for 3 different random seeds as mean (standard deviation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The first column represents the fraction of data used for training the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' BCSS Fraction Method mIoU Dice Accuracy 1/8 DeepLab-v3 [17] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='99 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='96) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='96 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='53 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='07) 1/8 CCT [39] 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='84 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='54) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='69) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='14 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='70) 1/8 CAC [40] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='67 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='32) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='97 (7.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='20 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='31) 1/4 DeepLab-v3 [17] 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='03 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='88) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='52 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='94) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='70 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='65) 1/4 CCT [39] 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='63 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='19) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='24 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='33) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='43 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='89) 1/4 CAC [40] 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='65) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='33 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='55) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='60 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='42) 1/4 CDCL [43] 1/4 CRCFP 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='98) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='21 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='74) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='87 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='89) 1/2 DeepLab-v3 [17] 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='26 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='19) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='33 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='98) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='07 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='031) 1/2 CCT [39] 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='78 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='56) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='63 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='28) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='94 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='17) 1/2 CAC [40] 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='44 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='48) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='67 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='16) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='50 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='92) 1/2 CDCL [43] 1/2 CRCFP 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='86 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='63) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='73 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='59) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='18 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='27) 1/1 DeepLab-v3 [17] 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='29 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='26) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='12) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='10 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='09) 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 Datasets We evaluated the proposed framework on two publicly available datasets, the Breast Cancer Semantic Segmentation (BCSS) [44] and Multi-organ Nucleus Segmentation Challenge (MoNuSeg) [45] dataset for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The data was obtained from the respective challenge pages hosted on Grand Challenge for Medical Image Analysis website (https://grand-challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='org/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' MoNuSeg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The MoNuSeg challenge was organised as a MICCAI 2018 satellite event and contains 21,623 annotated nuclei from 30 H&E stained images for training and contains 7223 annotated nuclei from 14 H&E stained images for testing purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Annotations were done by engineering students and then an expert pathologist served as quality control for the annotated nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Each image is of size 1000 × 1000 extracted from a WSI scanned at 40× resolution of an individual patient obtained from The Cancer Genome Atlas Program (TCGA) [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' WSIs are sampled from 18 different centres and 7 different organs including breast, liver, kidney, prostate, bladder, colon and stomach with various tumour stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' BCSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The BCSS challenge was conducted in 2021 and contains over 20,000 annotated regions of interest (ROI) from 151 H&E stained WSIs with the same number of patients from TCGA [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 25 annotators including pathologists, residents, and medical students helped annotate this large scale data into 25 refined categories which are later merged into 5 broad categories as tumour, stroma, inflammatory, necrosis, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For this work, we have used the same 5 broad categories by relabelling the regions and then split them into training and test centres according to the [44] where there were 14 centres for training and 7 centres for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2 Implementation Details 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3 Data Preparation In order to validate the CRCFP framework, we evaluated it against different label proportions of each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Where for BCSS different label proportions were collected from different centres (hospitals) to make training more susceptible to variation in colours enabling domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' DL methods often fail to perform well on samples from a different domain (centres), mainly due to domain shift, this also makes it a domain generalisation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Therefore, the training set was divided into portions by diving the total training centres as 1/1 (full), 1/2 (half), 1/4 (quarter), and 1/8 (one-eighth) centres where 1/8 results in training images coming from only 1 centre, while the test set remains intact as it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Similarly, for 1/4 (quarter) training images comes from 4 centres and 7 centres for 1/2 (half).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For MoNuSeg, different label proportions were based on training images themselves and are then divided into 1/1, 1/8, 1/16, and 1/32 9 SSCL proportions to make it comparable to the work of [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Further, this whole process is repeated using 3 different random seeds and then the results are reported using mean aggregation with standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='4 Evaluation metrics In order to compare our proposed method quantitatively with other state-of-the-art methods (SOTA), we have used different quantitative measures including accuracy, F1-score (Dice) and mean intersection over union (mIoU) for both the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='5 Network Architecture We used DeepLab-v3 [17] as base segmentation network with ResNet-50 [82] encoder pretrained on ImageNet [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Where the projector consists of two fully connected (FC) layers of size 128 with ReLU as an intermediate activation layer, FC → ReLU → FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Pixel classifiers consist of convolutional layers with a kernel of size 1 × 1 to reduce the number of channels to total classes with non-linear ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The final layers upsamples the output using bi-linear interpolation to match the input size as H × W × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='6 Experimental Settings The input size for the proposed framework for both labelled and unlabelled images was 320 × 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For contrastive learning, two patches xu1 and xu2 were randomly cropped from the unlabelled image with an overlap in the range of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0] and are resized to match the input dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For positive filtering mask λ was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='75 and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 as temperature for cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For cross-consistency training, number of auxiliary pixel classifiers were set to K = 4 for each perturbation type and for feature noise perturbation the parameters α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3 were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For feature dropout perturbation, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='75, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='9 were used as they can help remove approximately 10% to 30% of active regions from the feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Also, for simple Dropout the probability for Bernoulli distribution was set to δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' During training, a set of standard augmentations were applied to the input images including horizontal and vertical flipping, gaussian blur, colour and grey scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' PyTorch was used for implementing this framework where for optimisation we train the whole framework for 80 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For the initial 5 epochs, only supervised loss Lsup was used to train the whole framework as this provides a stable head start for the semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The batch size of 8 was used for labelled and unlabelled images with stochastic gradient descent (SGD) optimiser and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' As a common practice, poly learning rate decay policy was used where the learning rate is scaled using 1 − ( iter max _iter)power at each iteration with power = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Weights with respect to different losses Lsup, Lcont, Lcross and Lend were set to fixed values as wsup = 1, wcont = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1, wcross = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='01 and went = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='01 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' All models were trained with the same configurations for both datasets where two Nvidia GeForce 1080Ti GPUs are used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 5 Results The performance of our proposed method (CRCFP) compared to recent SOTA semi-supervised semantic segmentation methods including DeepLab [17], CCT [39], CAC [40] and CDCL [43] 1 is shown in Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' As these methods are implemented using different configurations and baseline segmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For a fair comparison, we have implemented these methods within a unified framework with the same segmentation baseline, experimental settings and data augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Table 1 shows the performance of our CRCFP model compared to supervised and semi-supervised methods on all matrices for the BCSS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Particularly, when 1/8 proportion of the training centres was used, it can be seen that in terms of mIoU our method performs ∼6% better than the supervised method and ∼3% better than the recent CAC [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Similarly, its worth noting that with 1/4 of the total centres, the CRCFP performance is almost similar to fully supervised method with all data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' On the other hand, the poor performance of CCT [39] can be attributed towards heavy perturbations applied directly to the features where it brings perturbed features from different contexts closer without pushing dissimilar apart whereas CAC [40] not only bring them closes but also pushes away the features from different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, it focuses more on encoder feature generalisation leaving pixel-classifier with less confident features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Figure 5 shows visual comparison of CRCFP with the SOTA algorithms, where, it can be observed that prediction maps of CRCFP are better as compared to the rest, specially highligted in the dashed red boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 1CDCL [43] cannot be applied to the multi-class problem as it divides the patches into foreground and background only for contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 10 SSCL Figure 5: Visual comparison of the CRCFP with different state-of-the-art methods for tissue region segmentation with 1/2 training data only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Dashed red box highlights superior performance of our method as compared to SOTA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Table 2 shows the performance of CRCFP surpassing other SOTA methods in all data proportions and metrics, especially in 1/32 proportion of the MoNuSeg dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' It can be seen that our CRCFP outperforms the CAC [40] by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='32% in mIoU with a smaller standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' It can also be observed that fully supervised models are more susceptible to domain generalisation problem from the table as in 1/32 proportion of the training images the performance of DeepLab-v3 [17] is 4% better than the 1/16 proportion of the training images whereas there is more data available in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' This is due to the fact that in a random sampling of training images some training images are better indicators of the testing distribution due to similarities in the same stain, organ and tumour stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, most of the SOTA semi-supervised algorithms solve this issue with the help of unlabelled data as it can be seen that the performance increase with the increase in data for all these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Figure 6 shows a visual comparison of CRCFP with SOTA methods where it can be seen that our approach predicts fewer false positives as compared to CDCL [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Further, in order to validate the contribution of each component (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', context-aware consistency, cross-consistency training and entropy minimisation) we conducted an extensive ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The ablation study is performed on the BCSS dataset due to its complexity and multi-class nature, where we studied the effect of using all data proportions for the different encoders and in stripping the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' While studying the effect of negative samples and the number of auxiliary pixel classifiers we used 1/8 data proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 Encoder To verify the performance boost by plugging in a bigger encoder in the base segmentation network, we replaced ResNet- 50 with ResNet-101 for all data proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Table 3 shows the performance of the proposed CRCFP framework with a 11 Input GT SupOnly CCT CAC Ours Tumor | Stroma / Inflammatory/ Necrosis / OthersSSCL Table 2: Comparison of the state-of-the-art methods with mIoU, dice score and accuracy aggregated for 3 different random seeds as mean (standard deviation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The first column represents the fraction of data used for training the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' MoNuSeg Fraction Method mIoU Dice Accuracy 1/32 DeepLab-v3 [17] 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='09 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='07) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='89 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='77) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='45 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='40) 1/32 CCT [39] 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='13 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='06) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='31 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='29) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='33 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='35) 1/32 CAC [40] 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='40 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='12) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='33 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='90) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='14 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='62) 1/32 CDCL [43] 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='72 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='83) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='95 (75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='95) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='66 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='57) 1/32 CRCFP 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='72 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='22) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='60 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='24) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='86 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='23) 1/16 DeepLab-v3 [17] 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='20 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='76) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='80 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='58) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='27 (75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='27) 1/16 CCT [39] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='99 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='08) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='56 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='29) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='36) 1/16 CAC [40] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='44 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='11) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='47 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='92) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='27 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='11) 1/16 CDCL [43] 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='63 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='15) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='40 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='90) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='99 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='37) 1/16 CRCFP 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='08 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='07) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='91 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='52) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='58 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='16) 1/8 DeepLab-v3 [17] 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='67 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='99) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='59 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='32) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='98 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='89) 1/8 CCT [39] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='13) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='00 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='54) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='63 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='42) 1/8 CAC [40] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='56 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='42) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='73 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='30) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='91 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='23) 1/8 CDCL [43] 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='07 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='45) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='63 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='14) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='29 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='48) 1/8 CRCFP 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='85) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='19 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='54) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='28 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='42) 1/1 DeepLab-v3 [17] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='29 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='16) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='49 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='11) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='52 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='09) Figure 6: Visual comparison of the CRCFP with different state-of-the-art techniques in nuclei image segmentation with 1/8 training data only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' GT represents the ground truth nuclei masks, and SupOnly shows the models trained with labelled training data only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Red pixels correspond to the ground truth while green shows the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Yellow pixels represent the overlap regions between the prediction and ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' bigger encoder and it can be seen that there is a performance boost overall for most of the methods, especially for CCT 12 Input GT SupOnly CCT CAC CDCL OursSSCL Table 3: Comparison of the state-of-the-art methods on mean (standard deviation) of mean intersection of union (mIoU), dice score and accuracy with baseline encoder as ResNet-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The first column represents the fraction of data used for training the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' BCSS Fraction Method mIoU Dice Accuracy 1/8 DeepLab-v3 [17] 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='50 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='61) 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='73 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='51) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='89 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='92) 1/8 CCT [39] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='71 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='64) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='66 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='96) 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='42 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='46) 1/8 CAC [40] 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='91 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='79) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='92 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='74) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='01 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='85) 1/8 CRCFP 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='15 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='76) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='27 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='72) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='57 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='82) 1/4 DeepLab-v3 [17] 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='18 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='88) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='30 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='70) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='37 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='25) 1/4 CCT [39] 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='63 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='98) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='35 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='26) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='94 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='73) 1/4 CAC [40] 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='48 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='73) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='52 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='47) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='78 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='84) 1/4 CRCFP 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='40) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='94 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='29) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='18 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='49) 1/2 DeepLab-v3 [17] 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='37 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='89) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='58) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='86) 1/2 CCT [39] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='65) 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='64 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='55) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='66 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='33) 1/2 CAC [40] 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='95 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='72) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='77 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='67) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='09 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='27) 1/2 CRCFP 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='09) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='57 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='09) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='67 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='12) 1/1 DeepLab-v3 [17] 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='33 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='04) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='22 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='73) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='68 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='58) [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, it can be observed that CRCFP with a smaller encoder (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', ResNet-50) still performs comparable/better than other SOTA techniques with a bigger encoder e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', in 1/8 proportion CAC [40] with ResNet-101 achieves mIoU of 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='91 where CRCFP with ResNet-50 achieves mIoU of 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='09 showing superiority of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Also, it is worth mentioning that with ResNet-101 the standard deviation we observed with ResNet-50 was reduced, owing to the fact that bigger encoders are more stable for semi-supervised learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Overall the CRCFP framework provides improved and stable performance with bigger encoders as compared to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2 Network Schemes We validated the contribution of each component by breaking down the whole framework with respect to different losses and called them network schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' We started with a baseline segmentation network i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', DeepLab-v3 with ResNet-50 as SupOnly, Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 consists of using context-aware consistency loss, Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2 consists of using context-aware consistency loss with entropy minimisation and finally Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3 is our proposed framework with context-aware consistency loss with cross-consistency training and entropy minimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Table 4 shows the schemes with respect to their respective losses being used, it can be seen that with each component’s addition we can see improvement in overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', in 1/8 data proportion, the addition of context-aware consistency brings about 4% of improvement while entropy minimisation further bumps it up by 1% and finally cross-consistent training beings about 2% of improvement accumulating the overall performance to ∼7% from baseline supervised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Also, for other data proportions the performance boost is not that much significant with the addition of these Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2 and Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3 as compared to Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, its worth mentioning that the standard deviation of Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2 and Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3 as compared to Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 is smaller which is due to the fact that these schemes brings confidence in prediction maps thus improving the overall performance with stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3 Negative Samples As increasing the negative samples in training contrastive learning framework boosts the performance of the underlying model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' This is done mostly by increasing the batch size to 2048 or 4096 where possible as the bigger the batch size the more samples you get for comparisons [65, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, where it is not possible, another workaround is to use a memory bank where negative samples from previous batches were stored for later use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Therefore, in order to get the upper bound of performance in our framework with respect to negative samples, we have experimented with different number of negative samples as seen in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' It can be noticed that with increasing negative samples, the performance increases for a while and then it reaches the plateau and then increases with very little gain as it can also be observed visually in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' This can be due to the fact that there might not be many variations to cover in the training set with more negative samples, thus reaching stable performance or very little performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Also, due to gradient checkpoint functionality in PyTorch adding more negative samples does not effect the training efficiency drastically but does consume more compute time and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Hence, based on these observations, for this study, we set the number of negative samples to 1200 for its memory vs accuracy trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 13 SSCL Table 4: CRCFP breakdown in different Schemes with respect to their loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' SupOnly correspond to baseline segmentation model with Lsup loss only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 corresponds to addition of Lcont loss on top of SupOnly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2 corresponds to addition of Lent on top of Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 and finally Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3 is addition of Lcons on top of Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Method Split Lsup Lcont Lent Lcons mIoU SupOnly 1/8 ✓ × × × 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='99 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='96) Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 1/8 ✓ ✓ × × 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='67 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='32) Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2 1/8 ✓ ✓ ✓ × 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='76 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='12) Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3 1/8 ✓ ✓ ✓ ✓ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='09 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='18) SupOnly 1/4 ✓ × × × 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='03 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='88) Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 1/4 ✓ ✓ × × 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='65 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='65) Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2 1/4 ✓ ✓ ✓ × 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='97 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='47) Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3 1/4 ✓ ✓ ✓ ✓ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='06 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='98) SupOnly 1/2 ✓ × × × 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='26 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='19) Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 1/2 ✓ ✓ × × 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='44 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='48) Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2 1/2 ✓ ✓ ✓ × 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='87 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='39) Scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='3 1/2 ✓ ✓ ✓ ✓ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='86 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='63) Table 5: Performance of CRCFP with respect different number of negatives samples used while training Lcont loss with BCSS data split of 1/8 # mIoU Dice Accuracy 100 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='62 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='10) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='46 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='87) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='38 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='14) 500 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='81 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='78) 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='86 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='88) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='05 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='30) 1200 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='09 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='18) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='84 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='59) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='20 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='31) 1600 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='16 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='70) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='81 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='05) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='68 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='07) 2400 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='60 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='09) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='14 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='80) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='58 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='43) 3200 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='34 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='25) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='83 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='01) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='06 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='59) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='4 Auxiliary Pixel Classifier To see the effect of a varying number of auxiliary pixel classifiers with respect to different perturbations we conducted experiments with K ∈ {1, 2, 4, 6, 8, 10} as seen in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' It can be seen that increasing the number of pixel classifiers per perturbation increases the performance but the upper bound is achieved soon after it reaches K = 4, from where the performance drops slightly as can be observed in the Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Increasing the number of perturbations can result in more aggressive penalisation of the model overall as it accumulates to K × 3 losses which can deviate the model from learning meaningful representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Based on this observation we set the number K = 4 for our study for the rest of the comparisons for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 6 Discussion Interpretable features from histology slides can be extracted by segmenting objects/structures from ROIs e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', nuclei, glands, stroma, tumours etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Intrepretable features can enable discovery of novel digital bio-markers with explanations for histology images for hard tasks like survival analysi [85, 10] and mutation prediction [86, 87, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Therefore, it is vital for the downstream tasks to have good quality and precise segmentation of region of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' For this purpose, utilising unlabelled data for representation learning not only improves performance but also improves the internal Table 6: Performance of CRCFP with respect different number of K auxiliary classifiers used while training Lcons loss with BCSS data split of 1/8 # mIoU Dice Accuracy 1 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='94 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='95) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='9 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='27) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='14 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='54) 2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='76 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='51) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='44 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='88) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='23 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='23) 4 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='09 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='18) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='84 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='59) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='20 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='31) 6 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='48 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='26) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='01 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='73) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='60 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='68) 8 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='72 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='88) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='38 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='29) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='25 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='89) 10 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='68 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='79) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='64 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='20) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='84 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='99) 14 SSCL Figure 7: Performance graph with respect varying number of negatives samples used while training Lcont loss with BCSS data split of 1/8 representations for better learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The qualitative and quantitative results along with the ablation study has shown superior performance of our proposed CRCFP with respect to other SOTA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' However, it’s worth exploring internal representations of the learned models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=',feature embeddings) to account for (1) Consistency in feature space (2) Cluster assumption, for the sake of validation of aforementioned claims in the introduction section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='1 Feature Space Visualisation In order to observe the consistency in feature space, feature embeddings were extracted from both our SSL based CRCFP trained on 1/2 proportion of the training data vs DeepLab-v3 trained on all data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', fully supervised), since they achieved same performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Extracted feature maps were upsampled to match the size of the input image (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', 320 × 320) and are then mapped to lower dimensions using UMAP [88] for visualisation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' It can be seen in Figure 9 that the feature embedding distributions are consistent with varying contexts specially in the 1st and 2nd column for our CRCFP model as compared to fully supervised ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Similarly, it can be observed in the other examples where the varying context is inherent due to the sequential overlap in patch tessellation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Whereas, the fully supervised model is susceptible to perturbations in contextual cues as can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' It is worth noting the last two columns where the shape of feature embedding distribution changes along with the orientation of same samples points from the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Specially, the ones shown in yellow dots as compared to our proposed framework where the distributions are almost consistent under these perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='2 Cluster Assumption Consistency regularisation based methods work on the basis of cluster assumption and have achieved SOTA results in semi-supervised classification and segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The main idea behind consistency regularisation is to have high and low density regions where samples closer to each other are likely to share the same label forming a high density region with a low average distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' While the class boundaries are likely to be aligned with the low density regions 15 →mloU --Dice -→Accuracy 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 PERFORMANC 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 100 500 1200 1600 2400 3200 # OF NEGATIVE SAMPLESSSCL Figure 8: Performance graph with respect varying number of pixel classifiers used while training Lcons loss with BCSS data split of 1/8 Figure 9: (a) BCSS dataset images with overlapping regions cropped sequentially from the same image to mimic changing contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' (b) UMAP visualisations of features embedding distributions extracted from a fully supervised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' (c) UMAP visualisations of feature embedding distributions extracted from a semi-supervised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Note that the feature embeddings are represented in the same UMAP space where dots with same colour represents feature embedding from the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 16 →mloU -Dice →-Accuracy 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 PERFORMANCE 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='0 1 2 4 6 8 10 # OF AUX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' CLASSIFIERSb) CSSCL Figure 10: (a) Example images from BCSS test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' (b) Respective masks showing the foreground and background pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' (c,d) Average euclidean distance L2 between the central patch of size 21 × 21 with four overlapping patches in the immediate neighbours in RGB colour space and feature space respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Note that for feature space visualisation encoder embeddings were upsampled to map input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The darker blue colour represents the low density regions corresponding to high average distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', high average distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' In order to observe cluster assumption, feature embeddings were extracted from CRCFP and were compared against RGB colour space as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Extracted feature maps were upsampled to match the size of the input image and then the average euclidean distance between each patch of size 21 × 21 centred around its 4 immediate spatial neighbours (left, right, top and bottom) was calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' It can be seen in Figure 10(d) that the class boundaries are much more aligned and apparent in the feature space as compared to the colour space where the boundaries doesn’t align well thus violating cluster assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' This can be due to the fact that the CNNs at higher layers tends to learn more semantic based features from the basic low-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Also, interestingly the background/fat represented in white colour in input images somewhat holds the high density regions because there is not much change in colour values for that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' While the rest of the tissue area is not very homogeneous in pixel values due to the presence of cells of various shapes and sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 7 Conclusions In this work, we haved presented a novel consistency based semi-supervised learning based semantic segmentation framework for region and nuclei segmentation in histology images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The proposed method is invariant to varying contexts and perturbations making it efficient and robust for semantic segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' We have shown that context- aware consistency learning can exploit unlabelled images efficiently with the help of cross-consistency training and entropy minimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Extensive experiments on two publicly available large histopathological datasets have shown the superiority of the CRCFP framework by achieving new SOTA results for semi-supervised semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Also, detailed ablation studies for different network parameters and components show the contribution of each network component, demonstrating the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Future directions include improvements to the proposed method with respect to improving the context-aware loss for minor classes and finding histology specific perturbation e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=', targeting stain variations, on a large multi-centric histopathological dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Large multi-centric data is vital for the validation of the study as the quality of downstream analysis is highly dependent on the segmented histology primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 17 a) b) c) d)SSCL 8 Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 9 Acknowledgements RMSB is funded by the Chancellor Scholarship from University of Warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' SEAR 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Sen-Ching Cheung, and Chen-Nee Chuah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' A semi- supervised learning for segmentation of gigapixel histopathology images from brain tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' In 2021 43rd Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC), pages 1920–1923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' [79] Hsien-Tzu Cheng, Chun-Fu Yeh, Po-Chen Kuo, Andy Wei, Keng-Chi Liu, Mong-Chi Ko, Kuan-Hua Chao, Yu-Ching Peng, and Tyng-Luh Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Self-similarity student for partial label histopathology image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' In European Conference on Computer Vision, pages 117–132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' [80] Minghao Chen, Hongyang Xue, and Deng Cai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Domain adaptation for semantic segmentation with maximum squares loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 2090–2099, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' [81] John N Weinstein, Eric A Collisson, Gordon B Mills, Kenna R Shaw, Brad A Ozenberger, Kyle Ellrott, Ilya Shmulevich, Chris Sander, and Joshua M Stuart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' The cancer genome atlas pan-cancer analysis project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Nature genetics, 45(10):1113–1120, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Human-interpretable image features derived from densely mapped cancer pathology slides predict diverse molecular phenotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Nature communications, 12(1):1–15, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' [88] Leland McInnes, John Healy, and James Melville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' Umap: Uniform manifold approximation and projection for dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' arXiv preprint arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content='03426, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FPT4oBgHgl3EQfqTWc/content/2301.13141v1.pdf'} diff --git a/VNE0T4oBgHgl3EQflwEn/content/tmp_files/2301.02489v1.pdf.txt b/VNE0T4oBgHgl3EQflwEn/content/tmp_files/2301.02489v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..72ae4bfe19f34e721b04022d7b133bf7d129bae8 --- /dev/null +++ b/VNE0T4oBgHgl3EQflwEn/content/tmp_files/2301.02489v1.pdf.txt @@ -0,0 +1,968 @@ +Has the chemical contribution a secondary role in +SERS? +Luis A. Guerra Hern´andez,† Andr´es A. Reynoso,† and Alejandro Fainstein∗,† +†Centro At´omico Bariloche and Instituto Balseiro, Comisi´on Nacional de Energ´ıa At´omica +(CNEA) - Universidad Nacional de Cuyo (UNCUYO), 8400 Bariloche, Argentina. +‡Instituto de Nanociencia y Nanotecnolog´ıa (INN-Bariloche), Consejo Nacional de +Investigaciones Cient´ıficas y T´ecnicas (CONICET), Argentina. +¶Departamento de F´ısica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain +E-mail: afains@cab.cnea.gov.ar +Abstract +It is an established understanding that the electromagnetic contribution (the plasmon- +mediated enhancement of the laser and scattered local electromagnetic fields) is the +main actor in Surface Enhanced Raman Scattering (SERS), with the so-called chemical +(molecule-related) contribution assuming only, if any, a supporting role. The conclusion +of our comprehensive resonant study of a broad range of nanosphere lithography based +metallic substrates, with covalently attached 4-mercaptobenzoic acid monolayers used +as probe (standard molecules which are non-resonant in solution), is that this accepted +understanding needs to be revised. We present a detailed resonant SERS study of +Metal-film over nanosphere (MFON) substrates which is done both by scanning the +laser wavelength, and by tuning the plasmon response through the nanosphere diameter +which is varied from 500 to 900 nm. Far and local field properties are characterized +through measures of optical reflectivity and SERS efficiency, respectively, and are +supported by numerical simulations. We demonstrate that the SERS efficiency depends +1 +arXiv:2301.02489v1 [physics.optics] 6 Jan 2023 + +indeed on the electromagnetic mechanism, determined by the plasmonic response of +the system, but we observe that it is also strongly defined by a chemical resonant +contribution related to a metal-to-ligand electronic transition of the covalently bound +probe molecule. Optimum amplification occurs when the plasmon modes intersect with +the ligand-to-metal chemical resonance, contributing synergically both mechanisms +together. Quite notably, however, the largest SERS signal is tuned with the metal- +to-ligand transition, and typically does not follow the wavelength dependence of the +plasmon modes when varying the nanosphere size. The same general trend is observed for +other nanosphere lithography based substrates, including sphere-segment void cavities +and hexagonally ordered triangular nanoparticles, using both Ag or Au as the plasmonic +metal, and also with a commercial substrate (Klarite). Interestingly, this extensive +comparative investigation shows in addition that metal-film-over-nanosphere substrates +are significantly better than the rest in terms of Raman efficiency and homogeneity. +We conclude that a deep understanding of both the electromagnetic and chemical +mechanisms is necessary to fully exploit these substrates for analytical applications. +Motivation +The plasmonic properties of metal nanostructures are of great interest since they exhibit +localized surface plasmons resonances (LSPR), with electromagnetic fields located on the +nanostructured surface, and with resonance energy depending on the material, size and shape +of the nanostructure. The interaction between photons impinging from the far field and +the LSPR leads to a confinement of the electromagnetic field, enhancing its magnitude and +opening the door to plasmon enhanced optical spectroscopies, including surface-enhanced +Raman spectroscopy (SERS).1–4The enhancement of either or both the laser and Raman +scattered fields by the plasmon resonances is identified as the electromagnetic contribution +to SERS.1–3,5–7 Raman scattering can also be amplified through electronic resonances either +intrinsic to the probed molecule (if electronic resonant transitions of the molecule are available +2 + +at the selected laser excitation), or related to the specific chemical interaction between the +molecule and the supporting substrate (a situation that can play a role particularly for +molecules that are not intrinsically resonant at the selected laser energy). A variety of +different mechanisms have been envisaged involving this latter type of electronic resonances, +mostly involving metal-to-ligand (ML) transitions of the bound molecule, and are generally +grouped in what is called the chemical contribution to SERS.8–17 Maybe in part because +of this complexity, and specificity for different systems, but also because it is assumed to +be much weaker and of lesser relevance than the electromagnetic plasmonic enhancement +(although some reports position it in the 105 − 107 enhancement range),18,19 the so-called +chemical contribution has remained in the backstage, at best assuming a secondary supporting +role in SERS. An enormous collection of experimental and theoretical work seems to support +this view, which has lead through the design of proper plasmonic substrates to great progress, +with successful applications that range from ultrasensitive analytical methods to single- +molecule spectroscopies,20,21 including the optical monitoring of single-molecule single-electron +transport.22 +To be fair, however, very few of the reported experimental investigations treat in depth +the issue of the resonant enhancement in SERS. Reportedly, less than one percent of the +published works address this problem.23 The reasons are simple to understand. To do such +a resonant investigation either the excitation wavelength needs to be tuned with enough +flexibility through the relevant wavelength range,24–27 or the nanostructures must allow +for a reproducible tuning of the plasmonic resonances across the available laser source +wavelength.23,29–32,46 Laser wavelength scans are rarely performed because they require the +availability of a large set of lasers or widely tunable sources, and access to a triple spectrometer +for efficient stray light rejection. Plasmon scans, in turn, require a flexible and controlled +tunability of the selected technology, and typically imply an important load of fabrication, +structural and optical characterization, and experimental work. Different groups have in any +case pursued either one of these strategies, and this has been done mostly interpreting the +3 + +results in the framework of the electromagnetic enhancement mechanism, broadly confirming +its relevance in the observed amplification. Notwithstanding this general agreement, it is also +interesting to note that the emerging phenomenology is not universal, nor simple. Painstaking +experiments based on an extensive number of nanosphere lithography nanoparticle substrates +of varying resonant energy, indicate that a correlation between the plasmon resonance and +the fixed laser wavelength indeed exists, although with a seemingly noisy correlation.23 Some +other experiments that report on precisely the same kind of substrates, but scanning the +laser wavelength, evidence what seems to be a clear blue shift of the Raman resonance respect +to the plasmon extinction maximum.26 This was interpreted as the convoluted resonant +effect of the incoming laser and the (red-shifted) out-going Stokes Raman scattered fields. +Other reports observe the reverse. That is a red shift of the Raman resonance respect to the +plasmon extinction maximum.24 This contrasting result has been interpreted as a supposedly +universal phenomena related to the dissipation-induced red-shift of the local field modeled +as due to the metal charges oscillating as a forced oscillator.25 In one case were the SERS +resonant enhancement was studied with great detail for individual nanoparticles on a mirror, +the overall intensity was shown to increase with the particle size and not when the plasmon +resonance matched the excitation laser.32 Notwithstanding this rather complex landscape +with contrasting evidence, when looked in detail, it is clear that the general agreement is +that the problem is understood in its essence, with the electromagnetic enhancement being +the main character of the plot,7 and only some voices raised arguing that such explanation is +not complete.15 +To illuminate this already extensively studied subject which, in our view, has nevertheless +remained rather obscure, we present what is in our understanding the first comprehensive +resonant SERS study in which experiments are carried out simultaneously at a wide variety of +excitation wavelengths and also tuning the surface plasmon resonance by controlling the SERS +substrate structure. This is done on substrates fabricated with nanosphere lithography33,34 +which are extensively studied and well known for their reproducibility and homogeneity. We +4 + +describe first the resonant wavelength plasmonic properties of metal-film over nanosphere +(MFON) substrates35,36 with M=Au and Ag, which are studied through optical reflectivity, +SERS measurements, and numerical simulations. The resonant wavelength of the localized +plasmons in these MFON substrates can be adjusted by changing the diameter of the +nanosphere, with wavelengths varying in the NIR-VIS spectral range (∼400-1100 nm). For +the SERS studies, a self-assembled monolayer of 4-mercaptobenzoic acid (4-MBA) was used +as probably the most extensively used Raman probe that forms an ordered and densely +packed monolayer on the surface of the metal.37,38 This Raman molecule in solution displays +electron transitions in the UV. That is, it is non-resonant in the spectral range where the +plasmons reside. Previous investigations have shown, however, that this and similar thiol- +bound molecules develop a resonant electronic ligand-to-metal transition (L-M ∼700 nm) +when covalently attached to either Au or Ag.39–42 Notably, we find that the maximum SERS +efficiency does not follow the plasmon dispersion but, on the contrary, is observed when a +plasmon mode becomes resonant with the ligand-to-metal electronic resonance. We extend +this investigation to other ordered plasmonic substrates, namely Au segment sphere void +cavities27,43–45 and triangular nanoparticles46 both fabricated by nanosphere lithography, and +a commercial Au substrate composed of square pyramidal pits (Klarite),47 with coincident +results. Our conclusion is that SERS is a single effect drawing on plasmon and metal-to-ligand +resonances which are intimately tied to each other and cannot straightforwardly be considered +separately. While this does not contradict the established conviction on the relevance of the +electromagnetic amplification, since determining the true SERS enhancement factor is critical +for analytical determinations,48 it becomes clear that the more complex unified perspective +will need to be considered for real world applications.15,17,49,50 +5 + +Samples and experimental set-up +Fabrication of periodic hexagonal arrays +Polystyrene spheres dispersed in a 1% water solution (Duke Scientific) are introduced between +a glass substrate covered with a 100 nm Au-film (Platypus) and a clean glass slide, with +a separation of about 300 µm. The Au-film is immersed in a 1 mM cysteamine ethanolic +solution overnight to enhance the polystyrene spheres adsorption. During the drying process +in an incubation chamber, a sweeping meniscus forms along the substrate, pulling the spheres +towards the substrate into a close-packed hexagonal monolayer. To fabricate the MFON +substrates this self-assembly of the template is followed by a physical vapor deposition of +Au or Ag (Vega and Camji S.A.I.C). The Au and Ag-films were vapor-deposited over 500- +900 nm polystyrene spheres, in a homemade evaporator, which achieves a residual pressure +of ∼1×10−7 torr. The control of the material fusion temperature is done manually, with a +current of 10-15 A. The triangular lattice of ordered nanoparticles is obtained starting from +50 nm thick Au-FON substrates, and after removal of the latex spheres by sonication in +acetone, isopropanol and Milli-Q water. Segment void sphere cavity substrates are fabricated +following the same procedure of deposition of the polystyrene nanospheres into a close-packed +hexagonal monolayer, followed by electrochemical deposition of Au from a Gold salt solution +(TG-25 RTU, Technic Inc.), with a deposition rate of 2.5mC/min, followed by removal of the +plystyrene spheres by sonication in a sequence of solvents.40,51 The studied square pyramidal +pit substrates were acquired from the company Klarite. +Substrate characterization and resonant SERS experiments +Reflectivity measurements were used to identify the plasmon modes of all the different studied +substrate arrays, using a fully automated Wollam WVASE32 variable angle spectroscopic +ellipsometer with focusing probes, presenting a 100 µm circular spot on the sample with +a numerical aperture of ∼0.02. SEM images were recorded with a FEI field-emission gun, +6 + +Nova NANO-SEM 230, operating at 10 kV and with a tilt angle of 45◦. Surface roughness +was characterized also in all the substrates through atomic force microscopy (AFM) with +an AFM Veeco Dimension 3100, with MESP tip. Signal homogeneity was monitored with +a LabRam HR Evolution Raman microscope using the He-Ne 633 nm laser line (close to +the M-L resonance at ∼ 675 nm) and taking 400 spectra in a 5 × 5µm2 square area with a +x100 microscope objective of NA=0.9. SERS measurements were performed using a triple- +stage Raman spectrometer (Horiba Jobin-Yvon T64000) operating in subtractive mode, and +equipped with a liquid nitrogen-cooled charge-coupled device (CCD). The excitation was +performed using the 514, 568, 647 and 676 nm lines of an Ar-Kr laser and a continuously +tunable Ti-Sapph laser between 680 and 780 nm. The position on the sample was manually +controlled, and the Raman signals were collected in a backscattering configuration with a +collection lens of focal length +10 cm. The entrance slit of the spectrometer was kept at +200 µm, as the Raman peaks for the metal-adsorbed molecule are relatively broad and do not +require high spectral resolution. Typical acquisition times were from 5 to 10 s, depending +on the wavelength and the sample. Typically 10 spectra were acquired for each wavelength +and substrate at different positions, with the average providing the SERS intensity and the +statistical dispersion the shown error bars. +Results and Discussion +Plasmonics modes and SERS in Au-film over nanosphere (AuFON) +substrates +Figure 1(a) presents a scheme of the AuFON substrate and the experimental configuration +with light incident at a small angle, and scattered light collected along the substrate normal. +Panel (b) in this same figure shows a high-resolution SEM image of a typical AuFON +fabricated with 500 nm diameter polystyrene spheres, taken with a tilt angle of 45◦. The +close-packed and highly ordered formation of the support nanosphere mask can be clearly +7 + +identified. A typical spectrum of 4-MBA immobilized on AuFON is displayed in Fig. 1(c), +with the aromatic-ring breathing vibration used to monitor the SERS intensity in the following +sections highlighted with grey background at 1076 cm−1. +µm +1 +(b) +(a) +400 +600 +800 1000 1200 1400 1600 1800 +SERS Int. [cts.] +Raman Shift [cm +-1] +4-MBA spectrum +1076 +(c) +4-MBA +Au/Ag +Latex sphere +SERS +Substrate +Laser +𝜃 +2 +Reflection +S +𝜙 +Figure 1: (a) Schematic of the 4-MBA SERS experiment on metal-film on nanosphere (MFON) +substrates. (b) High-resolution SEM of a typical nanostructure AuFON of 500 nm in diameter +of polystyrene sphere, tilt=45◦. (c) Typical spectrum of 4-MBA absorbed in AuFON. The +Raman mode at 1076 cm−1 used to monitor the SERS efficiency is highlighted. +The far-field response of the plasmon resonances in the studied AuFON substrates is +shown in Fig. 2 (left panel), for Au polystyrene nanosphere diameters in the range 500-900 nm. +The reflectivity experiments were performed with TE polarization and light incident at an +8 + +spot +HV +WD +mag +det +Landing E +um +2.0 +5.00 kV / +4.9 mm +80 000 x +TLD +5.00 keV +Nova NanosEMangle of 25◦, with wavelengths between 400-1000 nm. Each curve is vertically offset by +steps of 0.3 for clarity. We note that for λ <550 nm the reflectivity decreases due to the +interband transitions characteristic of Au. Three resonant absorptions can be identified in the +wavelength range of 600-900 nm, indicated by blue triangles, circles and squares, respectively. +These absorptions are associated with LSPR plasmons of the AuFON structures, and which +we label as M1, M2 and M3 for increasing energy (decreasing wavelength). We have observed +that these modes do not vary significantly with the polar (θ) and azimuthal (φ) angles. +Importantly, as expected the resonance wavelength of these modes strongly blue-shifts when +reducing the diameter of the polystyrene spheres. +The resonant SERS study for the different AuFON substrates is shown in Fig. 2(right). +The SERS intensity was monitored for the different samples through the amplitude of the +1076 cm−1 Raman peak. Each point is the average of ten measurements at different spots, with +the error bars indicating their dispersion. Raman intensities are given in counts per mW.s, +with all spectra acquired exactly under the same experimental conditions. In Fig.2(right) +the vertical red line indicates the wavelength of the ligand-to-metal transition (L-M), while +the blue symbols are guides to the eye identifying the plasmon resonances obtained from +the reflectivity curves in Fig.2(left). Two aspects can be highlighted from these results. +First, irrespective of the AuFON period, the most intense Raman signals are detected in all +studied cases very close to the M-L transition, with only a small red-shift of the resonance +maxima when the M1 plasmon mode approaches this resonance. Second, the maximum +detected signal augments when a plasmon mode is close to the M-L transition, and this +is particularly noteworthy for the M1 mode which leads to the strongest SERS (bottom +spectrum in Fig.2(right)). +9 + +400 +600 +800 +1000 +0,0 +0,3 +0,6 +0,9 +1,2 +1,5 +500 +600 +700 +800 +0 +50 +100 +150 +200 +250 +0 +20 +40 +60 +0 +10 +20 +30 +40 +0 +10 +20 +30 +40 +0 +10 +20 +30 +M3 +M2 +M1 +M3 +M2 +D=500nm +D=600nm +D=700nm +D=800nm +D=900nm +D=500nm +D=600nm +D=700nm +D=800nm +M1 +M-L + +D=900nm +Reflectivity + + + + Wavelength [nm] + + + + + + + + SERS Intensity [cts. /mW/s] + + + + + + + + + + +Figure 2: Left: Reflectivity of AuFON substrates with varying diameter of the polystyrene +spheres. +Curves are vertically offset by steps of 0.3 for clarity. +The dispersion of the +plasmon modes (M1, M2 and M2) is indicated with blue symbols and guides to the eye. +The experiments were done with TE polarization, and incident angle of 25◦. Right: Raman +intensity as a function of laser wavelength for the same substrates presented in the left panel. +The spectra were also acquired with parallel TE polarizations. The shown Gaussian curves +are guides to the eye. The symbols and connecting lines identify the plasmon resonances +determined by the reflectivity measurements shown in the left panel. The vertical line signals +the metal-to-ligand transition (M-L). +Plasmonics modes and SERS in Ag-film over nanosphere (AgFON) +substrates +The results presented in Fig.2 indicate that plasmons (that is, the electromagnetic enhance- +ment) play an important role on the SERS efficiency, but also suggest that this might not +10 + +400 +600 +800 +1000 +0.0 +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +2.1 +2.4 +2.7 +3.0 +3.3 +500 +600 +700 +800 +0 +500 +1000 +1500 +0 +250 +500 +750 +1000 +1250 +0 +200 +400 +600 +800 +0 +100 +200 +300 +400 +500 +SERS Intensity [cts. /mW/s] +Reflectivity +M3 +M2 +M1 +D=500nm +D=600nm +D=700nm +D=800nm +D=800nm +D=700nm +D=600nm +D=500nm +M1 +M2 +M3 + + +M-L +Wavelength [nm] + + + + + + + + + + + + + +Figure 3: Left: Reflectivity of AgFON substrates with varying diameter of the polystyrene +spheres. +Curves are vertically offset by steps of 0.3 for clarity. +The dispersion of the +plasmon modes (M1, M2 and M2) is indicated with blue symbols and guides to the eye. +The experiments were done with TE polarization, and incident angle of 25◦. Right: Raman +intensity as a function of laser wavelength for the same substrates presented in the left +panel. The spectra were also acquired with parallel TE polarizations. The shown black +dash-dotted curves are guides to the eye. The symbols and connecting lines identify the +plasmon resonances determined by the reflectivity measurements shown in the left panel. +The vertical line signals the metal-to-ligand transition (M-L). +be separable from the metal-to-ligand resonant contribution (that is, the so-called chemical +enhancement). To provide additional data on this phenomena we present in Fig.3 a similar +investigation but now performed on Ag instead of Au MFON substrates. Again the left panel +in Fig.3 presents the reflectivity measurements, while the right panel shows the corresponding +SERS intensity curves for AgFON substrates made with polystyrene NPs of sizes ranging +11 + +from 500 to 800nm. Similarly to the Au case, three plasmon resonances can be identified +that blue-shift with decreasing NP size (labeled as M1, M2 and M3 in the figure). Several +aspects of the SERS resonant curves can be mentioned. First, again Raman resonant maxima +are observed for all substrates at the same wavelength coincident with the M-L transition, +irrespective of the specific pattern of plasmon modes of the substrate. The absolute value of +the scattering efficiency increases when a plasmon mode is close to the M-L transition, but +the resonant peak does not follow the plasmon dispersion: it is fixed at the spectral position +of the M-L transition. Again as for the Au case the proximity of the M1 plasmon mode +seems to be specially relevant to provide the largest scattering efficiencies (bottom spectrum +in Fig.3(right). Second, for the Ag film on nanosphere substrates a second maximum can +be identified, in this case following the M3 plasmon mode (top two spectra in Fig.3(right)). +This difference when compared with the AuFON substrates can be traced to the inter-band +transitions existent in Au and absent in Ag,54 which are expected to quench the Raman +resonances due to absorption at wavelengths below ∼ 600nm. Note that part of the intensity +of the M3 resonance when compared to that due to the M1 mode can be ascribed to the +ω4 enhancement which between 700 and 550 nm varies by a factor of 2.5.55 Third, as is +standard in SERS Ag provides significant larger enhancement than Au when structurally +similar substrates are compared (at least a factor of 5 comparing Figs. 2 and 3). +Overall then, the results for AgFON substrates in Fig.3 confirm the main conclusions +obtained from those made from Au in Fig.2, namely: the electromagnetic plasmon-mediated +enhancement is clearly part of the menu, but the chemical contribution has no secondary +role in this phenomena, at least for the studied MFON substrates and for a quite universally +used covalently attached probe molecule. As we will see next, these conclusions are rather +general and valid for a much larger set of plasmonic substrates. +12 + + +µm +1 +Figure 4: Panels a-d present a comparative study of SERS resonant scans obtained under +the same conditions and using the same molecular probe as the experiments in Figs.2 and 3, +for AuFON, square pyramidal Au covered pits (commercial substrate Klarite), Au segment +sphere void cavities, and Au triangular nanoparticles, respectively. SEM images of the studied +substrates are shown at the right of each panel. AuFON (a), cavities (c) and Au NPs (d) were +all fabricated using nanosphere lithography methods with 500 nm polystyrene spheres. The +labels in each panel identify the plasmon modes determined from reflectivity measurements +equivalent to those presented for the MFON substrates in Figs.2 and 3. +Comparative study between different ordered plasmonic substrates +Figure 4 presents a comparative resonant SERS study of a diverse variety of ordered Au +plasmonic substrates, namely AuFON (a), square pyramidal pits of the commercial Klarite +13 + +LWT +(a) +AuFON +300 +200 +M2 +100 +S +SERS intensity [cts. /mW / +0 +P2! +(b) +Klarite +15 +10 +P3 +IP1 +5 +0 +Cavity +(c) +6 +4 +ID +M-L +2 +0 +(d) +AuNPS +0,3 +LSP +0,2 +0,1 +0,0 +500 +550 +600 +650 +700 +750 +800 +Wavelength [nm]spot +HV +WD +mag +det +μm +2.0 +5.00 kV 5.7 mm +80 000 x +TLD +Nova NanosEMsubstrate (b),47 segment sphere void cavities (c),27,43–45 and triangular nanoparticles (d).46 +Each panel in Fig.4 presents, at its right, a SEM image of the structure. Substrates (a), (c) +and (d) were all fabricated by nanosphere lithography using 500 nm polystyrene spheres. +As rugosity is known to have a potentially relevant effect on the SERS efficiency,51,56,57 it +was determined for all substrates (excluding the triangular Au NPs that are not extended +and thus are not directly comparable) using AFM scans finding comparatively similar values +in all cases. Grain sizes span from 0 to +120 nm with a peaked distribution with maxima +around 40-60 nm, with similar degree of roughness for the cavities and MFON substrates +and slightly lower frequency of occurrence of the grains for the commercial Klarite substrate. +The extended substrates were also monitored for their homogeneity in terms of Raman +efficiency, finding on 5 × 5µm2 square areas dispersions of around 17%, 21%, and 31% for +the AuFON, Klarite, and cavity structures, respectively. Assuming then that constructively +these substrates are comparable, we extract two important conclusions. First, again and in +all four cases, the M-L transition seems to be determinant to the spectral position where +maximum SERS efficiency is found. And second, by quite far (a factor of at least 20) the +AuFON substrate results in a much larger (and more homogeneous) Raman efficiency. For +comparison, we note here that experiments performed on identical 4-MBA self-assembled +monolayers on flat (non-structured) Au substrates lead to no observable Raman signals even +with 20 times larger acquisition times. To understand the physics behind the observed larger +electromagnetic SERS enhancement for the MFON substrates, and to identify the origin of +the M1-M3 plasmon modes and their contrasting efficiency for SERS, we briefly describe +next its modeling based on finite element methods. +Theoretical modeling of metal-film over nanosphere substrates +We evaluate the electromagnetic response of the nanostructured surface of a AuFON substrate +when it is excited by a plane wave by solving for the full-field Maxwell equations in 3D using +the finite element method. The standard dielectric function of Au is used as given in Ref. 54. +14 + +400 +600 +800 +1000 +0,0 +0,3 +0,6 +0,9 +1,2 +1,5 +400 +600 +800 +1000 +M2 +700nm +650nm +600nm +550nm +500nm +700nm +650nm +600nm +550nm + +Reflectivity +Wavelength [nm] +500nm +(a) (b) +M1 + + +M2 +M1 +0,0 +0,5 +1,0 +1,5 +2,0 +2,5 +3,0 + +IEavI (a.u.) +x 107 +Figure 5: Calculated wavelength dependence of the far-field reflectivity (a) and near-field +amplitude (b) for a plane wave incident on AuFON substrates of varying sphere size. These +substrates are modeled as solid Au half spheres on a planar and uniform Au film. The solid +connected symbols identify the plasmon modes. +Periodic boundary conditions are incorporated by ensuring perfect agreement in the triangular +mesh within each of the three pairs of partner faces defining the hexagonal arrangement. +The computed scattering matrix includes all allowed diffraction modes obtained using the +customary Ewald criteria for the working plane-wave wavelength and incident direction. To +first test the qualitative behavior of the mode, we perform calculations for an hexagonal +arrangement of solid Au semi-spheres over a uniform Au surface. This is shown in Fig.5, were +both the far field reflectivity (a) and the near field averaged on the surface (b) are shown as +15 + +a function of wavelength and for increasing sphere size (from bottom to top). The magnitude +of the near field determines the electromagnetic enhancement affecting the SERS efficiency, +and thus the two shown panels can be related to the similar ones in Figs.2. The agreement is +good in several features, namely: i) two main plasmon modes are observed in the relevant +wavelength range, ii) these disperse to larger wavelengths for increasing nanosphere size, and +iii) the smaller energy mode, identified as M1, leads to the largest averaged near field and +thus is expected to provide the stronger Raman signals as experimentally observed. +While the qualitative agreement between theory and experiment is reasonably good +with such a simplified description of the substrates, we have observed that the quantitative +determination of the plasmon energies and size dependence is strongly sensitive on the details +of the structure. Calculations with a more realistic description of the AuFON substrates +are presented in Fig.6, where the polystyrene spheres have been included (as a dielectric +material of index of refraction n = 1.59), and the film cover was allowed to have an ellipsoid +shape to account for a larger deposit of metal on the top (see a scheme in Fig.6(a)). For +this simulations we use 500 nm diameter spheres, the Au thickness was taken as 180 nm +at the top of the sphere, and light was incident at angles θ=25◦ and φ=0◦. The interstitial +spaces between spheres have been simulated with and without pyramidal Au arrangements, +without significant variations. In this case excellent agreement with the experimental results +is obtained as shown in Fig 6(b), both in the spectral position of the modes and the relative +magnitude of the averaged surface near-field associated to the M1 and M2 plasmon modes. +The strong sensitivity of the latter on the distance to the surface of the Au film is illustrated on +the right panel of Fig.6(b). The color maps in Fig.6(c) further clarify the contrasting response +of plasmons M1 and M2. The M1 one mode has very strong associated electromagnetic +fields very close to the surface and precisely in between neighbor spheres as known to be +relevant for nanoparticle dimers. This result also explains the comparative high efficiency of +the MFOM substrates when contrasted with the square pyramidal pit and segment sphere +16 + +400 +600 +800 +1000 +0,0 +0,2 +0,4 +0,6 +0,8 +400 +600 +800 +1000 +0,5 +1,0 +1,5 +2,0 +y [nm] +y [nm] +x [nm] +x [nm] +z [nm] +Reflectivity +Wavelength [ nm ] +(a) +(b) +(c) +M1 M2 +M1 + +z [nm] +z [nm] +y [nm] + Z=5nm + 30nm + 100nm + 150nm +M1 +M2 +M2 +lEavl (a.u.) +x [nm] +Figure 6: (a)Scheme of the ellipsoid shape of the deposited Au film, internal polystrene sphere, +and metal Au base used in the calculations.(b) Wavelength dependence of the calculated +far field reflectivity (left panel) and associated averaged magnitude of the surface near-field, +for an AuFON substrate of 500 nm spheres and 180 nm film thickness. In the right panel +different curves are presented for varying distance from the surface. M1 and M2 identify +the plasmon resonances. (c) Color maps of the spatial distribution of the magnitude of the +electromagnetic fields for plasmon modes M1 and M2, with incident polarization along x. +17 + +-500 +-S0Q +500 +2.0 +SOQ +TOQ +XJO-a +SOQ +J'2 +30Q +40Q +S +×JO-a +S2 +×JO-a +×1O:-S00 +-S0Q +500 +SOQ +TOQ +×JO-a +SOQ +30Q +40Q +×JO-a +X1O-a-500 +-S0Q +0 +500 +SOQ +-S0Q +SOQ +40Qvoid cavities as shown in Fig.4: the latter concentrate important parts of the fields in the +cavity’s empty space,27,43–45 while molecules are immobilized on the surface. +Conclusions +We have reported a comprehensive comparative study of ordered SERS plasmonic substrates +fabricated using nanosphere lithography and, to the best of our knowledge, the first full +resonant study in which both the laser wavelength and the plasmon resonances are inde- +pendently tuned to provide a complete experimental description of the resonant processes +at play. This was done relying on one of the standard (intrinsically non-resonant) probes +used for Raman enhancement studies, namely a self-assembled covalently bound monolayer +of 4-mercaptobenzoic acid (4-MBA) that is known to form an ordered and densely packed +monolayer on the surface of the metal. The concluding answer to the posed question, “has +the chemical contribution a secondary role in SERS?” is clearly no. +We conclude that the first and undoubtedly most important resonance is the surface +plasmon resonance of the metallic substrate (the electromagnetic mechanism). In fact, the +absence of nanostructuring of the metallic surfaces leads to no observable Raman signals. +However, another resonance of the system has also a key role in the enhancement, namely +the charge-transfer between the molecule and the Fermi level of the metal (usually termed as +the chemical mechanism). It has become experimentally clear that in order to adequately +explain the SERS enhancement, the combined molecule-metal system has to be considered.17 +Our experiments demonstrate that resonant experiments are indispensable to fully address +the SERS mechanisms involved, but most importantly that even these might not be enough. +It becomes clear that experiments done with a single laser and tuning the plasmon energies for +resonance might be misleading. And the reversed alternative, i.e. tuning the laser for a fixed +plasmonic substrate, can also provide an incomplete and potentially ambiguous picture. It is +to be expected that most experimental parameter that can be varied to probe a system will +18 + +have an influence via both mechanisms, electromagnetic and chemical, making the separation +of effects difficult.11 +Overall, the emerging picture calls for a unified description of Raman resonances in SERS, +as e.g., previously introduced in Ref. 16. Coincident with these discussed unified models, +our experiments evidence that in the studied substrates the plasmon transitions donate +intensity to charge-transfer transitions, and also suggest that the plasmonic resonances are +sufficiently broad to provide enhancement over a large wavelength range, implying that they +are not primarily responsible for the observed spectral features. In agreement with these +unified theories, our experiments indicate that both electromagnetic and so-called chemical +resonances are coupled and thus the resonant denominators cannot be divided out to consider +the contributions separately. +From our comparative study of ordered plasmonic substrates our conclusion is that, to +fully exploit the electromagnetic enhancement for the detection of surface-deposited molecules, +convex substrates (as MFON) seem to be much more efficient than concave ones (as the +studied sphere segment void cavities and the square pyramidal pits). And probably much +more important than that, we conclude that it is extremely important to take into account +and understand the so-called chemical mechanism for both fundamental reasons and for its +relevance to analytical applications. As previously alerted, since the resonant effects are +multiplicative, unexpected chemical enhancement could lead to analytical conclusions which +are quantitatively wrong.11 +Funding. +The authors acknowledge financial support from the ANPCyT (Argentina) under +grants PICT-2018-03255 and PICT-2020-SERIEA-03123. A.A.R. aknowledges support by +PAIDI 2020 Project No. P20-00548 with FEDER funds. +Disclosures. +The authors declare that there are no conflicts of interests related to this +article. +19 + +Data availability. +Data underlying the results presented in this paper are available from +the corresponding author upon reasonable request. +References +(1) M. Moskovits, Surface-enhanced spectroscopy Rev. Mod. Phys. 57, 783 (1985). +(2) G. C. Schatz, M. 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Uchida, Charge transfer resonance Raman +process in surface-enhanced Raman scattering from p-aminothiophenol adsorbed on +silver: Herzberg-Teller contribution, J. Phys. Chem. 98, 12702 (1994). +(54) P. B. Johnson and R. W. Christy, Optical Constants of the Noble Metals, Phys. Rev. B +6, 4370 (1972). +(55) E. C. Le Ru and P. G. Etchegoin, Rigorous justification of the —E—4 enhancement +factor in Surface Enhanced Raman Spectroscopy, Chem. Phys. Lett. 423, 63 (2006). +(56) J. Rodr´ıguez-Fern´andez, A. M. Funston, J. P´erez-Juste, R. A. ´Alvarez-Puebla, Luis. M. +Liz-Marz´an, and P. Mulvaney, The effect of surface roughness on the plasmonic response +of individual sub-micron gold spheres, Phys. Chem. Chem. Phys. 11, 5909 (2009). +(57) A. Tr¨ugler, J-C. Tinguely, G. Jakopic, U. Hohenester, J. R. Krenn, and A. Hohenau, +Near-field and SERS enhancement from rough plasmonic nanoparticles Phys. Rev. B +89, 165409 (2014). +26 + diff --git a/VNE0T4oBgHgl3EQflwEn/content/tmp_files/load_file.txt b/VNE0T4oBgHgl3EQflwEn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b327c863d0944094efc9f72d8862bc6d4ee5ce44 --- /dev/null +++ b/VNE0T4oBgHgl3EQflwEn/content/tmp_files/load_file.txt @@ -0,0 +1,800 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf,len=799 +page_content='Has the chemical contribution a secondary role in SERS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Luis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Guerra Hern´andez,† Andr´es A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Reynoso,† and Alejandro Fainstein∗,† †Centro At´omico Bariloche and Instituto Balseiro, Comisi´on Nacional de Energ´ıa At´omica (CNEA) - Universidad Nacional de Cuyo (UNCUYO), 8400 Bariloche, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' ‡Instituto de Nanociencia y Nanotecnolog´ıa (INN-Bariloche), Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' ¶Departamento de F´ısica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain E-mail: afains@cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='cnea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='ar Abstract It is an established understanding that the electromagnetic contribution (the plasmon- mediated enhancement of the laser and scattered local electromagnetic fields) is the main actor in Surface Enhanced Raman Scattering (SERS), with the so-called chemical (molecule-related) contribution assuming only, if any, a supporting role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The conclusion of our comprehensive resonant study of a broad range of nanosphere lithography based metallic substrates, with covalently attached 4-mercaptobenzoic acid monolayers used as probe (standard molecules which are non-resonant in solution), is that this accepted understanding needs to be revised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' We present a detailed resonant SERS study of Metal-film over nanosphere (MFON) substrates which is done both by scanning the laser wavelength, and by tuning the plasmon response through the nanosphere diameter which is varied from 500 to 900 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Far and local field properties are characterized through measures of optical reflectivity and SERS efficiency, respectively, and are supported by numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' We demonstrate that the SERS efficiency depends 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='02489v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='optics] 6 Jan 2023 indeed on the electromagnetic mechanism, determined by the plasmonic response of the system, but we observe that it is also strongly defined by a chemical resonant contribution related to a metal-to-ligand electronic transition of the covalently bound probe molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Optimum amplification occurs when the plasmon modes intersect with the ligand-to-metal chemical resonance, contributing synergically both mechanisms together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Quite notably, however, the largest SERS signal is tuned with the metal- to-ligand transition, and typically does not follow the wavelength dependence of the plasmon modes when varying the nanosphere size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The same general trend is observed for other nanosphere lithography based substrates, including sphere-segment void cavities and hexagonally ordered triangular nanoparticles, using both Ag or Au as the plasmonic metal, and also with a commercial substrate (Klarite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Interestingly, this extensive comparative investigation shows in addition that metal-film-over-nanosphere substrates are significantly better than the rest in terms of Raman efficiency and homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' We conclude that a deep understanding of both the electromagnetic and chemical mechanisms is necessary to fully exploit these substrates for analytical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Motivation The plasmonic properties of metal nanostructures are of great interest since they exhibit localized surface plasmons resonances (LSPR), with electromagnetic fields located on the nanostructured surface, and with resonance energy depending on the material, size and shape of the nanostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The interaction between photons impinging from the far field and the LSPR leads to a confinement of the electromagnetic field, enhancing its magnitude and opening the door to plasmon enhanced optical spectroscopies, including surface-enhanced Raman spectroscopy (SERS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='1–4The enhancement of either or both the laser and Raman scattered fields by the plasmon resonances is identified as the electromagnetic contribution to SERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='1–3,5–7 Raman scattering can also be amplified through electronic resonances either intrinsic to the probed molecule (if electronic resonant transitions of the molecule are available 2 at the selected laser excitation), or related to the specific chemical interaction between the molecule and the supporting substrate (a situation that can play a role particularly for molecules that are not intrinsically resonant at the selected laser energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' A variety of different mechanisms have been envisaged involving this latter type of electronic resonances, mostly involving metal-to-ligand (ML) transitions of the bound molecule, and are generally grouped in what is called the chemical contribution to SERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='8–17 Maybe in part because of this complexity, and specificity for different systems, but also because it is assumed to be much weaker and of lesser relevance than the electromagnetic plasmonic enhancement (although some reports position it in the 105 − 107 enhancement range),18,19 the so-called chemical contribution has remained in the backstage, at best assuming a secondary supporting role in SERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' An enormous collection of experimental and theoretical work seems to support this view, which has lead through the design of proper plasmonic substrates to great progress, with successful applications that range from ultrasensitive analytical methods to single- molecule spectroscopies,20,21 including the optical monitoring of single-molecule single-electron transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='22 To be fair, however, very few of the reported experimental investigations treat in depth the issue of the resonant enhancement in SERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Reportedly, less than one percent of the published works address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='23 The reasons are simple to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' To do such a resonant investigation either the excitation wavelength needs to be tuned with enough flexibility through the relevant wavelength range,24–27 or the nanostructures must allow for a reproducible tuning of the plasmonic resonances across the available laser source wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='23,29–32,46 Laser wavelength scans are rarely performed because they require the availability of a large set of lasers or widely tunable sources, and access to a triple spectrometer for efficient stray light rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Plasmon scans, in turn, require a flexible and controlled tunability of the selected technology, and typically imply an important load of fabrication, structural and optical characterization, and experimental work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Different groups have in any case pursued either one of these strategies, and this has been done mostly interpreting the 3 results in the framework of the electromagnetic enhancement mechanism, broadly confirming its relevance in the observed amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Notwithstanding this general agreement, it is also interesting to note that the emerging phenomenology is not universal, nor simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Painstaking experiments based on an extensive number of nanosphere lithography nanoparticle substrates of varying resonant energy, indicate that a correlation between the plasmon resonance and the fixed laser wavelength indeed exists, although with a seemingly noisy correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='23 Some other experiments that report on precisely the same kind of substrates, but scanning the laser wavelength, evidence what seems to be a clear blue shift of the Raman resonance respect to the plasmon extinction maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='26 This was interpreted as the convoluted resonant effect of the incoming laser and the (red-shifted) out-going Stokes Raman scattered fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Other reports observe the reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' That is a red shift of the Raman resonance respect to the plasmon extinction maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='24 This contrasting result has been interpreted as a supposedly universal phenomena related to the dissipation-induced red-shift of the local field modeled as due to the metal charges oscillating as a forced oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='25 In one case were the SERS resonant enhancement was studied with great detail for individual nanoparticles on a mirror, the overall intensity was shown to increase with the particle size and not when the plasmon resonance matched the excitation laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='32 Notwithstanding this rather complex landscape with contrasting evidence, when looked in detail, it is clear that the general agreement is that the problem is understood in its essence, with the electromagnetic enhancement being the main character of the plot,7 and only some voices raised arguing that such explanation is not complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='15 To illuminate this already extensively studied subject which, in our view, has nevertheless remained rather obscure, we present what is in our understanding the first comprehensive resonant SERS study in which experiments are carried out simultaneously at a wide variety of excitation wavelengths and also tuning the surface plasmon resonance by controlling the SERS substrate structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' This is done on substrates fabricated with nanosphere lithography33,34 which are extensively studied and well known for their reproducibility and homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' We 4 describe first the resonant wavelength plasmonic properties of metal-film over nanosphere (MFON) substrates35,36 with M=Au and Ag, which are studied through optical reflectivity, SERS measurements, and numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The resonant wavelength of the localized plasmons in these MFON substrates can be adjusted by changing the diameter of the nanosphere, with wavelengths varying in the NIR-VIS spectral range (∼400-1100 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' For the SERS studies, a self-assembled monolayer of 4-mercaptobenzoic acid (4-MBA) was used as probably the most extensively used Raman probe that forms an ordered and densely packed monolayer on the surface of the metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='37,38 This Raman molecule in solution displays electron transitions in the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' That is, it is non-resonant in the spectral range where the plasmons reside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Previous investigations have shown, however, that this and similar thiol- bound molecules develop a resonant electronic ligand-to-metal transition (L-M ∼700 nm) when covalently attached to either Au or Ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='39–42 Notably, we find that the maximum SERS efficiency does not follow the plasmon dispersion but, on the contrary, is observed when a plasmon mode becomes resonant with the ligand-to-metal electronic resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' We extend this investigation to other ordered plasmonic substrates, namely Au segment sphere void cavities27,43–45 and triangular nanoparticles46 both fabricated by nanosphere lithography, and a commercial Au substrate composed of square pyramidal pits (Klarite),47 with coincident results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Our conclusion is that SERS is a single effect drawing on plasmon and metal-to-ligand resonances which are intimately tied to each other and cannot straightforwardly be considered separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' While this does not contradict the established conviction on the relevance of the electromagnetic amplification, since determining the true SERS enhancement factor is critical for analytical determinations,48 it becomes clear that the more complex unified perspective will need to be considered for real world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='15,17,49,50 5 Samples and experimental set-up Fabrication of periodic hexagonal arrays Polystyrene spheres dispersed in a 1% water solution (Duke Scientific) are introduced between a glass substrate covered with a 100 nm Au-film (Platypus) and a clean glass slide, with a separation of about 300 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The Au-film is immersed in a 1 mM cysteamine ethanolic solution overnight to enhance the polystyrene spheres adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' During the drying process in an incubation chamber, a sweeping meniscus forms along the substrate, pulling the spheres towards the substrate into a close-packed hexagonal monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' To fabricate the MFON substrates this self-assembly of the template is followed by a physical vapor deposition of Au or Ag (Vega and Camji S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The Au and Ag-films were vapor-deposited over 500- 900 nm polystyrene spheres, in a homemade evaporator, which achieves a residual pressure of ∼1×10−7 torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The control of the material fusion temperature is done manually, with a current of 10-15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The triangular lattice of ordered nanoparticles is obtained starting from 50 nm thick Au-FON substrates, and after removal of the latex spheres by sonication in acetone, isopropanol and Milli-Q water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Segment void sphere cavity substrates are fabricated following the same procedure of deposition of the polystyrene nanospheres into a close-packed hexagonal monolayer, followed by electrochemical deposition of Au from a Gold salt solution (TG-25 RTU, Technic Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='), with a deposition rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='5mC/min, followed by removal of the plystyrene spheres by sonication in a sequence of solvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='40,51 The studied square pyramidal pit substrates were acquired from the company Klarite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Substrate characterization and resonant SERS experiments Reflectivity measurements were used to identify the plasmon modes of all the different studied substrate arrays, using a fully automated Wollam WVASE32 variable angle spectroscopic ellipsometer with focusing probes, presenting a 100 µm circular spot on the sample with a numerical aperture of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' SEM images were recorded with a FEI field-emission gun, 6 Nova NANO-SEM 230, operating at 10 kV and with a tilt angle of 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Surface roughness was characterized also in all the substrates through atomic force microscopy (AFM) with an AFM Veeco Dimension 3100, with MESP tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Signal homogeneity was monitored with a LabRam HR Evolution Raman microscope using the He-Ne 633 nm laser line (close to the M-L resonance at ∼ 675 nm) and taking 400 spectra in a 5 × 5µm2 square area with a x100 microscope objective of NA=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' SERS measurements were performed using a triple- stage Raman spectrometer (Horiba Jobin-Yvon T64000) operating in subtractive mode, and equipped with a liquid nitrogen-cooled charge-coupled device (CCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The excitation was performed using the 514, 568, 647 and 676 nm lines of an Ar-Kr laser and a continuously tunable Ti-Sapph laser between 680 and 780 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The position on the sample was manually controlled, and the Raman signals were collected in a backscattering configuration with a collection lens of focal length +10 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The entrance slit of the spectrometer was kept at 200 µm, as the Raman peaks for the metal-adsorbed molecule are relatively broad and do not require high spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Typical acquisition times were from 5 to 10 s, depending on the wavelength and the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Typically 10 spectra were acquired for each wavelength and substrate at different positions, with the average providing the SERS intensity and the statistical dispersion the shown error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Results and Discussion Plasmonics modes and SERS in Au-film over nanosphere (AuFON) substrates Figure 1(a) presents a scheme of the AuFON substrate and the experimental configuration with light incident at a small angle, and scattered light collected along the substrate normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Panel (b) in this same figure shows a high-resolution SEM image of a typical AuFON fabricated with 500 nm diameter polystyrene spheres, taken with a tilt angle of 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The close-packed and highly ordered formation of the support nanosphere mask can be clearly 7 identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' A typical spectrum of 4-MBA immobilized on AuFON is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' 1(c), with the aromatic-ring breathing vibration used to monitor the SERS intensity in the following sections highlighted with grey background at 1076 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' µm 1 (b) (a) 400 600 800 1000 1200 1400 1600 1800 SERS Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' [cts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='] Raman Shift [cm 1] 4-MBA spectrum 1076 (c) 4-MBA Au/Ag Latex sphere SERS Substrate Laser 𝜃 2 Reflection S 𝜙 Figure 1: (a) Schematic of the 4-MBA SERS experiment on metal-film on nanosphere (MFON) substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' (b) High-resolution SEM of a typical nanostructure AuFON of 500 nm in diameter of polystyrene sphere, tilt=45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' (c) Typical spectrum of 4-MBA absorbed in AuFON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The Raman mode at 1076 cm−1 used to monitor the SERS efficiency is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The far-field response of the plasmon resonances in the studied AuFON substrates is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' 2 (left panel), for Au polystyrene nanosphere diameters in the range 500-900 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The reflectivity experiments were performed with TE polarization and light incident at an 8 spot HV WD mag det Landing E um 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='00 kV / 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='9 mm 80 000 x TLD 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='00 keV Nova NanosEMangle of 25◦, with wavelengths between 400-1000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Each curve is vertically offset by steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='3 for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' We note that for λ <550 nm the reflectivity decreases due to the interband transitions characteristic of Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Three resonant absorptions can be identified in the wavelength range of 600-900 nm, indicated by blue triangles, circles and squares, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' These absorptions are associated with LSPR plasmons of the AuFON structures, and which we label as M1, M2 and M3 for increasing energy (decreasing wavelength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' We have observed that these modes do not vary significantly with the polar (θ) and azimuthal (φ) angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Importantly, as expected the resonance wavelength of these modes strongly blue-shifts when reducing the diameter of the polystyrene spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The resonant SERS study for the different AuFON substrates is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' 2(right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The SERS intensity was monitored for the different samples through the amplitude of the 1076 cm−1 Raman peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Each point is the average of ten measurements at different spots, with the error bars indicating their dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Raman intensities are given in counts per mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='s, with all spectra acquired exactly under the same experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='2(right) the vertical red line indicates the wavelength of the ligand-to-metal transition (L-M), while the blue symbols are guides to the eye identifying the plasmon resonances obtained from the reflectivity curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='2(left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Two aspects can be highlighted from these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' First, irrespective of the AuFON period, the most intense Raman signals are detected in all studied cases very close to the M-L transition, with only a small red-shift of the resonance maxima when the M1 plasmon mode approaches this resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Second, the maximum detected signal augments when a plasmon mode is close to the M-L transition, and this is particularly noteworthy for the M1 mode which leads to the strongest SERS (bottom spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='2(right)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' 9 400 600 800 1000 0,0 0,3 0,6 0,9 1,2 1,5 500 600 700 800 0 50 100 150 200 250 0 20 40 60 0 10 20 30 40 0 10 20 30 40 0 10 20 30 M3 M2 M1 M3 M2 D=500nm D=600nm D=700nm D=800nm D=900nm D=500nm D=600nm D=700nm D=800nm M1 M-L D=900nm Reflectivity Wavelength [nm] SERS Intensity [cts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' /mW/s] Figure 2: Left: Reflectivity of AuFON substrates with varying diameter of the polystyrene spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Curves are vertically offset by steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='3 for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The dispersion of the plasmon modes (M1, M2 and M2) is indicated with blue symbols and guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The experiments were done with TE polarization, and incident angle of 25◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Right: Raman intensity as a function of laser wavelength for the same substrates presented in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The spectra were also acquired with parallel TE polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The shown Gaussian curves are guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The symbols and connecting lines identify the plasmon resonances determined by the reflectivity measurements shown in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The vertical line signals the metal-to-ligand transition (M-L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Plasmonics modes and SERS in Ag-film over nanosphere (AgFON) substrates The results presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='2 indicate that plasmons (that is, the electromagnetic enhance- ment) play an important role on the SERS efficiency, but also suggest that this might not 10 400 600 800 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='3 500 600 700 800 0 500 1000 1500 0 250 500 750 1000 1250 0 200 400 600 800 0 100 200 300 400 500 SERS Intensity [cts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' /mW/s] Reflectivity M3 M2 M1 D=500nm D=600nm D=700nm D=800nm D=800nm D=700nm D=600nm D=500nm M1 M2 M3 M-L Wavelength [nm] Figure 3: Left: Reflectivity of AgFON substrates with varying diameter of the polystyrene spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Curves are vertically offset by steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='3 for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The dispersion of the plasmon modes (M1, M2 and M2) is indicated with blue symbols and guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The experiments were done with TE polarization, and incident angle of 25◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Right: Raman intensity as a function of laser wavelength for the same substrates presented in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The spectra were also acquired with parallel TE polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The shown black dash-dotted curves are guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The symbols and connecting lines identify the plasmon resonances determined by the reflectivity measurements shown in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The vertical line signals the metal-to-ligand transition (M-L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' be separable from the metal-to-ligand resonant contribution (that is, the so-called chemical enhancement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' To provide additional data on this phenomena we present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='3 a similar investigation but now performed on Ag instead of Au MFON substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Again the left panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='3 presents the reflectivity measurements, while the right panel shows the corresponding SERS intensity curves for AgFON substrates made with polystyrene NPs of sizes ranging 11 from 500 to 800nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Similarly to the Au case, three plasmon resonances can be identified that blue-shift with decreasing NP size (labeled as M1, M2 and M3 in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Several aspects of the SERS resonant curves can be mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' First, again Raman resonant maxima are observed for all substrates at the same wavelength coincident with the M-L transition, irrespective of the specific pattern of plasmon modes of the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The absolute value of the scattering efficiency increases when a plasmon mode is close to the M-L transition, but the resonant peak does not follow the plasmon dispersion: it is fixed at the spectral position of the M-L transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Again as for the Au case the proximity of the M1 plasmon mode seems to be specially relevant to provide the largest scattering efficiencies (bottom spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='3(right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Second, for the Ag film on nanosphere substrates a second maximum can be identified, in this case following the M3 plasmon mode (top two spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='3(right)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' This difference when compared with the AuFON substrates can be traced to the inter-band transitions existent in Au and absent in Ag,54 which are expected to quench the Raman resonances due to absorption at wavelengths below ∼ 600nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Note that part of the intensity of the M3 resonance when compared to that due to the M1 mode can be ascribed to the ω4 enhancement which between 700 and 550 nm varies by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='55 Third, as is standard in SERS Ag provides significant larger enhancement than Au when structurally similar substrates are compared (at least a factor of 5 comparing Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Overall then, the results for AgFON substrates in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='3 confirm the main conclusions obtained from those made from Au in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='2, namely: the electromagnetic plasmon-mediated enhancement is clearly part of the menu, but the chemical contribution has no secondary role in this phenomena, at least for the studied MFON substrates and for a quite universally used covalently attached probe molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' As we will see next, these conclusions are rather general and valid for a much larger set of plasmonic substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' 12 µm 1 Figure 4: Panels a-d present a comparative study of SERS resonant scans obtained under the same conditions and using the same molecular probe as the experiments in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='2 and 3, for AuFON, square pyramidal Au covered pits (commercial substrate Klarite), Au segment sphere void cavities, and Au triangular nanoparticles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' SEM images of the studied substrates are shown at the right of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' AuFON (a), cavities (c) and Au NPs (d) were all fabricated using nanosphere lithography methods with 500 nm polystyrene spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The labels in each panel identify the plasmon modes determined from reflectivity measurements equivalent to those presented for the MFON substrates in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Comparative study between different ordered plasmonic substrates Figure 4 presents a comparative resonant SERS study of a diverse variety of ordered Au plasmonic substrates, namely AuFON (a), square pyramidal pits of the commercial Klarite 13 LWT (a) AuFON 300 200 M2 100 S SERS intensity [cts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' /mW / 0 P2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' (b) Klarite 15 10 P3 IP1 5 0 Cavity (c) 6 4 ID M-L 2 0 (d) AuNPS 0,3 LSP 0,2 0,1 0,0 500 550 600 650 700 750 800 Wavelength [nm]spot HV WD mag det μm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='00 kV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='7 mm 80 000 x TLD Nova NanosEMsubstrate (b),47 segment sphere void cavities (c),27,43–45 and triangular nanoparticles (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='46 Each panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='4 presents, at its right, a SEM image of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Substrates (a), (c) and (d) were all fabricated by nanosphere lithography using 500 nm polystyrene spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' As rugosity is known to have a potentially relevant effect on the SERS efficiency,51,56,57 it was determined for all substrates (excluding the triangular Au NPs that are not extended and thus are not directly comparable) using AFM scans finding comparatively similar values in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Grain sizes span from 0 to 120 nm with a peaked distribution with maxima around 40-60 nm, with similar degree of roughness for the cavities and MFON substrates and slightly lower frequency of occurrence of the grains for the commercial Klarite substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The extended substrates were also monitored for their homogeneity in terms of Raman efficiency, finding on 5 × 5µm2 square areas dispersions of around 17%, 21%, and 31% for the AuFON, Klarite, and cavity structures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Assuming then that constructively these substrates are comparable, we extract two important conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' First, again and in all four cases, the M-L transition seems to be determinant to the spectral position where maximum SERS efficiency is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' And second, by quite far (a factor of at least 20) the AuFON substrate results in a much larger (and more homogeneous) Raman efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' For comparison, we note here that experiments performed on identical 4-MBA self-assembled monolayers on flat (non-structured) Au substrates lead to no observable Raman signals even with 20 times larger acquisition times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' To understand the physics behind the observed larger electromagnetic SERS enhancement for the MFON substrates, and to identify the origin of the M1-M3 plasmon modes and their contrasting efficiency for SERS, we briefly describe next its modeling based on finite element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Theoretical modeling of metal-film over nanosphere substrates We evaluate the electromagnetic response of the nanostructured surface of a AuFON substrate when it is excited by a plane wave by solving for the full-field Maxwell equations in 3D using the finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The standard dielectric function of Au is used as given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' 14 400 600 800 1000 0,0 0,3 0,6 0,9 1,2 1,5 400 600 800 1000 M2 700nm 650nm 600nm 550nm 500nm 700nm 650nm 600nm 550nm Reflectivity Wavelength [nm] 500nm (a) (b) M1 M2 M1 0,0 0,5 1,0 1,5 2,0 2,5 3,0 IEavI (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=') x 107 Figure 5: Calculated wavelength dependence of the far-field reflectivity (a) and near-field amplitude (b) for a plane wave incident on AuFON substrates of varying sphere size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' These substrates are modeled as solid Au half spheres on a planar and uniform Au film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The solid connected symbols identify the plasmon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Periodic boundary conditions are incorporated by ensuring perfect agreement in the triangular mesh within each of the three pairs of partner faces defining the hexagonal arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The computed scattering matrix includes all allowed diffraction modes obtained using the customary Ewald criteria for the working plane-wave wavelength and incident direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' To first test the qualitative behavior of the mode, we perform calculations for an hexagonal arrangement of solid Au semi-spheres over a uniform Au surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='5, were both the far field reflectivity (a) and the near field averaged on the surface (b) are shown as 15 a function of wavelength and for increasing sphere size (from bottom to top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The magnitude of the near field determines the electromagnetic enhancement affecting the SERS efficiency, and thus the two shown panels can be related to the similar ones in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The agreement is good in several features, namely: i) two main plasmon modes are observed in the relevant wavelength range, ii) these disperse to larger wavelengths for increasing nanosphere size, and iii) the smaller energy mode, identified as M1, leads to the largest averaged near field and thus is expected to provide the stronger Raman signals as experimentally observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' While the qualitative agreement between theory and experiment is reasonably good with such a simplified description of the substrates, we have observed that the quantitative determination of the plasmon energies and size dependence is strongly sensitive on the details of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Calculations with a more realistic description of the AuFON substrates are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='6, where the polystyrene spheres have been included (as a dielectric material of index of refraction n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='59), and the film cover was allowed to have an ellipsoid shape to account for a larger deposit of metal on the top (see a scheme in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='6(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' For this simulations we use 500 nm diameter spheres, the Au thickness was taken as 180 nm at the top of the sphere, and light was incident at angles θ=25◦ and φ=0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The interstitial spaces between spheres have been simulated with and without pyramidal Au arrangements, without significant variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' In this case excellent agreement with the experimental results is obtained as shown in Fig 6(b), both in the spectral position of the modes and the relative magnitude of the averaged surface near-field associated to the M1 and M2 plasmon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The strong sensitivity of the latter on the distance to the surface of the Au film is illustrated on the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The color maps in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='6(c) further clarify the contrasting response of plasmons M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The M1 one mode has very strong associated electromagnetic fields very close to the surface and precisely in between neighbor spheres as known to be relevant for nanoparticle dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' This result also explains the comparative high efficiency of the MFOM substrates when contrasted with the square pyramidal pit and segment sphere 16 400 600 800 1000 0,0 0,2 0,4 0,6 0,8 400 600 800 1000 0,5 1,0 1,5 2,0 y [nm] y [nm] x [nm] x [nm] z [nm] Reflectivity Wavelength [ nm ] (a) (b) (c) M1 M2 M1 z [nm] z [nm] y [nm] Z=5nm 30nm 100nm 150nm M1 M2 M2 lEavl (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=') x [nm] Figure 6: (a)Scheme of the ellipsoid shape of the deposited Au film, internal polystrene sphere, and metal Au base used in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' (b) Wavelength dependence of the calculated far field reflectivity (left panel) and associated averaged magnitude of the surface near-field, for an AuFON substrate of 500 nm spheres and 180 nm film thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' In the right panel different curves are presented for varying distance from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' M1 and M2 identify the plasmon resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' (c) Color maps of the spatial distribution of the magnitude of the electromagnetic fields for plasmon modes M1 and M2, with incident polarization along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' 17 500 S0Q 500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content="0 SOQ TOQ XJO-a SOQ J'2 30Q 40Q S ×JO-a S2 ×JO-a ×1O:-S00 S0Q 500 SOQ TOQ ×JO-a SOQ 30Q 40Q ×JO-a X1O-a-500 S0Q 0 500 SOQ S0Q SOQ 40Qvoid cavities as shown in Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='4: the latter concentrate important parts of the fields in the cavity’s empty space,27,43–45 while molecules are immobilized on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Conclusions We have reported a comprehensive comparative study of ordered SERS plasmonic substrates fabricated using nanosphere lithography and, to the best of our knowledge, the first full resonant study in which both the laser wavelength and the plasmon resonances are inde- pendently tuned to provide a complete experimental description of the resonant processes at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' This was done relying on one of the standard (intrinsically non-resonant) probes used for Raman enhancement studies, namely a self-assembled covalently bound monolayer of 4-mercaptobenzoic acid (4-MBA) that is known to form an ordered and densely packed monolayer on the surface of the metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The concluding answer to the posed question, “has the chemical contribution a secondary role in SERS?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' is clearly no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' We conclude that the first and undoubtedly most important resonance is the surface plasmon resonance of the metallic substrate (the electromagnetic mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' In fact, the absence of nanostructuring of the metallic surfaces leads to no observable Raman signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' However, another resonance of the system has also a key role in the enhancement, namely the charge-transfer between the molecule and the Fermi level of the metal (usually termed as the chemical mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' It has become experimentally clear that in order to adequately explain the SERS enhancement, the combined molecule-metal system has to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='17 Our experiments demonstrate that resonant experiments are indispensable to fully address the SERS mechanisms involved, but most importantly that even these might not be enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' It becomes clear that experiments done with a single laser and tuning the plasmon energies for resonance might be misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' And the reversed alternative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' tuning the laser for a fixed plasmonic substrate, can also provide an incomplete and potentially ambiguous picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' It is to be expected that most experimental parameter that can be varied to probe a system will 18 have an influence via both mechanisms, electromagnetic and chemical, making the separation of effects difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='11 Overall, the emerging picture calls for a unified description of Raman resonances in SERS, as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=', previously introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Coincident with these discussed unified models, our experiments evidence that in the studied substrates the plasmon transitions donate intensity to charge-transfer transitions, and also suggest that the plasmonic resonances are sufficiently broad to provide enhancement over a large wavelength range, implying that they are not primarily responsible for the observed spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' In agreement with these unified theories, our experiments indicate that both electromagnetic and so-called chemical resonances are coupled and thus the resonant denominators cannot be divided out to consider the contributions separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' From our comparative study of ordered plasmonic substrates our conclusion is that, to fully exploit the electromagnetic enhancement for the detection of surface-deposited molecules, convex substrates (as MFON) seem to be much more efficient than concave ones (as the studied sphere segment void cavities and the square pyramidal pits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' And probably much more important than that, we conclude that it is extremely important to take into account and understand the so-called chemical mechanism for both fundamental reasons and for its relevance to analytical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' As previously alerted, since the resonant effects are multiplicative, unexpected chemical enhancement could lead to analytical conclusions which are quantitatively wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='11 Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The authors acknowledge financial support from the ANPCyT (Argentina) under grants PICT-2018-03255 and PICT-2020-SERIEA-03123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' aknowledges support by PAIDI 2020 Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' P20-00548 with FEDER funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' The authors declare that there are no conflicts of interests related to this article.' metadata={'source': 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J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Krenn, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Hohenau, Near-field and SERS enhancement from rough plasmonic nanoparticles Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' B 89, 165409 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE0T4oBgHgl3EQflwEn/content/2301.02489v1.pdf'} diff --git a/WNFRT4oBgHgl3EQf9DhY/content/tmp_files/2301.13686v1.pdf.txt b/WNFRT4oBgHgl3EQf9DhY/content/tmp_files/2301.13686v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..abd0fd79154e3fe2a7af1d888d97efc4a6368984 --- /dev/null +++ b/WNFRT4oBgHgl3EQf9DhY/content/tmp_files/2301.13686v1.pdf.txt @@ -0,0 +1,3746 @@ +Detecting Unknown Encrypted Malicious Traffic in +Real Time via Flow Interaction Graph Analysis +Chuanpu Fu∗, Qi Li†‡, Ke Xu∗‡ +∗Department of Computer Science and Technology, Tsinghua University +†Institute for Network Sciences and Cyberspace, Tsinghua University ‡Zhongguancun Lab +Abstract—Nowadays traffic on the Internet has been widely +encrypted to protect its confidentiality and privacy. However, +traffic encryption is always abused by attackers to conceal their +malicious behaviors. Since the encrypted malicious traffic has +similar features to benign flows, it can easily evade traditional +detection methods. Particularly, the existing encrypted malicious +traffic detection methods are supervised and they rely on the prior +knowledge of known attacks (e.g., labeled datasets). Detecting +unknown encrypted malicious traffic in real time, which does not +require prior domain knowledge, is still an open problem. +In this paper, we propose HyperVision, a realtime unsuper- +vised machine learning (ML) based malicious traffic detection +system. Particularly, HyperVision is able to detect unknown +patterns of encrypted malicious traffic by utilizing a compact in- +memory graph built upon the traffic patterns. The graph captures +flow interaction patterns represented by the graph structural +features, instead of the features of specific known attacks. We de- +velop an unsupervised graph learning method to detect abnormal +interaction patterns by analyzing the connectivity, sparsity, and +statistical features of the graph, which allows HyperVision to de- +tect various encrypted attack traffic without requiring any labeled +datasets of known attacks. Moreover, we establish an information +theory model to demonstrate that the information preserved by +the graph approaches the ideal theoretical bound. We show the +performance of HyperVision by real-world experiments with 92 +datasets including 48 attacks with encrypted malicious traffic. The +experimental results illustrate that HyperVision achieves at least +0.92 AUC and 0.86 F1, which significantly outperform the state- +of-the-art methods. In particular, more than 50% attacks in our +experiments can evade all these methods. Moreover, HyperVision +achieves at least 80.6 Gb/s detection throughput with the average +detection latency of 0.83s. +I. +INTRODUCTION +Traffic encryption has been widely adopted to protect the +information delivered on the Internet. Over 80% websites +adopted HTTPS to prevent data breach in 2019 [16], [62]. +However, the cipher-suite for such protection is double-edged. +In particular, the encrypted traffic also allows attackers to con- +ceal their malicious behaviors, e.g., malware campaigns [2], +exploiting vulnerabilities [64], and data exfiltration [77]. The +ratio of encrypted malicious traffic on the Internet is growing +significantly [2], [3], [76] and exceeds 70% of the entire +malicious traffic [16]. +However, encrypted malicious traffic detection is not well +addressed due to the low-rate and diverse traffic patterns [2], +[39], [77]. The traditional signature based methods that lever- +age deep packet inspection (DPI) are invalid under the at- +tacks with the encrypted payloads [34]. Table I compares the +existing malicious traffic detection methods. Different from +plain-text malicious traffic, the encrypted traffic has similar +features to benign flows and thus can evade existing machine +learning (ML) based detection systems as well [2], [3], [62]. +Particularly, the existing encrypted traffic detection methods +are supervised, i.e., relying on the prior knowledge of known +attacks, and can only detect attacks with known traffic patterns. +They extract features of specific known attacks and use labeled +datasets of known malicious traffic for model training [2], +[3], [76]. Thus, they are unable to detect a broad spectrum +of attacks with encrypted traffic [39], [41], [64], [77], which +are constructed with unknown patterns [22]. Besides, these +methods are incapable of detecting both attacks constructed +with and without encrypted traffic and unable to achieve +generic detection because features of encrypted and non- +encrypted attack traffic are significantly different [2], [3]. +In a nutshell, the existing methods cannot achieve unsuper- +vised detection and they are unable to detect encrypted mali- +cious traffic with unknown patterns. In particular, the encrypted +malicious traffic has stealthy behaviors, which cannot be cap- +tured by these methods [2], [76] that detect attacks according +to the patterns of a single flow. However, it is still feasible to +detect such attack traffic because these attacks involve multiple +attack steps with different flow interactions among attackers +and victims, which are distinct from benign flow interactions +patterns [24], [26], [39], [46], [61]. For example, the encrypted +flow interactions between spam bots and SMTP servers are +significantly different from the legitimate communications [61] +even if the single flow of the attack is similar to the benign one. +Thus, this paper explores utilizing interaction patterns among +various flows for malicious traffic detection. +To the end, we propose HyperVision, a realtime detection +system that aims to capture footprints of encrypted malicious +traffic by analyzing interaction patterns among flows. In par- +ticular, it can detect encrypted malicious flows with unknown +footprints by identifying abnormal flow interactions, i.e., the +interaction patterns that are distinct from benign ones. To +achieve this, we build a compact graph to capture various +flow interaction patterns so that HyperVision can perform +detection on various encrypted traffic according to the graph. +The graph allows us to detect attacks without accessing packet +payloads, while retaining the ability of detecting traditional +(known) attacks with plain-text traffic. Therefore, HyperVision +can detect the malicious traffic with unknown patterns by +learning the graph structural features. Meanwhile, by learning +the graph structural features, it realizes unsupervised detection, +which does not require model training with labeled datasets. +However, it is challenging to build the graph for realtime +detection. We cannot simply use IP addresses as vertices and +traditional four-tuple of flows [19], [36] as edges to construct +the graph because the resulting dense graph cannot maintain +Network and Distributed System Security (NDSS) Symposium 2023 +27 February - 3 March 2023, San Diego, CA, USA +ISBN 1-891562-83-5 +https://dx.doi.org/10.14722/ndss.2023.23080 +www.ndss-symposium.org +arXiv:2301.13686v1 [cs.CR] 31 Jan 2023 + +TABLE I. +THE COMPARISON WITH THE EXISTING METHODS OF MALICIOUS TRAFFIC DETECTION. +Data Source +Categories +Data Sources +Typical Methods +Data for Detection +Design Goals +Detection Performance +Unlabeled +Datasets +Multi-Flow +Features +Generic +Detection +Realtime +Detection +Unknown +Attacks +Low +Latency +High +Throughput +Encrypted Traffic +Protocol Headers +TLS Extensions [16] +× +× +× +× +× +× +✓ +HTTPS Headers [3] +× +× +× +× +× +× +× +Related Flows +Time Series [76] +× +× +× +× +× +× +× +TLS Handshakes [2] +× +× +× +× +× +× +× +Flow Statistics [90] +✓ +× +× +✓ +× +× +✓ +Plain-text and +Encrypted Traffic +Network Logs +Intrusion Events [20] +✓ +× +× +× +✓ +× +× +Sampled Connections [8] +✓ +✓1 +× +✓ +× +× +✓ +Traffic Features +Per-Packet Features [56] +✓ +× +× +× +✓ +✓ +× +Per-Flow Features [5] +× +× +× +✓ +× +✓ +× +Flow Interaction Graph +✓ +✓ +✓ +✓ +✓ +✓ +✓ +1 Existing multi-flow features can only represent the features of specific flows, which cannot be used to represent complicated interaction patterns among various flows. +interaction patterns among various flows, e.g., incurring the +dependence explosion problem [87]. Inspired by the study of +the flow size distribution [25], [84], i.e., most flows on the +Internet are short while most packets are associated with long +flows, we utilize two strategies to record different sizes of +flows, and process the interaction patterns of short and long +flows separately in the graph. Specifically, it aggregates the +short flows based on the similarity of massive short flows +on the Internet, which reduces the density of the graph, and +performs distribution fitting for the long flows, which can +effectively preserve flow interaction information. +We design a four-step lightweight unsupervised graph +learning approach to detect encrypted malicious traffic by +utilizing the rich flow interaction information maintained on +the graph. First, we analyze the connectivity of the graph by +extracting the connected components and identify abnormal +components by clustering the high-level statistical features. +By excluding the benign components, we also significantly +reduce the learning overhead. Second, we pre-cluster the edges +according to the observed local adjacency in edge features. +The pre-clustering operations significantly reduce the feature +processing overhead and ensure realtime detection. Third, we +extract critical vertices by solving a vertex cover problem using +Z3 SMT solver [55] to minimize the number of clustering. Fi- +nally, we cluster each critical vertex according to its connected +edges, which are in the centers of the clusters produced by the +pre-clustering, and thus obtain the abnormal edges indicating +encrypted malicious traffic. +Moreover, to quantify the benefits of the graph based flow +recording of HyperVision over the existing approaches, we +develop a flow recording entropy model, an information theory +based framework that theoretically analyzes the amount of +information retained by the existing data sources of malicious +traffic detection systems. By using this framework, we show +that the existing sampling based and event based traffic data +sources (e.g., NetFlow [19] and Zeek [86]) cannot retain high- +fidelity traffic information. Thus, they are unable to record +flow interaction information for the detection. But the graph +in HyperVision captures near-optimal traffic information for +the graph learning based detection and the amount of the +information maintained in the graph approaches the theoretical +up-bound of the idealized data source with infinite storage +according to the data processing inequality [85]. Also, the +analysis results demonstrate that the graph in HyperVision +achieves higher information density (i.e., amount of traffic +information per unit of storage) than all existing data sources, +which is the foundation of the accurate and efficient detection. +We prototype HyperVision1 with Intel’s Data Plane De- +velopment Kit (DPDK) [37]. To extensively evaluate the +performance of the prototype, we replayed 92 attack datasets +including 80 new datasets collected in our virtual private +cloud (VPC) with more than 1,500 instances. In the VPC, we +collected 48 typical encrypted malicious traffic, including (i) +encrypted flooding traffic, e.g., flooding target links [41]; (ii) +web attacks, e.g., exploiting web vulnerabilities [64]; (iii) mal- +ware campaigns, including connectivity testing, dependency +update, and downloading. In the presence of the background +traffic by replaying the backbone network traffic [80], Hyper- +Vision achieves 13.9% ∼ 36.1% accuracy improvements over +five state-of-the-art methods. It detects all encrypted malicious +traffic in an unsupervised manner with more than 0.92 AUC, +0.86 F1, where 44 of the real-world stealthy traffic cannot be +identified by all the baselines, e.g., an advanced side-channel +attack exploiting the CVE-2020-36516 [26] and many newly +discovered cryptojacking attacks [7]. Moreover, HyperVision +achieves on average more than 100 Gb/s detection throughput +with the average detection latency of 0.83s. +In summary, the contributions of our paper are five-fold: +• We propose HyperVision, the first realtime unsupervised +detection for encrypted malicious traffic with unknown +patterns by utilizing the flow interaction graph. +• We develop several algorithms to build the in-memory +graph that allows us to accurately capture interaction +patterns among various flows. +• We design a lightweight unsupervised graph learning +method to detect encrypted traffic via graph features. +• We develop a theoretical analysis framework established +by information theory to show that the graph captures +near-optimal traffic interaction information. +• We prototype HyperVision and use the extensive experi- +ments with various real-world encrypted malicious traffic +to validate its accuracy and efficiency. +The rest of the paper is organized as follows: Section II in- +troduces the threat model of HyperVision. Section III presents +the high-level design of HyperVision. In section IV, V, and VI, +we describe the detailed designs. In Section VII, we conduct +the theoretical analysis. In Section VIII, we experimentally +evaluate the performances. Section IX reviews related works +and Section X concludes this paper. Finally, we present details +in Appendix. +1Source code and datasets: https://github.com/fuchuanpu/HyperVision. +2 + +Abnormal +Component Detection +Critical Vertex Detection +Edge Pre-Clustering +Interaction Pattern +Clustering +Flow Collection + +Flow Classification +Short Flows +Long Flows +Short Flow Aggregation +Similar Short Flows +Long Flow Distribution Fitting +Packet Feature Distribution +Flow Interaction Graph +Ongoing +Traffic +Raw Packet +Parser +1. Graph Construction +2. Graph Pre-Processing +Abnormal +Component +Timeout +Threshold +Long Flows + + +Connected Components +Attacker +Benign User +Malicious Flow +Benign Flow +Malicious Traffic +Identified Cluster +3. Abnormal Interaction Detection +Short Flows +Component +Statistical Features + + + + + +Critical +Vertex + + +Benign +Malicious + + +Benign + +Attacker +Victims + +Fig. 1. +The overview of HyperVision. +II. +THREAT MODEL AND DESIGN GOALS +We aim to develop a realtime system (i.e., HyperVision) +to detect encrypted malicious traffic. It performs detection ac- +cording to the traffic replicated by routers through port mirror- +ing [17], which ensures that the system will not interfere with +the traffic forwarding. After identifying encrypted malicious +traffic, it can cooperate with the existing on-path malicious +traffic defenses [48], [49], [88] to throttle the detected traffic. +To perform detection on encrypted traffic, we cannot parse and +analyze application layer headers and payloads. +In this paper, we focus on detecting active attacks con- +structed with encrypted traffic. We do not consider passive +attacks that do not generate traffic to victims, e.g., traffic +eavesdropping [68] and passive traffic analysis [70]. According +to the existing studies [10], [24], [29], [40], [46], [81], attack- +ers utilize reconnaissance steps to probe the information of +victims, e.g., the password of a victim [39], the TCP sequence +number of a TLS connection [26], [27], and the randomized +memory layout of a web server [75], which cannot be accessed +directly by attackers due to lack of prior knowledge. Note that, +these attacks are normally constructed with many addresses +owned or faked by attackers. +The design goals of HyperVision are as follows: First, +it should be able to achieve generic detection, i.e., detect +attacks constructed with encrypted or non-encrypted traffic, +which ensures that the attacks cannot evade detection by traffic +encryption [2], [77]. Second, it is able to achieve realtime +high-speed traffic processing, which means that it can identify +whether the passing through encrypted traffic is malicious, +while incurring low detection latency. Third, the performed +detection by HyperVision is unsupervised, which means that it +does not require any prior knowledge of encrypted malicious +traffic. That is, it should be able to deal with attacks with +unknown patterns, i.e., zero-day attacks, which have not been +disclosed [30]. Thus, we do not use any labeled traffic datasets +for ML training. These issues cannot be well addressed by the +existing detection methods [62]. +III. +OVERVIEW OF HYPERVISION +In this section, we develop HyperVision that is an unsuper- +vised detection system to capture malicious traffic in real time, +in particular, encrypted malicious traffic. Normally, patterns +of each flow in the encrypted malicious traffic, i.e., single- +flow patterns, may be similar to benign flows, which allow +them to evade the existing detection. However, the malicious +behaviors appearing in the interaction patterns between the +attackers and victims will be more distinct from the benign +ones. Thus, in HyperVision, we construct a compact graph +to maintain interaction patterns among various flows and +detect abnormal interaction patterns by learning the features of +the graph. HyperVision analyzes the graph structural features +representing the interaction patterns without prior knowledge +of known attack traffic and thus can achieve unsupervised +detection against various attacks. It realizes generic detection +by analyzing flows regardless of the traffic type and can detect +encrypted and non-encrypted malicious traffic. Figure 1 shows +three key parts of HyperVision, i.e., graph construction, graph +pre-processing, and abnormal interaction detection. +Graph Construction. HyperVision collects network flows for +graph construction. Meanwhile, it classifies the flows into +short and long ones and records their interaction patterns +separately for the purpose of reducing the density of the +graph. In the graph, it uses different addresses as vertices +that connect the edges associated with short and long flows, +respectively. It aggregates the massive similar short flows +to construct one edge for a group of short flows, and thus +reduces the overhead for maintaining flow interaction patterns. +Moreover, it fits the distributions of the packet features in the +long flows to construct the edges associated with long flows, +which ensures high-fidelity recorded flow interaction patterns, +while addressing the issue of coarse-grained flow features in +the traditional methods [36]. We will detail how HyperVision +maintains the high-fidelity flow interaction patterns in the in- +memory graph in Section IV. +Graph Pre-Processing. We pre-process the built interaction +graph to reduce the overhead of processing the graph by +extracting connected components and cluster the components +using high-level statistics. In particular, the clustering can +detect the components with only benign interaction patterns +accurately and thus filters these benign components to reduce +the scale of the graph. Moreover, we perform a pre-clustering +and use the generated cluster centers to represent the edges in +3 + +0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 +Flow Completion Time [log10 Scale] +0.0 +0.5 +1.0 +1.5 +2.0 +5.0 +PDF +Most Flows Are + Short-term +All +Long +Short +(a) FCT distribution. +0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 +Flow Length [log10 Scale] +0.0 +0.5 +1.0 +1.5 +2.0 +5.0 +PDF +Most Packets Are + in Long Flows +All +Long +Short +(b) Flow length distribution. +Fig. 2. +The real-world flow features distribution of short and long flows. +(a) Traditional flows as edges. +(b) Short flow aggregation. +Fig. 3. +HyperVision aggregates short flows to reduce the dense graph. +the identified clusters. We will detail the graph pre-processing +in Section V. +Malicious Traffic Detection Based on the Graph. We +achieve unsupervised encrypted malicious traffic detection by +analyzing the graph features. We identify critical vertices in the +graph by solving a vertex cover problem, which ensures that +the clustering based graph learning processes all edges with the +minimum number of clustering. For each selected vertex, we +cluster all connected edges according to their flow features and +structural features that represent the flow interaction patterns. +HyperVision can identify abnormal edges in real time by +computing the loss function of the clustering. We will describe +the details of graph learning based detection in Section VI. +IV. +GRAPH CONSTRUCTION +In this section, we present the design details of constructing +the flow interaction graph that maintains interaction patterns +among various flows. In particular, we classify different flows, +i.e., short and long flows, and aggregate short flows, and +perform the distribution fitting for long flows, respectively, for +efficient graph construction. In Section VII, we will show that +the graph retains the near-optimal information for detection. +A. Flow Classification +In order to efficiently analyze flows captured on the In- +ternet, we need to avoid the dependency explosion among +flows during the graph construction. We classify the collected +flows into short and long flows, according to the flow size +distribution [25] (see Figure 2), and then reduce the density of +the graph (shown in Figure 3). Figure 2 shows the distribution +of flow completion time (FCT) and flow length of the MAWI +Internet traffic dataset [80] in Jan. 2020. For simplicity, we use +the first 13 × 106 packets to plot the figure. According to the +figure, we observe that only 5.52% flows have FCT > 2.0s. +However, 93.70% packets in the dataset are long flows with +only 2.36% proportion. Inspired by the observation, we apply +different flow collection strategies for the short and long flows. +We poll the per-packet information from a data-plane high- +speed packet parsing engine and obtain their source and des- +tination addresses, port numbers, and per-packet features, in- +cluding protocols, lengths, and arrival intervals. These features +0 10 20 30 40 50 60 70 80 90 100 +Number of Buckets [10 Bytes] +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +PDF +Centralized Distribution +Avg. Num: 10.64 +Bucket Num. +(a) Number of packet length buckets. +0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 +Packet Length Bucket Size [log10 Scale] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +PDF +High Utilization +Avg. Size: 333.96 +Bucket Size +(b) Maximum bucket size. +Fig. 4. +The number and size of the buckets for feature distribution fitting. +can be extracted from both encrypted and plain-text traffic for +generic detection. We develop a flow classification algorithm to +classify the traffic (see Algorithm 1 in Appendix A). It main- +tains a timer TIME NOW, a hash table that uses HASH(SRC, +DST, SRC PORT, DST PORT) as key and the collected flows +indicated by the sequences of their per-packet features as +values. It traverses the hash table every JUDGE INTERVAL sec- +ond according to TIME NOW and judges the flow completion +when the last packet arrived before PKT TIMEOUT second +of TIME NOW. When the flows are completed, we classify +them as long flows if the flows have more than FLOW LINE +packets. Otherwise, we classify them as short flows. As shown +in Figure 2(b), we can accurately classify short and long +flows. The definitions of the hyper-parameters can be found in +Table VII (see Appendix A). Note that, we poll the state-less +per-packet information from data-plane, while not maintaining +flow states (e.g., a state machine [89]) on the data-plane to +prevent attackers manipulating the states, e.g., side-channel +attack [65] and evading detection [79]. +B. Short Flow Aggregation +We need to reduce the density of the graph for analysis. +As shown in Figure 3(a), the graph will be very dense for +analysis if we use traditional four-tuple flows as edges, which +is similar to the dependency explosion problem in provenance +analysis [83], [87]. We observe that most short flows have +almost the same per-packet feature sequences. For instance, the +encrypted flows of repetitive SSH cracking attempts originated +from specific attackers [39]. Thus, we perform the short flow +aggregation to represent similar flows using one edge after the +classification. +We design an algorithm to aggregate short flows (see +Algorithm 2 in Appendix A). A set of flows can be aggregated +when all the following requirements are satisfied: (i) the flows +have the same source and/or destination addresses, which +implies similar behaviors generated from the addresses; (ii) +the flows have the same protocol type; (iii) the number of the +flows is large enough, i.e., when the number of the short flows +reaches the threshold AGG LINE, which ensures that the flows +are repetitive enough. Next, we construct an edge for the short +flows, which preserves one feature sequence (i.e., protocols, +lengths, and arrival intervals) for all the flows, and their +four-tuples. As a result, four types of edges associated with +short flows exist on the graph, i.e., source address aggregated, +destination address aggregated, both addresses aggregated, and +without aggregation. Thus, a vertex connected to the edge can +denote a group of addresses or a single address. +Figure 3 compares the graph using traditional flows as +edges and our aggregated graph by using the real-world back- +bone traffic dataset, which is same to that used in Figure 2. The +diameter of a vertex indicates the number of addresses denoted +4 + +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +6.0 +7.0 +Number of Bytes [log10 Scale] +0.0 +0.3 +0.6 +0.9 +1.2 +1.5 +PDF +Small Components +Short Flow +Long Flows +(a) Component size distribution. +-2.0-1.00.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 +PCA Decomposed Features +-3.0 +-2.0 +-1.0 +0.0 +1.0 +2.0 +3.0 +Outlier Components +(b) Scatter of the components. +Fig. 5. +The statistical features of the components. +by the vertex and the depth of the color indicates the repeated +edges. In Figure 3(b), we observe that the algorithm reduces +93.94% vertices and 94.04% edges. The edge highlighted in +green indicates short flows (i.e., 2.38 Kpps, from PH) exploit- +ing a vulnerability. Note that, the flow aggregation reduces +the storage overhead, which makes it feasible to maintain the +in-memory graph for realtime detection. +C. Feature Distribution Fitting for Long Flows +Now we use histograms to represent the per-packet feature +distributions of a long flow which avoid preserving their long +per-packet feature sequences, since the features in long flows +are centrally distributed. Specifically, we maintain a hash table +to construct the histogram for each per-packet feature sequence +in each long flow. According to our empirical study, we set +the buckets widths for packet-length and arrival interval as 10 +bytes and 1 ms, respectively, to trade off between the fitting +accuracy and overhead. We calculate the hash code by dividing +the per-packet features by the bucket width and increase the +counter indexed by the hash code. Finally, we record the hash +codes and the associated counters as the histograms. Note that, +the coarse-grained flow statistics, e.g., numbers of packets [36], +are insufficient for encrypted malicious traffic detection [76], +which also lose the flow interaction information [18]. +Figure 4 shows the number of the used buckets and +the maximum bucket size for the long flows in the same +dataset shown in Figure 2. We confirm the centralized feature +distribution, i.e., most packets in the long flows have similar +packet lengths and arrival intervals. Specifically, in Figure 4(a), +we fit the distribution of packet length using only 11 buckets on +average, and most of the buckets collect more than 200 packets +(see Figure 4(b)), which demonstrate that the histogram based +fitting is effective with low storage overhead. Similarly, the +fitting for arrival interval uses 121 buckets on average and +realizes 71 packets per bucket high utilization. Besides, we use +the same method for protocol. We use the mask of protocols as +the hash code and use smaller numbers of buckets to realize +more efficient fitting due to the limited number of protocol +types. Note that, Flowlens [5] used a similar histogram to +efficiently utilize hardware flow tables on P4 switches. Instead, +we construct the histograms to accurately analyze long flows. +V. +GRAPH PRE-PROCESSING +In this section, we pre-process the flow interaction graph +to identify key components and pre-cluster the edges, which +can enable realtime graph learning based detection against +encrypted malicious traffic with unknown patterns. +A. Connectivity Analysis +To perform the connectivity analysis of the graph, we +obtain the connected components by using depth-first search +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +PCA Decomposed Long Flow Features +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +Adjacent Edges +Edge Features +(a) Adjacent long flows. +0.5 0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +PCA Decomposed Short Flow Features +0.0 +1.0 +2.0 +3.0 +(b) Adjacent short flows. +Fig. 6. +The sparsity of edges in the graph feature space. +Selected + + + +Flows in a +component + + + +Calculate the +subset of vertices +Cluster the edges +for selected vertices +Degree = 6 +Degree = 3 +Degree = 5 +Identify the edges +denoting attacks +Benign +Benign +Malicious + + + +Selected +Selected +Fig. 7. +Critical vertices identification via solving the vertex cover problem. +(DFS) and split the graph by the components. Figure 5(a) +presents the size distribution of the identified components of +the MAWI traffic dataset [80] collected in Jan. 2020. We +observe that most components contain few edges with similar +interaction patterns. Thus, we perform a clustering on the high- +level statistics for the connected components to capture the +abnormal components that have over one order of magnitude +clustering loss than normal components as clustering outliers. +Specifically, we extract five features to profile the components, +including: (i) the number of long flows; (ii) the number of +short flows; (iii) the number of edges denoting short flows; +(iv) the number of bytes in long flows; and (v) the number of +bytes in short flows. We perform a min-max normalization +and acquire the centers using the density based clustering, +i.e., DBSCAN [32]. For each component, we calculate the +Euclidean distance to its nearest center. We detect an abnormal +component when its distance is over the 99th percentile of all +the distances based on our empirical study. +Figure 5(b) shows an instance of the clustering, where the +diameters indicate the scale of the traffic on the components (in +the unit of bytes). We observe that most components are small, +and a high ratio of huge components is classified as abnormal. +All edges associated with the normal components are labeled +as benign traffic, and the edges associated with the abnormal +components will be further processed by the following steps. +B. Edge Pre-Clustering +Now we further need to process and pre-cluster the graph +for efficient detection. As shown in Figure 5, the abnormal +components in the graph have massive vertices and edges. In +particular, we cannot directly apply graph representation learn- +ing, e.g., graph neural network (GNN), for realtime detection. +Figure 6 shows the edges from the components in the graph +structural feature space. We observe that the distribution of +the edges is sparse, i.e., most edges are adjacent to massive +similar edges in the feature space. To utilize the sparsity, we +perform a pre-clustering using DBSCAN [32] that leverages +KD-Tree for efficient local search and select the cluster centers +of the identified clusters to represent all edges in each cluster +to reduce the overhead for graph processing. +Specifically, we extract eight and four graph structural +features (see Table V in Appendix A) for the edges associated +with short and long flow, respectively, e.g., the in-degree of +5 + +the source vertex of an edge associated with a long flow. +These degree features of malicious traffic are significantly +distinct from the benign ones, e.g., the vertices denoting +spam bots have higher out-degrees than benign clients due +to their frequent interactions with servers. Then, we perform +a min-max normalization for the features, and adopt a small +search range ϵ and a large minimum number of points for +DBSCAN clustering (see Section VIII-A for the setting of +hyper-parameters) to avoid including irrelevant edges in the +clusters, which may incur false positives. Moreover, some +edges cannot be clustered and should be treated as outliers, +which will be processed as clusters with only one edge. +VI. +MALICIOUS TRAFFIC DETECTION +In this section, we detect encrypted malicious traffic by +identifying abnormal interaction patterns on the graph. In +particular, we cluster edges connected to the same critical +vertex and detects outliers as malicious traffic (see Figure 7). +A. Identifying Critical Vertices +To efficiently learn the interaction patterns of the traffic, +we do not perform clustering for all edges directly but clus- +ter edges connected to critical vertices. For each connected +component, we select a subset of all vertices in the connected +component as the critical vertices according to the following +conditions: (i) the source and/or destination vertices of each +edge in the component are in the subset, which ensures that +all the edges are connected to more than one critical vertices +and clustered at least once; and (ii) the number of selected +vertices in the subset is minimized, which aims to minimize the +number of clustering to reduce the overhead of graph learning. +Finding such a subset of vertices is an optimization problem +and equivalent to the vertex cover problem [33], which was +proved to be NP Complete (NPC). We select all edges and +all vertices on each component to solve the problem. And we +reformulate the problem to a Satisfiability Modulo Theories +(SMT) problem that can be effectively solved by using Z3 +SMT solver [55]. Since we pre-cluster the massive edges and +reduce the scale of the problem (see Section V-B), the NPC +problem can be solved in real time. +B. Edge Feature Clustering for Detection +Now we cluster the edges connected to each critical vertex +to identify abnormal interaction patterns. In this step, we use +the structural features in Section V-B, and the flow features +extracted from the per-packet feature sequences of short flows +or the fitted feature distributions of long flows. All features are +shown in Table V (see Appendix A). We use the lightweight +K-Means algorithm to cluster the edges associated with short +and long flows, respectively, and calculate the clustering loss +that indicates the degree of maliciousness for malicious flow +detection. +losscenter(edge) = +min +Ci∈{C1,...,CK} ||Ci − f(edge)||2, +(1) +losscluster(edge) = TimeRange(C(edge)), +(2) +losscount(edge) = log2(Size(C(edge)) + 1), +(3) +loss(edge) =αlosscenter(edge) +−βlosscluster(edge) + γlosscount(edge), +(4) +where K is the number of obtained cluster centers, Ci is the +ith center, f(edge) is the feature vector, C(edge) contains all +edges in the cluster of edge produced by pre-clustering, and +TimeRange calculates the time range covered by the flows +denoted by the edges. +According to Equation (4), the loss has three parts: (i) +losscenter in (1) is the Euclidean distance to the cluster centers +which indicates the difference from other edges connected to +the critical vertex; (ii) losscluster in (2) indicates the time range +covered by the cluster identified by the pre-clustering in Sec- +tion V-B which implies long lasting interaction patterns tend to +be benign; (iii) losscount in (3) is the number of flows denoted +by the edges, which means a burst of massive flows implies +malicious behaviors. Moreover, we used weights: α, β, γ to +balance the loss terms. Finally, it detects the associated flows +as malicious when the loss function of the edge is larger than +a threshold. +VII. +THEORETICAL ANALYSIS +In this section, we develop a theoretical analysis frame- +work, i.e., flow recording entropy model, to analyze the in- +formation preserved in the graph of HyperVision for graph +learning based detection. The detailed analysis can be found +in Appendix C. +A. Information Entropy Based Analysis +We develop the framework that aims to quantitatively eval- +uate the information retained by the exiting traffic recording +modes, which decide the data representations for malicious +traffic detection, by using three metrics: (i) the amount of +information, i.e., the average Shannon entropy obtained by +recording one packet; (ii) the scale of data, i.e., the space +used to store the information; (iii) the density of information, +i.e., the amount of information on a unit of storage. By using +this framework, we model the graph based traffic recording +mode used by HyperVision as well as three typical types of +flow recording modes, i.e., (i) idealized mode that records and +stores the whole per-packet feature sequence; (ii) event based +mode (e.g., Zeek) that records specific events [2], [20]; and +(iii) sampling based mode (e.g., NetFlow) that records coarse- +grained flow information [8], [51]. +We model a flow, i.e., a sequence of per-packet features, +as a sequence of random variables represented by an ape- +riodic irreducible discrete-time Markov chain (DTMC). Let +G = {V, E} denote the state diagram of the DTMC, where +V is the set of states (i.e., the values of the variables) and +E denotes the edges. We define s = |V| as the number of +different states and use W = [wij]s×s to denote the weight +matrix of G. All of the weights are equal and normalized: +∀ 1 ≤ i, j, m, n ≤ s, (wij =wmn) ∨ (wij = 0 ∨ wmn = 0), +wi = +s +� +j=1 +wij, +1 = +s +� +i=1 +wi. +(5) +The state transition is performed based on the weights, i.e., +the transition probability matrix P = [Pij], Pij = wij/wi. +Therefore, the DTMC has a stationary distribution µ: +6 + +� +µP = µ, +1 = �s +j=1 µj +⇒ +µj = wj, +∀ 1 ≤ j ≤ s. +(6) +Assume that the stationary distribution is a binomial distri- +bution with the parameter: 0.1 ≤ p ≤ 0.9 to approach Gaussian +distribution with low skewness: +µ ∼ B(s, p) +App. +−→ N(sp, sp(1 − p)). +(7) +Based on the distribution, we obtain the entropy rate of +the DTMC which is the expected Shannon entropy increase +for each step in the state transition, i.e., the expected Shannon +entropy of each random variable in the sequence, (using nat +as unit, 1 nat ≈ 1.44 bit): +H[G] = +s +� +i=1 +µi +s +� +j=1 +pij ln 1 +pij = − +s +� +i=1 +s +� +j=1 +wij ln wij + +s +� +j=1 +wj ln wj += ln |E| − 1 +2 ln 2πsep(1 − p). +(8) +Moreover, for the real-world flow size distribution, we as- +sume that the length of the sequence of random variables obeys +a geometric distribution with high skewness, i.e., L ∼ G(q) +with a parameter: 0.5 ≤ q ≤ 0.9. H, L, and D denote the +expectation of the metrics, i.e., the amount of information, the +scale of data, and the density, respectively. +Idealized Recording Mode. The idealized recording mode has +infinite storage and captures optimal fidelity traffic information +by recording each random variable from the sequence without +any processing. Thus, the obtained information entropy of the +idealized mode grows at the entropy rate of the DTMC: +HIdeal = E[LH[G]] = 1 +q ln |E| − 1 +2q ln 2πsep(1 − p). +(9) +According to data processing inequality [85], the infor- +mation retained in the idealized recording mode reaches the +optimal value. It implies that processing of the observed per- +packet features denoted by the random variables may incur +information loss. In the following sections, we will show that +the other mode incurs information loss. +We can obtain the scale of data and the density of infor- +mation for the idealized recording mode as follows: +LIdeal = E[L] = 1 +q . +(10) +DIdeal = HIdeal +LIdeal = H[G]. +(11) +Graph Based Recording Mode of HyperVision. HyperVision +applies different strategies to process short and long flows for +the graph construction. Let K denote the threshold for classi- +fying the flows. When L < K, it collects all random variables +from the sequence for short flows. Otherwise, it collects the +histogram to fit the distribution for long flows. Then, we can +obtain the lower bound to estimate the information entropy in +the graph of HyperVision: +HH.V. = 1 − (Kq + 1)(1 − q)K +q +H[G] + 1 +4s(1 − q)K +[(1 + s) lnps + 2 ln 2πe + 2q ln K − 2s(1 + p + γ)]. +(12) +We can also obtain the expected data scale and the density: +LH.V. = s(1 − q)K + 1 − (Kq + 1)(1 − q)K +Cq +, +(13) +where C is the average number of flows denoted by an edge +associated with short flows. +DH.V. = HH.V. +LH.V. . +(14) +Sampling Based Recording Mode. Similarly, the sampling +based mode extracts and records flow statistics for the de- +tection. We analyze the accumulative statistics (e.g. the total +number of bytes) that are widely adopted [19], [36]. Let +⟨s1, s2, ..., sL⟩ denote the sequence of random variables, and +XSamp. = �L +i=1 si indicates the flow statistic to be recorded. +We can obtain a tight lower bound as an estimation for the +amount of information and the other metrics as follows: +HSamp. = H[XSamp.] = 1 +2 ln 2πesp(1 − p) + ln 2 +2 q(1 − q). (15) +LSamp. = 1. +(16) +DSamp. = HSamp. +(17) +Event Based Recording Mode. The event based recording +mode inspects each random variable in the sequence and +records events with a small probability. Since the observation +that the event based methods do not generate repetitive events +for a long flow with a larger s, for simplicity, we assume that +the probability is ps ∝ 1/s. Then, we can obtain the concise +closed-form solution of the amount of information, the scale of +data, and the density of information for event based recording +mode as follows: +HEve. = −2θ ln θ, +(18) +where θ = ζ +η, ζ = q − qps, and η = q − ps(q − 1). +LEve. = −ps +η . +(19) +DEve. = 2ζ +ps ln θ. +(20) +B. Analysis Results +We perform numerical studies to compare the flow record- +ing modes in real-world setting. We select three per-packet +features: protocol, length, and the arrival interval (in ms) as +the instances of the DTMC, then we measure the parameters +of the DTMC, i.e., |E| and |V| according to the first 106 packets +in the MAWI dataset on Jan. 2020 [80]. We also measure K, +C, and estimate the geometric distribution parameter q via the +second moment. We have the following three key results. +7 + +Length Param. q +0.5 +0.6 +0.7 +0.8 +0.9 +DTMC Param. p +0.1 +0.3 +0.5 +0.7 +0.9 +Entropy [nat] +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +6.0 +7.0 +Ideal +H.V. +Samp. +Eve. +(a) The entropy of the modes. +Length Param. q +0.5 +0.6 +0.7 +0.8 +0.9 +DTMC Param. p +0.1 +0.3 +0.5 +0.7 +0.9 +Data Scale [Num.] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Ideal +H.V. +Samp. +Eve. +(b) The data scale of the modes. +Length Param. q +0.5 +0.6 +0.7 +0.8 +0.9 +DTMC Param. p +0.1 +0.3 +0.5 +0.7 +0.9 +Density [nat / record] +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +6.0 +7.0 +Ideal +H.V. +Samp. +Eve. +(c) The density of the modes. +Length Param. q +0.5 +0.6 +0.7 +0.8 +0.9 +DTMC Param. p +0.1 +0.3 +0.5 +0.7 +0.9 +Density Increase [H.V. - Ideal] +0.0 +0.5 +1.0 +1.5 +2.0 +(d) The density improvement. +Fig. 8. +The traffic information retained by different recording modes on the feasible region of the parameters. +0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 +DTMC Param. p +5.0 +5.2 +5.4 +5.6 +5.8 +6.0 +Entropy [nat] +HyperVision +Ideal Mode +(a) Fix q and leave p as variable. +0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 +Flow Length Param. q +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +Entropy [nat] +HyperVision +Ideal Mode +(b) Fix p and leave q as variable. +Fig. 9. +HyperVision approaches the idealized flow recording mode on +information entropy. +TABLE II. +THE INTEGRAL OF THE DENSITY IN THE FEASIBLE REGION. +Per-Packet Features +Packet Length Time Interval +Protocol Type +�� +F DIdeal(p, q)dpdq +1.011▼32.10% +0.918▼32.00% 0.795▼32.51% +�� +F DSamp.(p, q)dpdq 0.965▼35.17% +0.963▼28.66% 0.800▼32.08% +�� +F DEve.(p, q)dpdq +0.588▼60.51% +0.588▼56.44% 0.588▼50.08% +�� +F DH.V.(p, q)dpdq +1.489▲47.27% +1.350▲35.51% 1.178▲48.18% +(1) HyperVision maintains more information using the graph +than the existing methods. Figure 8 shows the results on the +feasible region (F = {0.1 ≤ p ≤ 0.9, 0.5 ≤ q ≤ 0.9}). +We observe that HyperVision maintains at least 2.37 and 1.34 +times information entropy than traditional flow sampling and +event based flow recording. Thus, the traditional detection +methods cannot retain high-fidelity flow interaction informa- +tion. Actually, they only analyze the features of a single +flow, which can be evaded by encrypted traffic. According +to Figure 8(b), HyperVision has 69.69% data scale of the +sampling based mode. It implies that the data scale is the key +challenge for the existing methods to utilize flow interaction +patterns. We well address this issue by using the compact graph +for maintaining the interactions among flows. +(2) HyperVision maintains near-optimal information using the +graph. According to Figure 8(a), we observe that the informa- +tion maintained by the graph almost equals to the theoretical +optimum, with the difference ranging from 4.6 × 10−9 to 2.6 +nat. When the parameter of the geometric distribution of L +approaches 0.9, the flow information loss is larger because of +the increasing ratio of long flows that incur more information +loss. Figure 9 compares the information in HyperVision and +the idealized system when q = 0.59 and p = 0.8. We have +similar results. The gaps between the graph mode and the +optimal mode are only 0.056 and 0.021. +(3) HyperVision has higher information density than the ex- +isting methods. Figure 8(c) shows that HyperVision realizes +1.46, 1.54, and 2.39 times information density than the existing +methods, respectively. Although the idealized system realizes +the optimal amount of traffic information, the density is +only 78.55% of HyperVision in the worst case, as shown in +Figure 8(d). From Table II, we find that, for all kinds of per- +packet features, HyperVision can increase the density ranging +between 35.51% and 47.27% due to the different recording +strategies for short and long flows. +In summary, the flow interaction graph provides high- +fidelity and low-redundancy traffic information with obvious +flow interaction patterns, which ensures that HyperVision +achieves realtime and unsupervised detection, particularly, +detecting encrypted malicious traffic with unknown patterns. +VIII. +EXPERIMENTAL EVALUATION +A. Experiment Setup +Implementation. We prototype HyperVision with more than +8,000 Line of Code (LOC). The prototype is compiled by gcc +9.3.0 and cmake 3.16.3. We use DPDK [37] version 19.11.9 +encapsulated by libpcap++ [63] version 21.05 to implement +the high-speed data-plane module. The graph construction +module maintains the graph in memory for realtime detection. +The graph learning module detects the encrypted malicious +traffic on the interaction graph. It uses DBSCAN and K-Means +in mlpack [57] (version 3.4.2) for clustering and Z3 SMT +Solver [55] (version 4.8) to identify the critical vertices. +Testbed. We deploy HyperVision on a testbed built upon +DELL servers (PowerEdge R410, produced in 2012) with two +Intel Xeon E5645 CPUs (2 × 12 cores), Ubuntu 20.04.2 (Linux +5.11.0), Docker 20.10.7, 24GB memory, one Intel 82599ES 10 +Gb/s NIC, and two Intel 850nm SFP+ laser ports for optical +fiber connections. We configure 6GB huge page memory for +DPDK (3GB/NUMA Node) and bind 8 threads on 8 physical +cores for 16 NIC RX queues to parse the per-packet features +from high-speed traffic. We use 8 cores for in-memory graph +construction, and 7 cores are used for graph learning, the rest +one core is used as DPDK master core. +Datasets. We use real-world backbone network traffic datasets +from the vantage-G of WIDE MAWI project [80] in AS2500, +Tokyo Japan, Jan. ∼ Jun. 2020 as background traffic. The +vantage transits traffic from/to its BGP peers and providers +using 10 Gb/s fiber linked to its IXP (DIX-IE), and the traffic +is collected using port mirroring, which is consistent with +our threat model and the physical testbed described above. +We remove the attack traffic with obvious patterns in the +background traffic dataset according to the rules defined by the +existing studies [22], [43], [66], e.g., traffic will be detected as +scanning traffic if it has scanned over 10% IPv4 addresses [22]. +We generate the malicious traffic by constructing real attacks +or replaying the existing traces in our testbed. Specifically, we +collect malicious traffic in our virtual private cloud (VPC) with +8 + +more than 1,500 instances. We manipulate the instances to per- +form attacks according to the real-world measurements [22], +[24], [40], [42], [43], [54], [66] and the same settings in the +existing studies [11], [26], [41], [44]. We classify 80 new +datasets used in our experiments (see Table VI for details) into +four groups, three of which are encrypted malicious traffic: +• Traditional brute force attack. Although HyperVision fo- +cuses on encrypted traffic, we generate 28 kinds of tradi- +tional flooding attacks to verify its generic detection and +the correctness of baselines including 18 high-rate and 10 +low-rate attacks: (i) the brute scanning with the real packet +rates [22]; (ii) the source spoofing DDoS with various +rates [40]; (iii) the amplification attacks [43]; (iv) probing +vulnerable applications [21], [22]. We collected the traffic +in our VPC to avoid interference with real services. +• Encrypted flooding traffic. Different from the brute force +flooding, the encrypted flooding is generated by repetitive +attack behaviors which target specific applications: (i) the +link flooding generates encrypted low-rate flows, e.g., the +low-rate TCP attacks [44], [52] and the Crossfire attack [41], +to congest links; (ii) injecting encrypted flows that exploits +protocol vulnerabilities by flooding attack traffic and inject +packets into the channel [11], [26], [28]; (iii) the password +cracking performs slow attempts to hijack the encrypted +communication protocols [39], [50]. We perform SSH crack- +ing in the VPC with the scale of SSH servers in the ASes +reachable to AS2500. +• Encrypted web malicious traffic. Web malicious traffic is +normally encrypted by HTTPS. We collect the traffic gener- +ated by seven widely used web attacks including automatic +vulnerabilities discovery (including XSS, CSRF, various +injections) [64], SSL vulnerabilities detection [53], and +crawlers. We also collect the SMTP-over-TLS spam traffic +that lures victims to visit the phishing sites [61]. +• Malware generated encrypted traffic. The traffic of malware +campaigns is low-rate and encrypted, e.g., malware compo- +nent update or delivery [9], command and control (C&C) +channel [8], and data exfiltration [77]. We use the malware +infection statistics published in 2020 [42] and probed active +addresses from the adopted vantage [23], [59] to estimate +the number of visible victims. We use the same number +of instances to replay public malware traffic datasets [13], +[73] to mimic malware campaigns, which is similar to the +existing study [58]. +The malicious traffic is replayed with the background traffic +datasets on the physical testbed simultaneously according to +their original packet rates [80] which is the same as the existing +studies [30], [47], [51]. Specifically, each dataset contains +12∼15 million packets and the replay lasts 45s and the first +75% time does not contain malicious traffic for collecting flow +interactions and training the baselines. Note that, the rates of +the encrypted attack flows in our datasets are only 0.01 ∼ +8.79 Kpps which consume only 0.01% ∼ 0.72% bandwidth. +We will show that these stealthy attacks evade most baselines. +To eliminate the impact of the dataset bias, we also use 12 +existing datasets including the Kitsune datasets [56], the CIC- +DDoS2019 datasets [14], and the CIC-IDS2017 datasets [15], +which are collected in the real-world. These detailed results +can be found in Appendix B2. In particular, the traffic in +two CIC datasets [14], [15] lasts 6∼8 hours under multiple +TABLE III. +THE AVERAGE ACCURACY ON THE GROUPS OF DATASETS. +Method +Metric +Traditional +Attacks +Flooding +Enc. Traffic +Enc. Web +Attacks +Malware +Traffic +Overall +Jaqen +AUC +0.913▼7% +0.782▼19% +N/A1 +N/A +0.867▼12% +F1 +0.819▼16% 0.495▼46% +N/A +N/A +0.705▼26% +FlowLens +AUC +0.939▼4% +0.757▼22% 0.685▼30% 0.768▼22% 0.752▼36% +F1 +0.799▼18% 0.651▼29% 0.384▼59% 0.411▼57% 0.451▼41% +Whisper +AUC +0.951▼3% +0.932▼4% +0.958▼2% +0.648▼34% 0.752▼23% +F1 +0.705▼27% 0.461▼50% 0.546▼42% 0.357▼62% 0.407▼57% +Kitsune +AUC +0.748▼24% +- 2 +0.759▼22% +- +0.751▼23% +F1 +0.419▼57% +- +0.366▼61% +- +0.402▼58% +DeepLog +AUC +0.716▼27% 0.621▼26% 0.767▼22% 0.653▼34% 0.666▼32% +F1 +0.513▼47% 0.508▼45% 0.572▼40% 0.628▼34% 0.597▼37% +H.V. +AUC +0.988▲8% +0.974▲4% +0.985▲2% +0.993▲29% 0.988▲13% +F1 +0.978▲19% 0.927▲42% 0.957▲67% 0.970▲54% 0.960▲36% +1 The results are N/A because Jaqen is designed for detection of volumetric attacks. +2 - means that the average AUC is lower than 0.60, which is nearly the result of +random guessing. +attacks, which aims to verify the long-run performances of +HyperVision (see Appendix B3). Moreover, we validate the +robustness of HyperVision against evasion attacks with obfus- +cation techniques, which can be found in Appendix B4. +Baselines. We use five state-of-the-art generic malicious traffic +detection methods as baselines: +• Jaqen (sampling based recording and signature based de- +tection). Jaqen [51] uses Sketches to obtain flow statistics +and applies the threshold based detection. We prototype +Jaqen on the testbed, and adjust the signatures for each +statistic and each attack to obtain the best accuracy. +• FlowLens (sampling based recording and ML based de- +tection). FlowLens [5] uses sampled flow distribution and +supervised learning, i.e., random forest. We use the hyper- +parameter setting with the best accuracy used in the paper +to retrain the ML model. +• Whisper (flow-level features and ML based detection). +Whisper [30], [31] extracts the frequency domain features +of flows and uses clustering to learn the features. We deploy +Whisper on the physical testbed without modifications and +then retrain the clustering model. +• Kitsune (packet-level features and DL based detection). +Kitsune extracts per-packet features and uses autoencoders +to learn the features which is an unsupervised method [56]. +We use its default hyper-parameters and retrain the model. +• DeepLog (event based recording and DL based detection). +DeepLog is a general log analyzer using LSTN RNN [20]. +We use the logs of connections for detection and its original +hyper-parameter setting to achieve the best accuracy. +Note that, in the baselines above, we do not include DPI- +based encrypted malicious traffic detection because they are +unable to investigate encrypted payloads [34]. Also, we do not +compare the task-specific detection methods [3], [76] because +they cannot achieve acceptable detection accuracy. Features in +FlowLens, Kitsune, and Whisper are similar to them, e.g., flow +features [3], packet header features [2], and time-series [76]. +Metrics. We mainly use AUC and F1 score because they +are most widely used in the literature [8], [20], [30], [35], +[56], [75], [91]. Also, we use other six metrics to validate the +improvements of HyperVision, including precision, recall, F2, +ACC, FPR, and EER. +9 + +TABLE IV. +DETECTION ACCURACY OF HYPERVISION AND THE BASELINES ON TRADITIONAL BRUTE FORCE ATTACKS. +Method +Metric +Brute Scanning +Amplification Attack +Source Spoofing DDoS +ICMP +NTP +SSH +SQL +DNS +HTTP HTTPS +NTP +DNS CharG. SSDP RIPv1 Mem. CLDAP +SYN +RST +UDP +ICMP +Jaqen +AUC 0.9478 0.9989 0.9706 0.9851 0.9989 0.9774 0.9988 0.9822 0.9590 0.9860 0.9907 0.9011 0.9586 0.9537 0.9976 0.9985 0.9682 0.9995 +F1 +0.9710 0.9356 0.9835 0.9924 0.9965 0.9884 0.9299 0.9457 0.8816 0.7986 0.7054 0.6549 0.8500 0.7931 0.9614 0.9236 0.5603 0.9861 +FlowLens AUC 0.9906 0.9021 0.9961 0.9993 0.9985 0.9874 0.9226 0.9784 0.8001 0.9998 0.9907 0.9833 0.9786 0.9993 0.9912 0.9918 0.9999 0.6351 +F1 +0.9181 0.6528 0.8899 0.9996 0.9992 0.9936 0.9572 0.9794 0.7127 0.9991 0.8918 0.9889 0.9691 0.9986 0.8638 0.8173 0.9990 0.2632 +Whisper +AUC 0.9499 0.9796 0.9562 0.9811 0.9832 0.9658 0.9827 0.9125 0.9645 0.8489 0.9662 0.9761 0.8954 0.9402 0.9563 0.9658 0.8956 0.9489 +F1 +0.7004 0.7585 0.8869 0.7022 0.6748 0.7182 0.7489 0.8248 0.8435 0.4686 0.6195 0.6396 0.6956 0.8620 0.7587 0.8778 0.4857 0.4192 +Kitsune +AUC 0.4522 0.7252 +- 2 +0.7439 0.7228 0.7380 0.9614 0.7340 0.9994 0.9998 0.9989 0.4343 0.3993 0.7592 0.6210 0.4086 0.8534 0.7913 +F1 +- 1 +0.3459 +- +0.5033 0.4923 0.4798 0.4878 0.4461 0.5031 0.4609 0.4360 +- +- +0.3838 0.3361 +- +0.4539 0.4153 +DeepLog +AUC 0.6717 0.8232 0.8377 0.6518 0.8261 0.6617 0.5545 0.7475 0.7428 0.7462 0.7458 0.7487 0.7480 0.7483 0.7564 0.2470 0.7012 0.7521 +F1 +0.3566 0.4178 0.5266 0.2695 0.4050 0.2668 0.3653 0.5108 0.7201 0.5705 0.4313 0.3368 0.3321 0.3424 0.6074 +- +0.4370 0.3428 +H.V. +AUC 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9989 0.9998 0.9969 0.9999 0.9999 0.9999 0.9996 0.9928 +F1 +0.9939 0.9928 0.9960 0.9932 0.9831 0.9808 0.9892 0.9998 0.9998 0.9992 0.9956 0.9984 0.9983 0.9996 0.9993 0.9571 0.9981 0.9295 +1 We highlight the best accuracy in • and the worst accuracy in •. We mark - for the F1 when the AUC is lower than 0.50, which is the accuracy of random guessing. +2 Kitsune did not finish the detection within 90 min (i.e., meaningless for defenses). And H.V. is short for HyperVision. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True Positive Rate +Jaqen +FlowLens +Whisper +Kitsune +DeepLog +H.V. +(a) ROC of detecting NTP DDoS. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +False Positive Rate +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True Positive Rate +Jaqen +FlowLens +Whisper +Kitsune +DeepLog +H.V. +(b) ROC of detecting HTTP scan. +0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 +Precision +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Recall +Jaqen +FlowLens +Whisper +Kitsune +DeepLog +H.V. +(c) PRC of detecting NTP DDoS. +0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 +Precision +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Recall +Jaqen +FlowLens +Whisper +Kitsune +DeepLog +H.V. +(d) PRC of detecting SYN DDoS. +Fig. 10. +ROC and PRC of HyperVision and all the baselines. +Hyper-parameter Selection. We conduct four-fold cross val- +idation to avoid overfitting and hyper-parameter bias. Specif- +ically, the datasets are equally partitioned into four subsets. +Each subset is used once as a validation set to tune the +hyper-parameters via the empirical study and the remaining +three subsets are used as testing sets. Finally, four results are +averaged to produce final results. Moreover, our ablation study +shows that the different threshold settings incur at most 5.2% +accuracy loss. Therefore, the hyper-parameter selection has +limited impacts on the detection results. +B. Accuracy Evaluation +Table III summarizes the detection accuracy and the im- +provements of HyperVision over the existing methods. In gen- +eral, HyperVision achieves average F1 ranging between 0.927 +and 0.978 and average AUC ranging between 0.974 and 0.993 +on the 80 datasets, which are 35% and 13% improvements over +the best accuracy of the baselines. In 44 datasets, none of the +baselines achieves F1 higher than 0.80, which means that they +are not effective to detect the attacks. Due to the page limits, +we do not show the failed detection results of these baselines. +Traditional Brute Force Attacks. First, we measure the +performance of the baselines by using the flooding attacks with +short flows. Although HyperVision is designed for encrypted +malicious traffic detection, we find that it can also detect tra- +ditional attacks accurately. The results are shown in Table IV. +HyperVision has 0.992 ∼ 0.999 AUC and 0.929 ∼ 0.999 F1, +which achieves at most 13.4% and 1.3% improvement of F1 +and AUC over the best performance of the baselines. The ROC +and PRC results are illustrated in Figure 10. According to +Figure 10(a) and 10(b), we observe that HyperVision has less +false positives while achieving similar accuracy. Figure 10(c) +and Figure 10(d) show that the PRC of HyperVision is largely +better than the baselines, which means that it has a higher +precision when all methods reach the same recall. +Second, by comparing HyperVision with Jaqen, we can see +that HyperVision can realize higher accuracy (i.e., a 19.4% F1 +improvement) than Jaqen with the best threshold set manually. +That is, the unsupervised method allows reducing manual +design efforts. Moreover, it has 56.3% AUC improvement +over the typical supervised ML based method (FlowLens). +Note that, we assume that HyperVision cannot acquire labeled +datasets for training, which is more realistic. Also, it outper- +forms Whisper with 11.6% AUC, which is an unsupervised +detection in high-speed network. We observe that Kitsune and +DeepLog have lower accuracy because they cannot afford high- +speed backbone traffic. +Third, we measure the detection accuracy of probing +vulnerable applications. As shown in Figure 11, we see that +HyperVision can detect the low-rate attacks with 0.920 ∼ +0.994 F1 and 0.916 ∼ 0.999 AUC under 6 ∼ 268 attackers +with 17.6 ∼ 97.9 Kpps total bandwidth. It also achieves +at most 46.8% F1 and 27.3% AUC improvements over the +baselines that have a more significant accuracy decrease than +the high-rate attacks. For example, FlowLens only achieves +averagely 0.684 F1, which is only 77% under the high-rate +attacks. Although Jaqen can be deployed on programmable +switches, its thresholds are invalided by the low-rate attacks. +And Whisper is unable to detect the attacks with two datasets. +Moreover, Kitsune and DeepLog cannot detect the attacks +because of the low rate of malicious packets (≤ 1.2%). +The reason why HyperVision can detect the slow probing +while maintaining the similar accuracy to the high-rate attacks +is that the graph preserves flow interaction patterns. Although +the flows from a single attacker are slow, e.g., at least 244 pps, +HyperVision can record and analyze their interaction patterns. +Specifically, each flow in the stealthy attack traffic can be +represented by an edge in the graph, while the vertices in the +graph indicate the addresses generating the traffic. Thus, the +10 + +SMTP +NetBios +Telnet +VLC +SNMP +RDP +HTTP +DNS +ICMP +SSH +Jaqen +FlowLens +Whisper +Kitsune +DeepLog +H.V. +0.9864 0.8117 0.6214 0.8829 0.9864 0.6280 +- +0.9975 0.9674 0.7123 +0.9649 0.9615 0.9693 0.9518 0.9649 0.9639 +- +0.9480 0.9688 0.9226 +0.9611 0.9553 0.9672 0.9540 0.9611 0.9594 +- +0.9530 +- +0.9549 +0.9484 0.9363 0.9410 0.9309 0.9484 +- +0.8526 0.8539 0.8959 +- +0.6501 0.6504 0.6496 0.8515 0.6501 0.6496 0.6751 0.8486 0.6503 0.8248 +0.9707 0.9587 0.9439 0.9902 0.9999 0.9751 0.9161 0.9706 0.9882 0.9868 +(a) AUC of detecting probing vulnerable application. +Jaqen +FlowLens +Whisper +Kitsune +DeepLog +H.V. +0.8354 0.5484 0.4740 0.5333 0.8354 0.4565 +- +0.9748 0.9616 0.4997 +0.6211 0.7040 0.8569 0.6416 0.6211 0.8439 +- +0.6561 0.8861 0.9572 +0.7048 0.8013 0.8583 0.7633 0.7048 0.8528 +- +0.8030 +- +0.7525 +0.3232 0.3724 0.4601 0.3710 0.3232 +- +0.5236 0.2406 0.4909 +- +0.3569 0.6211 0.7046 0.8333 0.3569 0.7068 0.7128 0.8530 0.7176 0.6645 +0.9207 0.9469 0.9664 0.9790 0.9944 0.9791 0.9471 0.9332 0.9869 0.9323 +(b) F1 of detecting probing vulnerable application. +Fig. 11. +Heatmap of accuracy for probing vulnerabilities. +traffic can be captured by identifying vertices with large out- +degrees (i.e., a large number of edges). Moreover, the brute +force attacks validate that our method is effective to capture +the DDoS traffic because it utilizes the short flow aggregation +to construct the edge associated with short flows and avoids +inspecting each short spoofing flow. Besides, the experiment +results also show that the critical vertices denote the addresses +of major active flows, e.g., web servers, DNS servers, and +scanners. Note that, we exclude the results of the baselines that +cannot detect encrypted traffic with lower rates in the following +sections due to the page limits. +Encrypted Flooding Traffic. Figure 12 shows the detection +accuracy under flooding attacks using encrypted traffic. Gen- +erally, HyperVision achieves 0.856 ∼ 0.981 F1 and 0.917 +∼ 0.998 AUC, which are 58.7% and 25.3% accuracy im- +provements over the baselines that can detect such attacks. +Specifically, as shown in Figure 12(a) and 12(b), we observe +that HyperVision can accurately detect the link flooding traffic +consists of various encrypted traffic with different parameters. +For instance, it can detect the Crossfire attack using HTTPS +web requests generated by different sizes of botnets [41] +with at most 0.939 F1. The massive web traffic generated by +bots, which is low-rate (≤ 4Kbps) and encrypted, evades the +detection of Whisper and FlowLens (F1 ≤ 0.8). As shown in +Figure 14(a), HyperVision can detect the attack efficiently by +splitting the botnet clusters into a single connected component +to exclude the interference from the similar benign web traffic, +where the inner layer denotes botnets and the outer denotes +decoy servers. +Moreover, we find that HyperVision can detect low-rate +TCP DoS attacks that use burst encrypted video traffic for +at most 0.995 AUC and 0.938 F1. Although Whisper has +slightly better AUC in some cases, we find that it cannot +achieve high accuracy on all scenarios. As a result, it has +only 55.5% AUC in the worse case. Moreover, HyperVision +can aggregate the short flows in the SSH connection injection +attacks and achieves more than 0.95 F1. The attacks exploiting +protocol vulnerabilities realize low-rate packet injection and +evade the detection of FlowLens (i.e., AUC ≤ 0.774, F1 ≤ +0.513). Figure 12(c) and 12(d) illustrate that HyperVision can +identify slow and persisted password attempts for the channels +Size 100 +Size 200 +Size 500 +0.2s Burst +0.5s Burst +1.0s Burst +ACK Inj. +IPID Inj. +IPID Port +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +AUC +Crossfire Attack +Low-rate TCP DoS +SSH. Conn. Injection +Flowlens +Whisper +H.V. +(a) AUC of detecting encrypted link-flooding and encrypted channel injection. +Size 100 +Size 200 +Size 500 +0.2s Burst +0.5s Burst +1.0s Burst +ACK Inj. +IPID Inj. +IPID Port +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +F1 +Crossfire Attack +Low-rate TCP DoS +SSH Conn. Injection +(b) F1 of detecting encrypted link-flooding and encrypted channel injection. +35 v. +257 v. +486 v. +19 v. +43 v. +83 v. +Num. Victim +0.6 +0.7 +0.8 +0.9 +1.0 +AUC +SSH +Telnet +(c) F1 of password cracking. +35 v. +257 v. +486 v. +19 v. +43 v. +83 v. +Num. Victim +0.4 +0.6 +0.8 +1.0 +F1 +SSH +Telnet +(d) AUC of password cracking. +Fig. 12. +Detection accuracy of encrypted flooding traffic. +Padding +Oracle +XSS +[Xsssniper] +SSL Vul. +[SSLScan] +Param. Inj. +[Commix] +Code Inj. +[Commix] +Agent Inj. +[Commix] +CVE- +2014-6271 +CVE- +2013-2028 +CSRF +[Bolt] +Crawler +[Scrapy] +Spam +[1 Bot] +Spam +[50 Bots] +Spam +[100 Bots] +0.80 +0.85 +0.90 +0.95 +1.00 +AUC +Whisper Avg. AUC +H.V. Avg. AUC +Whisper +H.V. +(a) AUC of detecting encrypted web attack traffic. +Padding +Oracle +XSS +[Xsssniper] +SSL Vul. +[SSLScan] +Param. Inj. +[Commix] +Code Inj. +[Commix] +Agent Inj. +[Commix] +CVE- +2014-6271 +CVE- +2013-2028 +CSRF +[Bolt] +Crawler +[Scrapy] +Spam +[1 Bot] +Spam +[50 Bots] +Spam +[100 Bots] +0.50 +0.60 +0.70 +0.80 +0.90 +1.00 +F1 +Whisper Avg. F1 +H.V. Avg. F1 +Whisper +H.V. +(b) F1 of detecting encrypted web attack traffic. +Fig. 13. +Accuracy of encrypted web attack traffic detection. +(a) Crossfire. +(b) SSH cracking. +(c) XSS detection. +(d) P2P botnet. +Fig. 14. +Subgraph with various encrypted malicious traffic. +with over 0.881 F1 and 0.917 AUC, which are 1.19 and 1.28 +times improvements over FlowLens and Whisper. The reason is +that HyperVision maintains the interaction patterns of attackers +using the graph, e.g., the massive short flows for login attempts +shown as red edges in Figure 14(b). +11 + +Magic +Trickster +Plankton +Penetho +Zsone +CCleaner +Feiwo +Mobidash +Adload +WebComp +Koler +Svpeng +Ransombo +Wannalocker +Dridex +BitCoinM +TrojanM +CoinMiner +THBot +Emotet +Snojan +Trickbot +Sality +Mazarbot +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +AUC +Spyware +Adware +Ransomeware +Miner +Botware +Avg. AUC +AUC +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +F1 +Avg. F1 +F1 +Fig. 15. +HyperVision can detect various encrypted malware traffic. +10 +15 +20 +25 +30 +35 +40 +45 +50 +Throughput [Gb/s] +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +PDF +Avg: 28.2 Gb/s +Jan. 2020 +(a) Graph construction throughput. +10 +20 +30 +40 +50 +60 +70 +Maximum Throughput [Gb/s] +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +PDF +Jan. +Feb. +Apr. +Jun. +(b) Max construction throughput. +0 +50 +100 +150 +200 +250 +300 +Throughput [Gb/s] +0.00 +0.25 +0.50 +0.75 +1.00 +PDF [×10 2] +Avg: 121 Gb/s +Jan. 2020 +(c) Graph detection throughput. +0 +25 +50 +75 100 125 150 175 200 +Stable Throughput [Gb/s] +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +PDF +Jan. +Feb. +Apr. +Jun. +(d) Stable detection throughput. +Fig. 16. +Throughput of graph construction and detection. +0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 +Latency [s] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +PDF +Jan. 2020 +Jun. 2020 +(a) Graph construction latency. +Flow +Class. +Proc. +Long +Proc. +Short +Flow +Class. +Proc. +Long +Proc. +Short +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Latency [s] +Jan. 2020 +Jun. 2020 +(b) Construct latency composition. +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +6.0 +7.0 +Latency [s] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +PDF +99th Percentile +Avg: 0.82 s +Jan. 2020 +(c) Graph detection latency. +Total +Comp. +Identify +Pre +Cluster. +Critical +Vertex +Cluster. +3.5 +3.0 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +Ltency [10x s] +(d) Detection latency composition. +Fig. 17. +Latency of graph construction and detection. +Encrypted Web Malicious Traffic. Figure 13 presents the +detection accuracy of the encrypted traffic generated by various +web vulnerabilities discovery. HyperVision achieves 0.985 +average AUC and 0.957 average F1 (i.e., 2.8% and 75.2% +increase compared to Whisper). The flow based ML detection +cannot detect web encrypted malicious traffic because the traf- +fic has single-flow patterns that are almost same to benign web +access flows. HyperVision can accurately detect the encrypted +web malicious traffic, because, as shown in Figure 14(c), it +captures the traffic from the frequent interactions as the edges +associated with long flows, and identifies the malicious traffic +(denoted by red edges) generated by the attacker (denoted +by the green vertex) by clustering the edges associated with +benign web traffic that are connected to the same critical vertex +(denoted by the red solid vertex). +Encrypted Malware Traffic. We show the detection accuracy +of encrypted malware traffic in Figure 15. Note that, the +0 10 20 30 40 50 60 70 80 90 100 +Time [s] +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +9.5 +10.0 +Memory Usage [GB] +Overall +Graph +(a) Runtime memory usages. +Head +Suri. +Zeek +H.V. +Head +Suri. +Zeek +H.V. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Storage Usage [10x MB] +Jan. 2020 Benign +Jun. 2020 RST DoS +(b) Graph storage usages. +Fig. 18. +Hardware resource usages of HyperVision. +encrypted malware traffic is hard to detect for the baselines +because it is slow and persistent. However, HyperVision ac- +curately detects the malware campaigns with at least 0.964 +AUC and 0.891 F1. Specifically, it captures the C&C servers +of spyware for exfiltration as abnormal critical vertices that +are connected by massive infected hosts in the graph. As +a result, it detects the encrypted malicious traffic of the +malware with at least 0.942 F1. For example, to detect Sality +P2P botnet shown in Figure 14(d), HyperVision collects the +interactions among similar P2P bots, aggregates the encrypted +short flows as edges, and finally clusters the edges with higher +loss than benign interaction patterns. Similarly, it can capture +the static servers of adware, malware component delivery +servers, the infected miner pools as abnormal vertices. Note +that, the low-rate malicious flows (at least 0.814 pps) are +represented as the edges associated with short flows connected +to critical vertices. Meanwhile, the massive long flows with +almost 100% encrypted packet proportion are represented as +the edges associated with long flows to the vertices. Therefore, +a critical vertex connected with the edges indicates the malware +campaign that is significantly different from benign vertices +with large degrees, e.g., benign websites. +C. Performance Results +Throughput. We truncate the packets to the first 200 bytes +on the physical testbed and increase the sending rates until +the graph construction module reaches maximum throughput. +Figure 16 shows the throughput of the graph construction +and the detection. Figure 16(a) presents the distribution of +average throughput within a 1.0s time window. We observe +that HyperVision constructs the graph for 28.21 Gb traffic per +second. Figure 16(b) presents the maximum throughput in each +time window with all the backbone traffic datasets used in +the experiments. HyperVision achieves 32.43 ∼ 39.71 peak +throughput on average. Moreover, we measure the throughput +of the graph learning module, which inspects flow interactions. +According to Figure 16(c), we observe that it can analyze +121.14 Gb traffic per second on average. Note that, the de- +tection throughput is 4.2 times higher than the construction so +that the detection can analyze the recorded traffic iteratively to +consider the past interaction information. We observe that the +average throughput exhibits a bimodal distribution. The peak +of low throughput (around 75 Gb/s) is caused by lacking the +information on the graph for analyzing during cold start stages. +12 + +Figure 16(d) illustrates the throughput when the performance +of the system is stable. We observe that it achieves 80.6 ∼ +148.9 Gb/s throughput. Note that, the throughput on Apr. and +Jun. 2020 datasets is lower because of their low original traffic +volume. +Latency. We measure the latency caused by graph construction +and detection. Figure 17(a) presents the PDF of the maximum +latency for constructing each edge within a 1.0s window. We +observe that HyperVision has 1.09s ∼ 1.04s average construc- +tion latency with an upper bound of 1.93s. The distribution +is a significant bimodal one because the receive side scaling +(RSS) on the Intel NIC is unbalanced on the threads. The +light-load threads have only 0.75s latency. We analyze the +composition of the latency in Figure 17(b) (where the error bar +is 10th and 90th percentile) and find that the flow classification, +short flow aggregation, and long flow distribution fitting share +50.95%, 35.03%, and 14.0% latency, respectively. We measure +the average detection latency. Figure 17(c) shows that the +learning module has a 0.83s latency on average with a 99th +percentile of 4.48s. We also analyze the latency in each +step (see Figure 17(d)). We see that 75.8% of the latency +comes from pre-clustering (i.e., 0.66s on average). However, +the pre-clustering step reduces the processing overhead of +the subsequent processing, i.e., selecting critical vertex and +clustering, for 5.5 × 10−3s (0.64%) and 3.4 × 10−3s (0.40%). +Resource Consumption. Figure 18(a) presents the memory +usage of HyperVision. Note that, the DPDK huge pages require +6GB memory and thus we measure the consumption when the +usage reaches 6GB. We observe that the increasing rate of +memory for maintaining the graph is only 13.1 MB/s. Finally, +HyperVision utilizes 1.78 GB memory to maintain the flow +interaction patterns extracted from 2.82 TB ongoing traffic. +HyperVision incurs low memory consumption because the fea- +ture distribution fitting for long flow and short flow aggregation +make the in-memory graph compact which ensures low-latency +detection and long-term recording. Moreover, the memory +consumption of the learning algorithm is 1.452 ∼ 1.619 GB. +HyperVision can export the graph to disk for forensic analysis. +Figure 18(b) shows the storage used for recording the first +45s traffic of the MAWI dataset by different methods, i.e., +HyperVision, event based network monitors (i.e., Suricata [74] +and Zeek [86]), and raw packet headers. We observe that +HyperVision achieves 8.99%, 55.7%, 98.1% storage reduction +over the baselines, respectively. Meanwhile, our analysis shows +that HyperVision retains more traffic information than the +existing tools (see Section VII). Thus, the graph based analysis +is more efficient than these existing tools. +IX. +RELATED WORK +Graph Based Anomaly Detection. Graph based structures +have been used for task-specific traffic detection. These meth- +ods heavily rely on DPI and thus cannot be applied to detect +encrypted traffic [76]. Kwon et al. analyzed the download re- +lationship graph to identify malware downloading [45], which +is similar to WebWitness [60]. Eshete et al. constructed HTTP +interaction graphs to detect malware static resources [24], +and Invernizzi et al. used a graph constructed from plain- +text traffic to identify malware infrastructures [38]. Different +from these works, HyperVision constructs the interaction graph +without parsing specific application layer headers and thus +achieves task-agnostic encrypted traffic detection. Note that, +the provenance graph based attack forensic analysis [83], [87] +is orthogonal to our traffic detection. +DTMC Based Anomaly Detection. Discrete-Time Markov +Chain (DTMC) has been used to model the behaviors of +users/devices [1], [71], [72]. These methods aim to predict +behaviors of users and devices by utilizing DTMC. For in- +stance, Peek-a-Boo predicted user activities [1], Aegis pre- +dicted user behaviors for abnormal event detection [72], and +6thSense predicted sensor behaviors for detecting sensor-based +attacks [71]. Different to these methods, our work utilizes +DTMC to quantify the benefits of building the compact graph +for detecting various unknown attacks. +ML Based Malicious Traffic Detection. ML based detec- +tion can detect zero-day attacks [12] and achieve higher +accuracy than the traditional signature based methods [89]. +For example, Fu et al. leveraged frequency domain features +to realize realtime detection [30]. Barradas et al. developed +Flowlens to extract flow distribution features on data-plane +and detect attacks by applying random forest [5]. Stealthwatch +detected attacks by analyzing flow features extracted from +NetFlow [16]. Mirsky et al. developed Kitsune to learn the +per-packet features by adopting auto-encoders [56]. For task- +specific methods, Nelms et al. [60], Invernizzi et al. [38], +and Bilge et al. [8] detected traffic in the different stages of +malware campaigns by using statistical ML. Bartos et al. [6] +and Tang et al. [75] detect malformed HTTP request traffic. +Holland et al. [35] developed an automatic pipeline for traffic +detection. All these methods cannot effectively detect attacks +based on encrypted traffic. +Task-Specific Encrypted Traffic Detection. The existing +encrypted traffic detection relies on domain knowledge for +short-term flow-level features [2], [16], [62]. For example, +Zheng et al. leveraged SDN to achieve crossfire attack de- +tection [90], and Xing et al. designed the primitives for the +programmable switch to detect link flooding attacks [82]. +For encrypted malware traffic, Bilge et al. +[8] leveraged +the traffic history to detect C&C server, and Tegeler et al. +developed supervised learning using time-scale flow features +extracted from malware binaries [76]. Anderson et al. studies +the feasibility of detecting malware encrypted communication +via malformed TLS headers [3]. To the best of our knowledge, +our HyperVision is the first system that enables unsupervised +detection for the encrypted traffic with unknown patterns. +Encrypted Traffic Classification. HyperVision aims to iden- +tify the malicious behaviors according to encrypted traffic. It +is different from encrypted traffic classifications that decide if +the traffic is generated by certain applications or users [69]. +For instance, Rimmer et al. leveraged DL for web fingerprint, +which de-anonymizes Tor traffic by classifying encrypted web +traffic [67]. Siby et al. showed that classifying encrypted +DNS traffic can jeopardize the user privacy [70]. Similarly, +Bahramali et al. classified the encrypted traffic of instant mes- +saging applications [4]. Ede et al. designed semi-supervised +learning for mobile applications fingerprinting [78]. All these +classifications are orthogonal to HyperVision. +13 + +X. +CONCLUSION +In this paper, we present HyperVision, an ML based +realtime detection system for encrypted malicious traffic with +unknown patterns. HyperVision utilizes a compact in-memory +graph to retain flow interaction patterns, while not requiring +prior knowledge on the traffic. Specifically, HyperVision uses +two different strategies to represent the interaction patterns +generated by short and long flows and aggregates the informa- +tion of these flows. We develop an unsupervised graph learning +method to detect the traffic by utilizing the connectivity, +sparsity, and statistical features in the graph. Moreover, we +establish an information theory based analysis framework to +demonstrate that HyperVision preserves near-optimal informa- +tion of flows for effective detection. 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Zhu et al., “You do (not) belong here: detecting DPI evasion attacks +with context learning,” in CoNEXT. +ACM, 2020, pp. 183–197. +APPENDIX +A. Details of Implementations +We present the details of the flow classification and short +flow aggregation algorithm in Algorithm 1 and 2, respectively. +The features used for edge pre-clustering and clustering are +shown in Table V. And Table VII shows the hyper-parameters +used in HyperVision and the recommended values. +15 + +TABLE V. +THE FEATURES OF EDGES USED IN HYPERVISION. +Edge Group Data +Description +Edge Denoting Short Flows +structural +bool +Denoting short flows with the same source address. +bool +Denoting short flows with the same source port. +bool +Denoting short flows with the same destination address. +bool +Denoting show flows with the same destination port. +int +The in-degree of the connected source vertex. +int +The out-degree of the connected source vertex. +int +The in-degree of the connected destination vertex. +int +The out-degree of the connected destination vertex. +statistical +int +The number of flows denoted by the edge. +int +The length of the feature sequence associated with the edge. +int +The sum of packet lengths in the feature sequence. +int +The mask of protocols in the feature sequence. +float +The mean of arrival intervals in the feature sequence. +Edge Denoting Long Flows +structural +int +The in-degree of the connected source vertex. +int +The out-degree of the connected source vertex. +int +The in-degree of the connected destination vertex. +int +The out-degree of the connected destination vertex. +statistical +float +The flow completion time of the denoted long flow. +float +The packet rate of the denoted long flow. +int +The number of packets in the denoted long flow. +int +The maximum bin size for fitting packet length distribution. +int +The length associated with the maximum bin size. +int +The maximum bin size for fitting protocol distribution. +int +The protocol associated with the maximum bin size. +TABLE VI. +DETAILS OF MALICIOUS TRAFFIC DATASETS. +Class +Dataset +Label +Description +Att.1 +Vic. +B.W.2 Enc. +Ratio +Malware Related Encrypted Traffic +Spyware +Magic. +Magic Hound spyware. +2 +479 +0.34 0.13% +Trickster +Encrypted C&C connections. +2 +793 +0.63 10.0% +Plankton +Pulling components from CDN. +3 +579 +59.2 23.8% +Penetho +Wifi cracking APK spyware. +1 +516 +3.57 100% +Zsone +Multi-round encrypted uploads. +1 +479 +5.98 93.0% +CCleaner +Unwanted software downloads. +4 +466 +28.1 4.09% +Adware +Feiwo +Encrypted ad API calls. +3 +1.00K 19.8 100% +Mobidash +Periodical statistic ad updates. +3 +624 +6.08 100% +WebComp. +WebCompanion click tricker. +3 +281 +8.38 55.2% +Adload +Static resources for PPI adware. +1 +280 +1.04 1.09% +Ransom- +ware +Svpeng +Periodical C&C interactions (10s). +2 +403 +1.21 1.26% +Koler +Invalid TLS connections. +3 +333 +2.22 100% +Ransombo +Executable malware downloads. +5 +369 +58.6 42.7% +WannaL. +Wannalocker delivers components. +2 +275 +7.49 30.3% +Dridex +Victim locations uploading. +1 +429 +4.10 100% +Miner +BitCoinM. +Abnormal encrypted channels. +1 +1.54K 0.79 100% +TrojanM. +Long SSL connections to C&C. +3 +1.37K 2.39 89.4% +CoinM. +Periodical connections to pool. +1 +1.40K 0.21 100% +Botware +THBot +Getting C&C server addresses. +4 +103 +1.72 2.71% +Emotet +Communication to C&C servers. +6 +1.17K 1.43 68.6% +Snojan +PPI malware downloading. +3 +326 +8.94 100% +Trickbot +Connecting to alternative C&C. +4 +347 +0.57 100% +Mazarbot +Long C&C connections to cloud. +3 +409 +6.13 30.9% +Sality +A P2P botware. +20 +247 +2.19 100% +Encrypted Flooding Traffic +Link Flooding +CrossfireS. +We use the botnet cluster sizes +and the ratio of decony servers +(HTTPS) in [41]. +100 +313 +197 +100% +CrossfireM. +200 +313 +278 +100% +CrossfireL. +500 +313 +503 +100% +LrDoS 0.2 We use the traffic of an encrypted +video application and the settings +in WAN experiments [44] +1 +1 +5.57 100% +LrDoS 0.5 +1 +1 +3.25 100% +LrDoS 1.0 +1 +1 +1.90 100% +SSH +Inject +ACK Inj. +SSH injection via ACK rate-limits. +1 +2 +1.78 +- +IPID Inj. +SSH injection via IPID counters. +1 +2 +0.28 +- +IPID Port +Requires of the SSH injection. +1 +1 +1.83 +- +Password +Cracking +Telnet S. +Telnet servers in AS38635. +1 +19 +0.63 100% +Telnet M. +Telnet servers in AS2501. +1 +43 +1.70 100% +Telnet L. +Telnet servers in AS2500. +1 +83 +2.76 100% +SSH S. +SSH servers in AS9376. +1 +35 +1.39 100% +SSH M. +SSH servers in AS2500. +1 +257 +2.49 100% +SSH L. +SSH servers in AS2501. +1 +486 +5.53 100% +Encrypted Web Traffic +Web Attacks +Oracle +TLS padding Oracle. +1 +1 +3.99 100% +XSS +Xsssniper XSS detection. +1 +1 +31.8 100% +SSLScan +SSL vulnerabilities detection. +1 +1 +15.0 100% +Param.Inj. +Commix parameter injection. +1 +1 +17.1 100% +Cookie.Inj. +Commix cookie injection. +1 +1 +39.6 100% +Agent.Inj. +Commix agent-based injection. +1 +1 +19.7 100% +WebCVE +Exploiting CVE-2013-2028. +1 +1 +2.30 100% +WebShell +Exploiting CVE-2014-6271. +1 +1 +11.2 100% +CSRF +Bolt CSRF detection. +1 +1 +7.73 100% +Crawl +A crawler using scrapy. +1 +1 +29.7 100% +SMTP +SSL +Spam1 +Spam using SMTP-over-SSL. +1 +1 +36.2 100% +Spam50 +Encrypted spam with 50 bots. +50 +1 +61.7 100% +Spam100 +Brute spam using 100 bots. +100 +1 +88.9 100% +Traditional Brute Force Attack +Brute Scanning +ICMP +We use the brute force scanning +rates identified by darknet +in [22]. We reproduce the +scan using Zmap which targets +the peers and customers +of AS 2500. +1 +211K +5.61 +- +NTP +1 +99.3K 3.87 +- +SSH +1 +205K +5.79 +- +SQL +1 +112K +3.04 +- +DNS +1 +198K +6.61 +- +HTTP +1 +93.7K 2.68 +- +HTTPS +1 +209K +4.89 +- +Source +Spoof +SYN +We use the protocol types and +the packet rates in [40]. +6.50K +1 +11.41 +- +RST +32.5K +1 +5.79 +- +UDP +6.50K +1 +54.3 +- +ICMP +3.20K +1 +0.13 +- +Amplification +Attack +NTP +We use the packet rates and +the vulnerable protocols +observed in [40]. +And we use the number of +the reflectors in [43]. +650 +1 +95.8 +- +DNS +200 +1 +82.7 +- +CharGen +200 +1 +175 +- +SSDP +1.30K +1 +7.23 +- +RIPv1 +500 +1 +7.04 +- +Memcache +1.60K +1 +63.5 +- +CLDAP +1.30K +1 +36.8 +- +Probing Vulnerable +Application +Lr. SMTP +We use the sending rates of +vulnerable application discovery +disclosed by a darknet [22]. We +estimate the number of scanners +by the number of visible active +addresses from the vantage +(i.e., realword measurements) +and the size of the darknet. +11 +158K +7.97 +- +Lr.NetBios +28 +444K +17.3 +- +Lr.Telnet +156 +1.23M 49.0 +- +Lr.VLC +22 +352K +20.5 +- +Lr.SNMP +6 +110K +6.51 +- +Lr.RDP +172 +1.30M 53.0 +- +Lr.HTTP +94 +640K +38.0 +- +Lr.DNS +28 +428K +25.0 +- +Lr.ICMP +268 +1.82M 63.3 +- +Lr.SSH +72 +994K +5.63 +- +1 Att. and Vic. indicate the number of attackers and victims. +2 B.W. is short for total bandwidth in the unit of Mb/s. +TABLE VII. +RECOMMENDED HYPER-PARAMETER CONFIGURATION. +Group +Hyper-Parameter +Description +Value +Graph +Construction +PKT TIMEOUT +Flow completion time threshold. +10.0s +FLOW LINE +Flow classification threshold. +15 +AGG LINE +Flow aggregation threshold. +20 +Graph Pre- +Processing +ϵ +DBSCAN hyper-parameters for +clustering components and edges. +4 × 10−3 +minPoint +40 +Traffic +Detection +K +K-means hyper-parameter. +10 +T +Loss threshold for malicious traffic. +10.0 +α +Balancing the terms in +the loss function. +0.1 +β +0.5 +γ +1.7 +Algorithm 1: Secure flow classification. +Input: Per-packet features: PktInfo, the hash table for flow collecting: +FlowHashTable. +Output: Classified flows: ShortFlow and LongFlow. +1 time now := PktInfo[0].time, last check := time now. +2 for pkt in PktInfo do +// Aggregate packets into flows. +3 +if Hash(pkt) not in FlowHashTable then +4 +FlowHashTable adds an entry for pkt. +5 +FlowHashTable[Hash(pkt)] appends pkt. +6 +if time now − last check > JUDGE INTERVAL then +7 +for flow in FlowHashTable do +// Judge the completion of flows. +8 +if time now − flow[−1].time > PKT TIMEOUT then +// Classify the flow via the number of packets. +9 +if flow.size > FLOW LINE then +10 +ShortFlow adds flow. +11 +else +12 +LongFlow adds flow. +13 +FlowHashTable clears the states of flow. +14 +last check ← time now. // Record the time of checking. +15 +time now ← pkt.time. // Update the timer. +Algorithm 2: Short flow aggregation. +Input: Short flows: ShortFlow. +Output: Constructed edges: ShortEdge. +1 Initialize ProtoHashTable as an empty table. +// Select candidate protocols for the aggregation. +2 for flow in ShortFlow do +// Calculate the protocol mask of a short flow. +3 +flow proto := (flow[0].proto|...|...|flow[−1].proto). +4 +if Hash(flow proto) not in ProtoHashTable then +5 +ProtoHashTable adds an entry for flow proto. +6 +Append flow to ProtoHashTable[Hash(flow proto)]. +// Perform the source aggregation. +7 for flows in ProtoHashTable with same protocols do +8 +SrcAddrTable collects the flows with same sources in flows. +9 +for sflow in SrcAddrTable do +// The flows can be aggregated and denoted by one edge. +10 +if sflow.size > AGG LINE then +11 +edge.features := sflow[0].features. +12 +edge.source := sflow[0].source. +13 +if an unique destination in sflow then +// Source and destination aggregation. +14 +edge.destination saves the unique destination. +15 +else +// Source aggregation only. +16 +Record each destination in sflow. +17 +Add the constructed edge to ShortEdge. +18 +SrcAddrTable evicts sflow. +19 +DstAddrTable collects flows with same destinations. +20 +Inspect the flows with the same destinations similarly. +// Process short flows which cannot be aggregated. +21 +ShortEdge adds flows in SrcAddrTable and DstAddrTable. +16 + +B. Details of Experiments +1) Details of Datasets: We present the detailed properties +of the 80 newly collected datasets in Table VI, including the +number of attackers and victims, the packet rates of attack +flows, and the ratios of encrypted traffic. All the datasets +are collected and labeled using the same method as MAWI +datasets [80] and CIC datasets [14], [15]. +2) Detection Accuracy of Other Datasets: We use 12 +existing datasets to eliminate the impact of dataset bias. +Overall, HyperVision achieves 7.8%, 11.0%, 5.1% F1 im- +provements over the best accuracy of the baselines on Kitsune +datasets [56], CIC-IDS2017 datasets [15], and CIC-DDoS2019 +datasets [14], respectively. From the Kitsune datasets, we +validate the correctness of the deployed baselines. +3) Long-run Performances: By using the CIC datasets [14], +[15], we validate the long-run performances of HyperVision. +Specifically, the experiments show that HyperVision achieves +over 0.95 F1 and 0.99 AUC in long-run detection (6∼8 hours). +The results also verify that the accumulation of detection errors +cannot interfere with HyperVision, and HyperVision can detect +multiple attacks simultaneously even in the presence of attacks +with changed addresses. Moreover, the memory consumption +of the compact graph is bounded by 15.6 GB. +4) Robustness Against Obfuscation Techniques: We vali- +date our method under evasion attacks with different obfusca- +tion techniques according to a recent study [30], i.e., injecting +three kinds of benign traffic. The results demonstrate that the +accuracy decrease incurred by the obfuscation is bounded by +4.3% F1. Specifically, when benign TLS traffic, UDP video +traffic, and normal ICMP traffic is injected into brute force +attack traffic, the average F1 decreases by 1.7%, 0.9%, and +2.4%, respectively. +The reason why the obfuscation techniques incur negligible +accuracy decrease is that they only manipulate patterns of a +single flow. HyperVision can still detect the evasion attacks by +learning the interaction patterns among various flows. +C. Details of Theoretical Analysis +1) Analysis of Event based Mode: Let random variable +IEve. indicate if the event based mode records an event for a +flow denoted by a random variable sequence, ⟨s1, s2, . . . , sL⟩, +L ∼ G(q). And we assume that the mode can merge repetitive +events. First, we obtain the probability distribution of the +random variable IEve.: +P[IEve. = 1] = 1 − P[IEve. = 0], +P[IEve. = 0] = +∞ +� +l=1 +P[L = l] · P[IEve. = 0|L = l] += +∞ +� +l=1 +(1 − q)l−1 · q · (1 − ps)l += +q(1 − ps) +1 − (1 − q)(1 − ps). +(21) +Then, we obtain the entropy of the random variable IEve.: +HEve. = H[IEve.] = +−P[IEve. = 0] ln P[IEve. = 0] − P[IEve. = 1] ln P[IEve. = 1]. (22) +We observe that ∂H[IEve.] +∂q +≈ 0 when q > 0.5. Thus, we use +the second-order taylor series of q to approach HEve.: +HEve. = +2q(1 − ps) ln[ +(ps−1)q +ps(q−1)−q ] +ps(q − 1) − q += −2θ ln θ, +(23) +where θ = ζ +η, ζ = q − qps, and η = q − ps(q − 1). Similarly, +we obtain the expected data scale LEve. and the information +density DEve.: +LEve. = P[IEve. = 1] = +ps +ps(1 − q) + q = −ps +η , +DEve. = HEve. +LEve. = 2ζ +ps · ln θ. +(24) +Here, we complete the analysis for the event based mode. +2) Analysis of Sampling based Mode: We use XSamp. +to denote the random variable to be recorded as the flow +information in the sampling based mode which is the sum +of the observed per-packet features denoted by the random +variable sequence. We can obtain the distribution of XSamp. +as follows: +XSamp. = +L +� +i=1 +si, +si ∼ B(s, p) ⇒ XSamp. ∼ B(Ls, p). +(25) +The amount of the information recorded by the sampling +based mode is the Shannon entropy of XSamp.. We decompose +the entropy as conditional entropy and mutual information: +HSamp. = H[XSamp.] += H[XSamp.|L] + I(XSamp.; L). +(26) +We assume that the mutual information between the se- +quence length L and the accumulative statistic XSamp. is close +to zero. It implies the impossibility of inferring the statistic +from the length of the packet sequence. Then we obtain a +lower bound of the entropy as an estimation which is verified +to be a tight bound via numerical analysis: +� +� +� +HSamp. = H[XSamp.|L] += +∞ +� +l=1 +P[L = l] · H[XSamp.|L = l] +H[XSamp.|L = l] += 1 +2 ln 2πelsp(1 − p), +⇒ HSamp. = 1 +2 ln 2πesp(1 − p) + q +2 +∞ +� +l=1 +(1 − q)l−1 ln l. +(27) +We observed that the second-order taylor series can accu- +rately approach the second term of the entropy: +HSamp. = 1 +2 ln 2πesp(1 − p) + ln 2 +2 q(1 − q). +(28) +Finally, we obtain the expected data scale and the informa- +tion density similar to the analysis for the event based mode +and complete the analysis for the sampling based mode. +17 + +3) Analysis of Graph based Mode in HyperVision: Hyper- +Vision applies different recording strategies for short and long +flows, i.e., when L > K it extracts the histogram for long +flow feature distribution fitting, and when L ≤ K it records +detailed per-packet features and aggregates short flows. Let +XH.V. denote the random set of the recorded information. For +short flows, all the random variables are collected in XH.V.. +For long flows, XH.V. collects s counters of the histogram +for each state on the state diagram of the DTMC. First, we +decompose the entropy of the graph based recording mode as +the terms for short and long flows: +HH.V. = H[XH.V.|L] = +∞ +� +l=1 +P[L = l] · H[XH.V.|L = l] += H[X S +H.V.|L] + H[X L +H.V.|L] +(29) +� +� +� +� +� +� +� +H[X S +H.V.|L] += +K +� +l=1 +P[L = l] · H[XH.V.|L = l] +H[X L +H.V.|L] += +∞ +� +l=K+1 +P[L = l] · H[XH.V.|L = l]. +Short Flow Information. HyperVision records detailed per- +packet feature sequences for short flows which is the same as +the brute recording in the idealized mode. Thus, the increasing +rate of information equals the entropy rate of the DTMC: +H[XH.V.|L = l] = l · H[G], +(30) +H[X S +H.V.|L] = +K +� +l=1 +P[L = l] · l · H[G] += q · H[G] · +K +� +l=1 +(1 − q)l−1 · l += 1 − (Kq + 1)(1 − q)K +q +· H[G]. +(31) +Long Flow Information. When L > K, the random set +collects the counters for distribution fitting. When the DTMC +has s states, the histogram has s counters υ1, υ2, . . . , υs, i.e., +XH.V. = {υ1, υ2, . . . , υs}. We assume that the counters are +independent: +υi = +L +� +j=1 +δj, +δj = +� +1, +if sj is the ith state +0, +else. +(32) +We observe that ⟨υ1, υ2, . . . , υs⟩ is a binomial process: +υi ∼ B(L, P[si = i]) +∼ B(L, Ci +spi(1 − p)s−i). +(33) +To obtain the closed-form solution, we use (sp)ie−sp +i! +as an +estimation of Ci +spi(1−p)s−i. Moreover, the length of the per- +packet feature sequence of a long flow is relatively large which +implies υi approaches a Poisson distribution: +υi ∼π(L · P[si = i]) +∼π(λi), +λi = (sp)ie−sp +i! +. +(34) +Basing on the distribution of the collected counters, we +obtain the entropy of the random set: +� +� +� +H[υi|L = l] += +1 +2 ln 2πel (sp)ie−sp +i! +H[X L +H.V.|L = l] += +s� +i=1 +H[υi|L = l], +(35) +H[X L +H.V.|L] = +∞ +� +l=K+1 +P[L = l] · H[X L +H.V.|L = l] += +∞ +� +l=K+1 +q(1 − q)l−1 · +s +� +i=1 +1 +2 ln 2πel (sp)ie−sp +i! += (1 − q)K +2 +[s ln 2πe + s(s + 1) +2 +ln sp +− sp2 − +s +� +i=1 +ln i!] + qs +2 [ +∞ +� +l=K+1 +(1 − q)l−1 ln l]. +The assumption of q > 0.5 implies Kth order taylor series +can accurately approach the last term in (35). Moreover, we +utilize the quadric term of s in the taylor series of �s +i=1 ln i! +to approach the entropy of long flows (γ is Euler–Mascheroni +constant): +H[X L +H.V.|L] = 1 +4s(1 − q)K[(1 + s) ln ps+ +2 ln 2πe + 2q ln K − 2s(1 + p + γ)]. +(36) +Finally, we take (31) and (36) in (29) and complete the +analysis for the entropy of the graph based recording mode. +Similarly, we obtain the expected data scale by analyzing the +conditions of short and long flows separately: +LH.V. = E[LS +H.V.|L] + E[LL +H.V.|L] += +K +� +l=1 +P[L = l] · L +C + +∞ +� +l=K+1 +s · P[L = l] += s(1 − q)K + 1 − (Kq + 1)(1 − q)K +Cq +, +(37) +where C is the average number of flows denoted by an edge +associated with short flows. Also, we obtain the expected +information density by its definition: DH.V. = HH.V./LH.V. +and complete the analysis for the graph based recording mode +used by HyperVision. +18 + diff --git a/WNFRT4oBgHgl3EQf9DhY/content/tmp_files/load_file.txt b/WNFRT4oBgHgl3EQf9DhY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d623fd81ce58429d35a78d6423a0c4c2cb61904b --- /dev/null +++ b/WNFRT4oBgHgl3EQf9DhY/content/tmp_files/load_file.txt @@ -0,0 +1,2958 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf,len=2957 +page_content='Detecting Unknown Encrypted Malicious Traffic in Real Time via Flow Interaction Graph Analysis Chuanpu Fu∗, Qi Li†‡, Ke Xu∗‡ ∗Department of Computer Science and Technology, Tsinghua University †Institute for Network Sciences and Cyberspace, Tsinghua University ‡Zhongguancun Lab Abstract—Nowadays traffic on the Internet has been widely encrypted to protect its confidentiality and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' However, traffic encryption is always abused by attackers to conceal their malicious behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Since the encrypted malicious traffic has similar features to benign flows, it can easily evade traditional detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Particularly, the existing encrypted malicious traffic detection methods are supervised and they rely on the prior knowledge of known attacks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', labeled datasets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Detecting unknown encrypted malicious traffic in real time, which does not require prior domain knowledge, is still an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In this paper, we propose HyperVision, a realtime unsuper- vised machine learning (ML) based malicious traffic detection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Particularly, HyperVision is able to detect unknown patterns of encrypted malicious traffic by utilizing a compact in- memory graph built upon the traffic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The graph captures flow interaction patterns represented by the graph structural features, instead of the features of specific known attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We de- velop an unsupervised graph learning method to detect abnormal interaction patterns by analyzing the connectivity, sparsity, and statistical features of the graph, which allows HyperVision to de- tect various encrypted attack traffic without requiring any labeled datasets of known attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, we establish an information theory model to demonstrate that the information preserved by the graph approaches the ideal theoretical bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We show the performance of HyperVision by real-world experiments with 92 datasets including 48 attacks with encrypted malicious traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The experimental results illustrate that HyperVision achieves at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='92 AUC and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='86 F1, which significantly outperform the state- of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In particular, more than 50% attacks in our experiments can evade all these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, HyperVision achieves at least 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 Gb/s detection throughput with the average detection latency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='83s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' INTRODUCTION Traffic encryption has been widely adopted to protect the information delivered on the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Over 80% websites adopted HTTPS to prevent data breach in 2019 [16], [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' However, the cipher-suite for such protection is double-edged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In particular, the encrypted traffic also allows attackers to con- ceal their malicious behaviors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', malware campaigns [2], exploiting vulnerabilities [64], and data exfiltration [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The ratio of encrypted malicious traffic on the Internet is growing significantly [2], [3], [76] and exceeds 70% of the entire malicious traffic [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' However, encrypted malicious traffic detection is not well addressed due to the low-rate and diverse traffic patterns [2], [39], [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The traditional signature based methods that lever- age deep packet inspection (DPI) are invalid under the at- tacks with the encrypted payloads [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Table I compares the existing malicious traffic detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Different from plain-text malicious traffic, the encrypted traffic has similar features to benign flows and thus can evade existing machine learning (ML) based detection systems as well [2], [3], [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Particularly, the existing encrypted traffic detection methods are supervised, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', relying on the prior knowledge of known attacks, and can only detect attacks with known traffic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' They extract features of specific known attacks and use labeled datasets of known malicious traffic for model training [2], [3], [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, they are unable to detect a broad spectrum of attacks with encrypted traffic [39], [41], [64], [77], which are constructed with unknown patterns [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Besides, these methods are incapable of detecting both attacks constructed with and without encrypted traffic and unable to achieve generic detection because features of encrypted and non- encrypted attack traffic are significantly different [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In a nutshell, the existing methods cannot achieve unsuper- vised detection and they are unable to detect encrypted mali- cious traffic with unknown patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In particular, the encrypted malicious traffic has stealthy behaviors, which cannot be cap- tured by these methods [2], [76] that detect attacks according to the patterns of a single flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' However, it is still feasible to detect such attack traffic because these attacks involve multiple attack steps with different flow interactions among attackers and victims, which are distinct from benign flow interactions patterns [24], [26], [39], [46], [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For example, the encrypted flow interactions between spam bots and SMTP servers are significantly different from the legitimate communications [61] even if the single flow of the attack is similar to the benign one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, this paper explores utilizing interaction patterns among various flows for malicious traffic detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' To the end, we propose HyperVision, a realtime detection system that aims to capture footprints of encrypted malicious traffic by analyzing interaction patterns among flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In par- ticular, it can detect encrypted malicious flows with unknown footprints by identifying abnormal flow interactions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the interaction patterns that are distinct from benign ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' To achieve this, we build a compact graph to capture various flow interaction patterns so that HyperVision can perform detection on various encrypted traffic according to the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The graph allows us to detect attacks without accessing packet payloads, while retaining the ability of detecting traditional (known) attacks with plain-text traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Therefore, HyperVision can detect the malicious traffic with unknown patterns by learning the graph structural features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Meanwhile, by learning the graph structural features, it realizes unsupervised detection, which does not require model training with labeled datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' However, it is challenging to build the graph for realtime detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We cannot simply use IP addresses as vertices and traditional four-tuple of flows [19], [36] as edges to construct the graph because the resulting dense graph cannot maintain Network and Distributed System Security (NDSS) Symposium 2023 27 February - 3 March 2023, San Diego, CA, USA ISBN 1-891562-83-5 https://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='14722/ndss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='23080 www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='ndss-symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='org arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='13686v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='CR] 31 Jan 2023 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' THE COMPARISON WITH THE EXISTING METHODS OF MALICIOUS TRAFFIC DETECTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Data Source ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Categories ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Data Sources ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Typical Methods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Data for Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Design Goals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Detection Performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Unlabeled ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Datasets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Multi-Flow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Generic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Realtime ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Unknown ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Attacks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Latency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='High ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Throughput ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Encrypted Traffic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Protocol Headers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='TLS Extensions [16] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='HTTPS Headers [3] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Related Flows ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Time Series [76] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='TLS Handshakes [2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Flow Statistics [90] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Plain-text and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Encrypted Traffic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Network Logs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Intrusion Events [20] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Sampled Connections [8] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Traffic Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Per-Packet Features [56] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Per-Flow Features [5] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Flow Interaction Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 Existing multi-flow features can only represent the features of specific flows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' which cannot be used to represent complicated interaction patterns among various flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' interaction patterns among various flows, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', incurring the dependence explosion problem [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Inspired by the study of the flow size distribution [25], [84], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', most flows on the Internet are short while most packets are associated with long flows, we utilize two strategies to record different sizes of flows, and process the interaction patterns of short and long flows separately in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, it aggregates the short flows based on the similarity of massive short flows on the Internet, which reduces the density of the graph, and performs distribution fitting for the long flows, which can effectively preserve flow interaction information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We design a four-step lightweight unsupervised graph learning approach to detect encrypted malicious traffic by utilizing the rich flow interaction information maintained on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' First, we analyze the connectivity of the graph by extracting the connected components and identify abnormal components by clustering the high-level statistical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' By excluding the benign components, we also significantly reduce the learning overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Second, we pre-cluster the edges according to the observed local adjacency in edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The pre-clustering operations significantly reduce the feature processing overhead and ensure realtime detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Third, we extract critical vertices by solving a vertex cover problem using Z3 SMT solver [55] to minimize the number of clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fi- nally, we cluster each critical vertex according to its connected edges, which are in the centers of the clusters produced by the pre-clustering, and thus obtain the abnormal edges indicating encrypted malicious traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, to quantify the benefits of the graph based flow recording of HyperVision over the existing approaches, we develop a flow recording entropy model, an information theory based framework that theoretically analyzes the amount of information retained by the existing data sources of malicious traffic detection systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' By using this framework, we show that the existing sampling based and event based traffic data sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', NetFlow [19] and Zeek [86]) cannot retain high- fidelity traffic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, they are unable to record flow interaction information for the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' But the graph in HyperVision captures near-optimal traffic information for the graph learning based detection and the amount of the information maintained in the graph approaches the theoretical up-bound of the idealized data source with infinite storage according to the data processing inequality [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Also, the analysis results demonstrate that the graph in HyperVision achieves higher information density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', amount of traffic information per unit of storage) than all existing data sources, which is the foundation of the accurate and efficient detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We prototype HyperVision1 with Intel’s Data Plane De- velopment Kit (DPDK) [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' To extensively evaluate the performance of the prototype, we replayed 92 attack datasets including 80 new datasets collected in our virtual private cloud (VPC) with more than 1,500 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In the VPC, we collected 48 typical encrypted malicious traffic, including (i) encrypted flooding traffic, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', flooding target links [41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (ii) web attacks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', exploiting web vulnerabilities [64];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (iii) mal- ware campaigns, including connectivity testing, dependency update, and downloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In the presence of the background traffic by replaying the backbone network traffic [80], Hyper- Vision achieves 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9% ∼ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1% accuracy improvements over five state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' It detects all encrypted malicious traffic in an unsupervised manner with more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='92 AUC, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='86 F1, where 44 of the real-world stealthy traffic cannot be identified by all the baselines, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', an advanced side-channel attack exploiting the CVE-2020-36516 [26] and many newly discovered cryptojacking attacks [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, HyperVision achieves on average more than 100 Gb/s detection throughput with the average detection latency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='83s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In summary, the contributions of our paper are five-fold: We propose HyperVision, the first realtime unsupervised detection for encrypted malicious traffic with unknown patterns by utilizing the flow interaction graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We develop several algorithms to build the in-memory graph that allows us to accurately capture interaction patterns among various flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We design a lightweight unsupervised graph learning method to detect encrypted traffic via graph features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We develop a theoretical analysis framework established by information theory to show that the graph captures near-optimal traffic interaction information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We prototype HyperVision and use the extensive experi- ments with various real-world encrypted malicious traffic to validate its accuracy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The rest of the paper is organized as follows: Section II in- troduces the threat model of HyperVision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Section III presents the high-level design of HyperVision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In section IV, V, and VI, we describe the detailed designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In Section VII, we conduct the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In Section VIII, we experimentally evaluate the performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Section IX reviews related works and Section X concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Finally, we present details in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1Source code and datasets: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='com/fuchuanpu/HyperVision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2 Abnormal Component Detection Critical Vertex Detection Edge Pre-Clustering Interaction Pattern Clustering Flow Collection \uf050\uf050\uf050\uf050\uf050\uf050\uf050\uf04f\uf050\uf04f\uf050\uf050 Flow Classification Short Flows Long Flows Short Flow Aggregation Similar Short Flows Long Flow Distribution Fitting Packet Feature Distribution Flow Interaction Graph Ongoing Traffic Raw Packet Parser 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Graph Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Graph Pre-Processing Abnormal Component Timeout Threshold Long Flows \uf04f \uf04f Connected Components Attacker Benign User Malicious Flow Benign Flow Malicious Traffic Identified Cluster 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Abnormal Interaction Detection Short Flows Component Statistical Features \uf050 \uf04f \uf050 \uf050 \uf050 Critical Vertex \uf04f \uf04f Benign Malicious \uf050 \uf04f Benign \uf050 Attacker Victims \uf04f Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The overview of HyperVision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' THREAT MODEL AND DESIGN GOALS We aim to develop a realtime system (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', HyperVision) to detect encrypted malicious traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' It performs detection ac- cording to the traffic replicated by routers through port mirror- ing [17], which ensures that the system will not interfere with the traffic forwarding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' After identifying encrypted malicious traffic, it can cooperate with the existing on-path malicious traffic defenses [48], [49], [88] to throttle the detected traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' To perform detection on encrypted traffic, we cannot parse and analyze application layer headers and payloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In this paper, we focus on detecting active attacks con- structed with encrypted traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We do not consider passive attacks that do not generate traffic to victims, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', traffic eavesdropping [68] and passive traffic analysis [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' According to the existing studies [10], [24], [29], [40], [46], [81], attack- ers utilize reconnaissance steps to probe the information of victims, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the password of a victim [39], the TCP sequence number of a TLS connection [26], [27], and the randomized memory layout of a web server [75], which cannot be accessed directly by attackers due to lack of prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, these attacks are normally constructed with many addresses owned or faked by attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The design goals of HyperVision are as follows: First, it should be able to achieve generic detection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', detect attacks constructed with encrypted or non-encrypted traffic, which ensures that the attacks cannot evade detection by traffic encryption [2], [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Second, it is able to achieve realtime high-speed traffic processing, which means that it can identify whether the passing through encrypted traffic is malicious, while incurring low detection latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Third, the performed detection by HyperVision is unsupervised, which means that it does not require any prior knowledge of encrypted malicious traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' That is, it should be able to deal with attacks with unknown patterns, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', zero-day attacks, which have not been disclosed [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, we do not use any labeled traffic datasets for ML training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' These issues cannot be well addressed by the existing detection methods [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' OVERVIEW OF HYPERVISION In this section, we develop HyperVision that is an unsuper- vised detection system to capture malicious traffic in real time, in particular, encrypted malicious traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Normally, patterns of each flow in the encrypted malicious traffic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', single- flow patterns, may be similar to benign flows, which allow them to evade the existing detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' However, the malicious behaviors appearing in the interaction patterns between the attackers and victims will be more distinct from the benign ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, in HyperVision, we construct a compact graph to maintain interaction patterns among various flows and detect abnormal interaction patterns by learning the features of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision analyzes the graph structural features representing the interaction patterns without prior knowledge of known attack traffic and thus can achieve unsupervised detection against various attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' It realizes generic detection by analyzing flows regardless of the traffic type and can detect encrypted and non-encrypted malicious traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 1 shows three key parts of HyperVision, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', graph construction, graph pre-processing, and abnormal interaction detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Graph Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision collects network flows for graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Meanwhile, it classifies the flows into short and long ones and records their interaction patterns separately for the purpose of reducing the density of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In the graph, it uses different addresses as vertices that connect the edges associated with short and long flows, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' It aggregates the massive similar short flows to construct one edge for a group of short flows, and thus reduces the overhead for maintaining flow interaction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, it fits the distributions of the packet features in the long flows to construct the edges associated with long flows, which ensures high-fidelity recorded flow interaction patterns, while addressing the issue of coarse-grained flow features in the traditional methods [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We will detail how HyperVision maintains the high-fidelity flow interaction patterns in the in- memory graph in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Graph Pre-Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We pre-process the built interaction graph to reduce the overhead of processing the graph by extracting connected components and cluster the components using high-level statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In particular, the clustering can detect the components with only benign interaction patterns accurately and thus filters these benign components to reduce the scale of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, we perform a pre-clustering and use the generated cluster centers to represent the edges in 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Flow Completion Time [log10 Scale] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 PDF Most Flows Are Short-term All Long Short (a) FCT distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 Flow Length [log10 Scale] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 PDF Most Packets Are in Long Flows All Long Short (b) Flow length distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The real-world flow features distribution of short and long flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (a) Traditional flows as edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (b) Short flow aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision aggregates short flows to reduce the dense graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' the identified clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We will detail the graph pre-processing in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Malicious Traffic Detection Based on the Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We achieve unsupervised encrypted malicious traffic detection by analyzing the graph features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We identify critical vertices in the graph by solving a vertex cover problem, which ensures that the clustering based graph learning processes all edges with the minimum number of clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For each selected vertex, we cluster all connected edges according to their flow features and structural features that represent the flow interaction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision can identify abnormal edges in real time by computing the loss function of the clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We will describe the details of graph learning based detection in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' GRAPH CONSTRUCTION In this section, we present the design details of constructing the flow interaction graph that maintains interaction patterns among various flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In particular, we classify different flows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', short and long flows, and aggregate short flows, and perform the distribution fitting for long flows, respectively, for efficient graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In Section VII, we will show that the graph retains the near-optimal information for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Flow Classification In order to efficiently analyze flows captured on the In- ternet, we need to avoid the dependency explosion among flows during the graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We classify the collected flows into short and long flows, according to the flow size distribution [25] (see Figure 2), and then reduce the density of the graph (shown in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 2 shows the distribution of flow completion time (FCT) and flow length of the MAWI Internet traffic dataset [80] in Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For simplicity, we use the first 13 × 106 packets to plot the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' According to the figure, we observe that only 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='52% flows have FCT > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' However, 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='70% packets in the dataset are long flows with only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='36% proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Inspired by the observation, we apply different flow collection strategies for the short and long flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We poll the per-packet information from a data-plane high- speed packet parsing engine and obtain their source and des- tination addresses, port numbers, and per-packet features, in- cluding protocols, lengths, and arrival intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' These features 0 10 20 30 40 50 60 70 80 90 100 Number of Buckets [10 Bytes] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='10 PDF Centralized Distribution Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Num: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='64 Bucket Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (a) Number of packet length buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Packet Length Bucket Size [log10 Scale] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 PDF High Utilization Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Size: 333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='96 Bucket Size (b) Maximum bucket size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The number and size of the buckets for feature distribution fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' can be extracted from both encrypted and plain-text traffic for generic detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We develop a flow classification algorithm to classify the traffic (see Algorithm 1 in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' It main- tains a timer TIME NOW, a hash table that uses HASH(SRC, DST, SRC PORT, DST PORT) as key and the collected flows indicated by the sequences of their per-packet features as values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' It traverses the hash table every JUDGE INTERVAL sec- ond according to TIME NOW and judges the flow completion when the last packet arrived before PKT TIMEOUT second of TIME NOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' When the flows are completed, we classify them as long flows if the flows have more than FLOW LINE packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Otherwise, we classify them as short flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' As shown in Figure 2(b), we can accurately classify short and long flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The definitions of the hyper-parameters can be found in Table VII (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, we poll the state-less per-packet information from data-plane, while not maintaining flow states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', a state machine [89]) on the data-plane to prevent attackers manipulating the states, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', side-channel attack [65] and evading detection [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Short Flow Aggregation We need to reduce the density of the graph for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' As shown in Figure 3(a), the graph will be very dense for analysis if we use traditional four-tuple flows as edges, which is similar to the dependency explosion problem in provenance analysis [83], [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We observe that most short flows have almost the same per-packet feature sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For instance, the encrypted flows of repetitive SSH cracking attempts originated from specific attackers [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, we perform the short flow aggregation to represent similar flows using one edge after the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We design an algorithm to aggregate short flows (see Algorithm 2 in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' A set of flows can be aggregated when all the following requirements are satisfied: (i) the flows have the same source and/or destination addresses, which implies similar behaviors generated from the addresses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (ii) the flows have the same protocol type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (iii) the number of the flows is large enough, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', when the number of the short flows reaches the threshold AGG LINE, which ensures that the flows are repetitive enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Next, we construct an edge for the short flows, which preserves one feature sequence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', protocols, lengths, and arrival intervals) for all the flows, and their four-tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' As a result, four types of edges associated with short flows exist on the graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', source address aggregated, destination address aggregated, both addresses aggregated, and without aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, a vertex connected to the edge can denote a group of addresses or a single address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 3 compares the graph using traditional flows as edges and our aggregated graph by using the real-world back- bone traffic dataset, which is same to that used in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The diameter of a vertex indicates the number of addresses denoted 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Number of Bytes [log10 Scale] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 PDF Small Components Short Flow Long Flows (a) Component size distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 PCA Decomposed Features 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Outlier Components (b) Scatter of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The statistical features of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' by the vertex and the depth of the color indicates the repeated edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In Figure 3(b), we observe that the algorithm reduces 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='94% vertices and 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='04% edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The edge highlighted in green indicates short flows (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='38 Kpps, from PH) exploit- ing a vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, the flow aggregation reduces the storage overhead, which makes it feasible to maintain the in-memory graph for realtime detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Feature Distribution Fitting for Long Flows Now we use histograms to represent the per-packet feature distributions of a long flow which avoid preserving their long per-packet feature sequences, since the features in long flows are centrally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, we maintain a hash table to construct the histogram for each per-packet feature sequence in each long flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' According to our empirical study, we set the buckets widths for packet-length and arrival interval as 10 bytes and 1 ms, respectively, to trade off between the fitting accuracy and overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We calculate the hash code by dividing the per-packet features by the bucket width and increase the counter indexed by the hash code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Finally, we record the hash codes and the associated counters as the histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, the coarse-grained flow statistics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', numbers of packets [36], are insufficient for encrypted malicious traffic detection [76], which also lose the flow interaction information [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 4 shows the number of the used buckets and the maximum bucket size for the long flows in the same dataset shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We confirm the centralized feature distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', most packets in the long flows have similar packet lengths and arrival intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, in Figure 4(a), we fit the distribution of packet length using only 11 buckets on average, and most of the buckets collect more than 200 packets (see Figure 4(b)), which demonstrate that the histogram based fitting is effective with low storage overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Similarly, the fitting for arrival interval uses 121 buckets on average and realizes 71 packets per bucket high utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Besides, we use the same method for protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We use the mask of protocols as the hash code and use smaller numbers of buckets to realize more efficient fitting due to the limited number of protocol types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, Flowlens [5] used a similar histogram to efficiently utilize hardware flow tables on P4 switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Instead, we construct the histograms to accurately analyze long flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' GRAPH PRE-PROCESSING In this section, we pre-process the flow interaction graph to identify key components and pre-cluster the edges, which can enable realtime graph learning based detection against encrypted malicious traffic with unknown patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Connectivity Analysis To perform the connectivity analysis of the graph, we obtain the connected components by using depth-first search 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 PCA Decomposed Long Flow Features 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 Adjacent Edges Edge Features (a) Adjacent long flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 PCA Decomposed Short Flow Features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 (b) Adjacent short flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The sparsity of edges in the graph feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Selected \uf050 \uf050 \uf050 Flows in a component \uf04f \uf04f \uf04f Calculate the subset of vertices Cluster the edges for selected vertices Degree = 6 Degree = 3 Degree = 5 Identify the edges denoting attacks Benign Benign Malicious \uf050 \uf050 \uf04f Selected Selected Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Critical vertices identification via solving the vertex cover problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (DFS) and split the graph by the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 5(a) presents the size distribution of the identified components of the MAWI traffic dataset [80] collected in Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We observe that most components contain few edges with similar interaction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, we perform a clustering on the high- level statistics for the connected components to capture the abnormal components that have over one order of magnitude clustering loss than normal components as clustering outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, we extract five features to profile the components, including: (i) the number of long flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (ii) the number of short flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (iii) the number of edges denoting short flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (iv) the number of bytes in long flows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' and (v) the number of bytes in short flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We perform a min-max normalization and acquire the centers using the density based clustering, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', DBSCAN [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For each component, we calculate the Euclidean distance to its nearest center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We detect an abnormal component when its distance is over the 99th percentile of all the distances based on our empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 5(b) shows an instance of the clustering, where the diameters indicate the scale of the traffic on the components (in the unit of bytes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We observe that most components are small, and a high ratio of huge components is classified as abnormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' All edges associated with the normal components are labeled as benign traffic, and the edges associated with the abnormal components will be further processed by the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Edge Pre-Clustering Now we further need to process and pre-cluster the graph for efficient detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' As shown in Figure 5, the abnormal components in the graph have massive vertices and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In particular, we cannot directly apply graph representation learn- ing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', graph neural network (GNN), for realtime detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 6 shows the edges from the components in the graph structural feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We observe that the distribution of the edges is sparse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', most edges are adjacent to massive similar edges in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' To utilize the sparsity, we perform a pre-clustering using DBSCAN [32] that leverages KD-Tree for efficient local search and select the cluster centers of the identified clusters to represent all edges in each cluster to reduce the overhead for graph processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, we extract eight and four graph structural features (see Table V in Appendix A) for the edges associated with short and long flow, respectively, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the in-degree of 5 the source vertex of an edge associated with a long flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' These degree features of malicious traffic are significantly distinct from the benign ones, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the vertices denoting spam bots have higher out-degrees than benign clients due to their frequent interactions with servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Then, we perform a min-max normalization for the features, and adopt a small search range ϵ and a large minimum number of points for DBSCAN clustering (see Section VIII-A for the setting of hyper-parameters) to avoid including irrelevant edges in the clusters, which may incur false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, some edges cannot be clustered and should be treated as outliers, which will be processed as clusters with only one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' MALICIOUS TRAFFIC DETECTION In this section, we detect encrypted malicious traffic by identifying abnormal interaction patterns on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In particular, we cluster edges connected to the same critical vertex and detects outliers as malicious traffic (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Identifying Critical Vertices To efficiently learn the interaction patterns of the traffic, we do not perform clustering for all edges directly but clus- ter edges connected to critical vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For each connected component, we select a subset of all vertices in the connected component as the critical vertices according to the following conditions: (i) the source and/or destination vertices of each edge in the component are in the subset, which ensures that all the edges are connected to more than one critical vertices and clustered at least once;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' and (ii) the number of selected vertices in the subset is minimized, which aims to minimize the number of clustering to reduce the overhead of graph learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Finding such a subset of vertices is an optimization problem and equivalent to the vertex cover problem [33], which was proved to be NP Complete (NPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We select all edges and all vertices on each component to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' And we reformulate the problem to a Satisfiability Modulo Theories (SMT) problem that can be effectively solved by using Z3 SMT solver [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Since we pre-cluster the massive edges and reduce the scale of the problem (see Section V-B), the NPC problem can be solved in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Edge Feature Clustering for Detection Now we cluster the edges connected to each critical vertex to identify abnormal interaction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In this step, we use the structural features in Section V-B, and the flow features extracted from the per-packet feature sequences of short flows or the fitted feature distributions of long flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' All features are shown in Table V (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We use the lightweight K-Means algorithm to cluster the edges associated with short and long flows, respectively, and calculate the clustering loss that indicates the degree of maliciousness for malicious flow detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' losscenter(edge) = min Ci∈{C1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=',CK} ||Ci − f(edge)||2, (1) losscluster(edge) = TimeRange(C(edge)), (2) losscount(edge) = log2(Size(C(edge)) + 1), (3) loss(edge) =αlosscenter(edge) −βlosscluster(edge) + γlosscount(edge), (4) where K is the number of obtained cluster centers, Ci is the ith center, f(edge) is the feature vector, C(edge) contains all edges in the cluster of edge produced by pre-clustering, and TimeRange calculates the time range covered by the flows denoted by the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' According to Equation (4), the loss has three parts: (i) losscenter in (1) is the Euclidean distance to the cluster centers which indicates the difference from other edges connected to the critical vertex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (ii) losscluster in (2) indicates the time range covered by the cluster identified by the pre-clustering in Sec- tion V-B which implies long lasting interaction patterns tend to be benign;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (iii) losscount in (3) is the number of flows denoted by the edges, which means a burst of massive flows implies malicious behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, we used weights: α, β, γ to balance the loss terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Finally, it detects the associated flows as malicious when the loss function of the edge is larger than a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' THEORETICAL ANALYSIS In this section, we develop a theoretical analysis frame- work, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', flow recording entropy model, to analyze the in- formation preserved in the graph of HyperVision for graph learning based detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The detailed analysis can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Information Entropy Based Analysis We develop the framework that aims to quantitatively eval- uate the information retained by the exiting traffic recording modes, which decide the data representations for malicious traffic detection, by using three metrics: (i) the amount of information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the average Shannon entropy obtained by recording one packet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (ii) the scale of data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the space used to store the information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (iii) the density of information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the amount of information on a unit of storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' By using this framework, we model the graph based traffic recording mode used by HyperVision as well as three typical types of flow recording modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', (i) idealized mode that records and stores the whole per-packet feature sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (ii) event based mode (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', Zeek) that records specific events [2], [20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' and (iii) sampling based mode (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', NetFlow) that records coarse- grained flow information [8], [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We model a flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', a sequence of per-packet features, as a sequence of random variables represented by an ape- riodic irreducible discrete-time Markov chain (DTMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Let G = {V, E} denote the state diagram of the DTMC, where V is the set of states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the values of the variables) and E denotes the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We define s = |V| as the number of different states and use W = [wij]s×s to denote the weight matrix of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' All of the weights are equal and normalized: ∀ 1 ≤ i, j, m, n ≤ s, (wij =wmn) ∨ (wij = 0 ∨ wmn = 0), wi = s � j=1 wij, 1 = s � i=1 wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (5) The state transition is performed based on the weights, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the transition probability matrix P = [Pij], Pij = wij/wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Therefore, the DTMC has a stationary distribution µ: 6 � µP = µ, 1 = �s j=1 µj ⇒ µj = wj, ∀ 1 ≤ j ≤ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (6) Assume that the stationary distribution is a binomial distri- bution with the parameter: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 ≤ p ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 to approach Gaussian distribution with low skewness: µ ∼ B(s, p) App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' −→ N(sp, sp(1 − p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (7) Based on the distribution, we obtain the entropy rate of the DTMC which is the expected Shannon entropy increase for each step in the state transition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the expected Shannon entropy of each random variable in the sequence, (using nat as unit, 1 nat ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='44 bit): H[G] = s � i=1 µi s � j=1 pij ln 1 pij = − s � i=1 s � j=1 wij ln wij + s � j=1 wj ln wj = ln |E| − 1 2 ln 2πsep(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (8) Moreover, for the real-world flow size distribution, we as- sume that the length of the sequence of random variables obeys a geometric distribution with high skewness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', L ∼ G(q) with a parameter: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 ≤ q ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' H, L, and D denote the expectation of the metrics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the amount of information, the scale of data, and the density, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Idealized Recording Mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The idealized recording mode has infinite storage and captures optimal fidelity traffic information by recording each random variable from the sequence without any processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, the obtained information entropy of the idealized mode grows at the entropy rate of the DTMC: HIdeal = E[LH[G]] = 1 q ln |E| − 1 2q ln 2πsep(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (9) According to data processing inequality [85], the infor- mation retained in the idealized recording mode reaches the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' It implies that processing of the observed per- packet features denoted by the random variables may incur information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In the following sections, we will show that the other mode incurs information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We can obtain the scale of data and the density of infor- mation for the idealized recording mode as follows: LIdeal = E[L] = 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (10) DIdeal = HIdeal LIdeal = H[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (11) Graph Based Recording Mode of HyperVision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision applies different strategies to process short and long flows for the graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Let K denote the threshold for classi- fying the flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' When L < K, it collects all random variables from the sequence for short flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Otherwise, it collects the histogram to fit the distribution for long flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Then, we can obtain the lower bound to estimate the information entropy in the graph of HyperVision: HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 1 − (Kq + 1)(1 − q)K q H[G] + 1 4s(1 − q)K [(1 + s) lnps + 2 ln 2πe + 2q ln K − 2s(1 + p + γ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (12) We can also obtain the expected data scale and the density: LH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = s(1 − q)K + 1 − (Kq + 1)(1 − q)K Cq , (13) where C is the average number of flows denoted by an edge associated with short flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' LH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (14) Sampling Based Recording Mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Similarly, the sampling based mode extracts and records flow statistics for the de- tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We analyze the accumulative statistics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' the total number of bytes) that are widely adopted [19], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Let ⟨s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', sL⟩ denote the sequence of random variables, and XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = �L i=1 si indicates the flow statistic to be recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We can obtain a tight lower bound as an estimation for the amount of information and the other metrics as follows: HSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = H[XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='] = 1 2 ln 2πesp(1 − p) + ln 2 2 q(1 − q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (15) LSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (16) DSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = HSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (17) Event Based Recording Mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The event based recording mode inspects each random variable in the sequence and records events with a small probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Since the observation that the event based methods do not generate repetitive events for a long flow with a larger s, for simplicity, we assume that the probability is ps ∝ 1/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Then, we can obtain the concise closed-form solution of the amount of information, the scale of data, and the density of information for event based recording mode as follows: HEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = −2θ ln θ, (18) where θ = ζ η, ζ = q − qps, and η = q − ps(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' LEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = −ps η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (19) DEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 2ζ ps ln θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (20) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Analysis Results We perform numerical studies to compare the flow record- ing modes in real-world setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We select three per-packet features: protocol, length, and the arrival interval (in ms) as the instances of the DTMC, then we measure the parameters of the DTMC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', |E| and |V| according to the first 106 packets in the MAWI dataset on Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020 [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We also measure K, C, and estimate the geometric distribution parameter q via the second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We have the following three key results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 7 Length Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 DTMC Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 Entropy [nat] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Ideal H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Samp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (a) The entropy of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Length Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 DTMC Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 Data Scale [Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 Ideal H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Samp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (b) The data scale of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Length Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 DTMC Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 Density [nat / record] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Ideal H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Samp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (c) The density of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Length Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 DTMC Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 Density Increase [H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' - Ideal] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 (d) The density improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The traffic information retained by different recording modes on the feasible region of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 DTMC Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' p 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Entropy [nat] HyperVision Ideal Mode (a) Fix q and leave p as variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='90 Flow Length Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' q 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 Entropy [nat] HyperVision Ideal Mode (b) Fix p and leave q as variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision approaches the idealized flow recording mode on information entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' THE INTEGRAL OF THE DENSITY IN THE FEASIBLE REGION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Per-Packet Features Packet Length Time Interval Protocol Type �� F DIdeal(p, q)dpdq 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='011▼32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='10% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='918▼32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='00% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='795▼32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='51% �� F DSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (p, q)dpdq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='965▼35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='17% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='963▼28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='66% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='800▼32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='08% �� F DEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (p, q)dpdq 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='588▼60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='51% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='588▼56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='44% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='588▼50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='08% �� F DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (p, q)dpdq 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='489▲47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='27% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='350▲35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='51% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='178▲48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='18% (1) HyperVision maintains more information using the graph than the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 8 shows the results on the feasible region (F = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 ≤ p ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 ≤ q ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We observe that HyperVision maintains at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='37 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='34 times information entropy than traditional flow sampling and event based flow recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, the traditional detection methods cannot retain high-fidelity flow interaction informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Actually, they only analyze the features of a single flow, which can be evaded by encrypted traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' According to Figure 8(b), HyperVision has 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='69% data scale of the sampling based mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' It implies that the data scale is the key challenge for the existing methods to utilize flow interaction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We well address this issue by using the compact graph for maintaining the interactions among flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (2) HyperVision maintains near-optimal information using the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' According to Figure 8(a), we observe that the informa- tion maintained by the graph almost equals to the theoretical optimum, with the difference ranging from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 × 10−9 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' When the parameter of the geometric distribution of L approaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9, the flow information loss is larger because of the increasing ratio of long flows that incur more information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 9 compares the information in HyperVision and the idealized system when q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='59 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We have similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The gaps between the graph mode and the optimal mode are only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='056 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (3) HyperVision has higher information density than the ex- isting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 8(c) shows that HyperVision realizes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='46, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='54, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='39 times information density than the existing methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Although the idealized system realizes the optimal amount of traffic information, the density is only 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='55% of HyperVision in the worst case, as shown in Figure 8(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' From Table II, we find that, for all kinds of per- packet features, HyperVision can increase the density ranging between 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='51% and 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='27% due to the different recording strategies for short and long flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In summary, the flow interaction graph provides high- fidelity and low-redundancy traffic information with obvious flow interaction patterns, which ensures that HyperVision achieves realtime and unsupervised detection, particularly, detecting encrypted malicious traffic with unknown patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' EXPERIMENTAL EVALUATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Experiment Setup Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We prototype HyperVision with more than 8,000 Line of Code (LOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The prototype is compiled by gcc 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 and cmake 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We use DPDK [37] version 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 encapsulated by libpcap++ [63] version 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='05 to implement the high-speed data-plane module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The graph construction module maintains the graph in memory for realtime detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The graph learning module detects the encrypted malicious traffic on the interaction graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' It uses DBSCAN and K-Means in mlpack [57] (version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2) for clustering and Z3 SMT Solver [55] (version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8) to identify the critical vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We deploy HyperVision on a testbed built upon DELL servers (PowerEdge R410, produced in 2012) with two Intel Xeon E5645 CPUs (2 × 12 cores), Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 (Linux 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0), Docker 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7, 24GB memory, one Intel 82599ES 10 Gb/s NIC, and two Intel 850nm SFP+ laser ports for optical fiber connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We configure 6GB huge page memory for DPDK (3GB/NUMA Node) and bind 8 threads on 8 physical cores for 16 NIC RX queues to parse the per-packet features from high-speed traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We use 8 cores for in-memory graph construction, and 7 cores are used for graph learning, the rest one core is used as DPDK master core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We use real-world backbone network traffic datasets from the vantage-G of WIDE MAWI project [80] in AS2500, Tokyo Japan, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' ∼ Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020 as background traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The vantage transits traffic from/to its BGP peers and providers using 10 Gb/s fiber linked to its IXP (DIX-IE), and the traffic is collected using port mirroring, which is consistent with our threat model and the physical testbed described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We remove the attack traffic with obvious patterns in the background traffic dataset according to the rules defined by the existing studies [22], [43], [66], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', traffic will be detected as scanning traffic if it has scanned over 10% IPv4 addresses [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We generate the malicious traffic by constructing real attacks or replaying the existing traces in our testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, we collect malicious traffic in our virtual private cloud (VPC) with 8 more than 1,500 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We manipulate the instances to per- form attacks according to the real-world measurements [22], [24], [40], [42], [43], [54], [66] and the same settings in the existing studies [11], [26], [41], [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We classify 80 new datasets used in our experiments (see Table VI for details) into four groups, three of which are encrypted malicious traffic: Traditional brute force attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Although HyperVision fo- cuses on encrypted traffic, we generate 28 kinds of tradi- tional flooding attacks to verify its generic detection and the correctness of baselines including 18 high-rate and 10 low-rate attacks: (i) the brute scanning with the real packet rates [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (ii) the source spoofing DDoS with various rates [40];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (iii) the amplification attacks [43];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (iv) probing vulnerable applications [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We collected the traffic in our VPC to avoid interference with real services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Encrypted flooding traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Different from the brute force flooding, the encrypted flooding is generated by repetitive attack behaviors which target specific applications: (i) the link flooding generates encrypted low-rate flows, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the low-rate TCP attacks [44], [52] and the Crossfire attack [41], to congest links;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (ii) injecting encrypted flows that exploits protocol vulnerabilities by flooding attack traffic and inject packets into the channel [11], [26], [28];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (iii) the password cracking performs slow attempts to hijack the encrypted communication protocols [39], [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We perform SSH crack- ing in the VPC with the scale of SSH servers in the ASes reachable to AS2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Encrypted web malicious traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Web malicious traffic is normally encrypted by HTTPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We collect the traffic gener- ated by seven widely used web attacks including automatic vulnerabilities discovery (including XSS, CSRF, various injections) [64], SSL vulnerabilities detection [53], and crawlers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We also collect the SMTP-over-TLS spam traffic that lures victims to visit the phishing sites [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Malware generated encrypted traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The traffic of malware campaigns is low-rate and encrypted, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', malware compo- nent update or delivery [9], command and control (C&C) channel [8], and data exfiltration [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We use the malware infection statistics published in 2020 [42] and probed active addresses from the adopted vantage [23], [59] to estimate the number of visible victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We use the same number of instances to replay public malware traffic datasets [13], [73] to mimic malware campaigns, which is similar to the existing study [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The malicious traffic is replayed with the background traffic datasets on the physical testbed simultaneously according to their original packet rates [80] which is the same as the existing studies [30], [47], [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, each dataset contains 12∼15 million packets and the replay lasts 45s and the first 75% time does not contain malicious traffic for collecting flow interactions and training the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, the rates of the encrypted attack flows in our datasets are only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='01 ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='79 Kpps which consume only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='01% ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='72% bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We will show that these stealthy attacks evade most baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' To eliminate the impact of the dataset bias, we also use 12 existing datasets including the Kitsune datasets [56], the CIC- DDoS2019 datasets [14], and the CIC-IDS2017 datasets [15], which are collected in the real-world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' These detailed results can be found in Appendix B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In particular, the traffic in two CIC datasets [14], [15] lasts 6∼8 hours under multiple TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' THE AVERAGE ACCURACY ON THE GROUPS OF DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Method Metric Traditional Attacks Flooding Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Traffic Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Web Attacks Malware Traffic Overall Jaqen AUC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='913▼7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='782▼19% N/A1 N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='867▼12% F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='819▼16% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='495▼46% N/A N/A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='705▼26% FlowLens AUC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='939▼4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='757▼22% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='685▼30% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='768▼22% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='752▼36% F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='799▼18% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='651▼29% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='384▼59% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='411▼57% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='451▼41% Whisper AUC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='951▼3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='932▼4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='958▼2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='648▼34% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='752▼23% F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='705▼27% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='461▼50% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='546▼42% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='357▼62% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='407▼57% Kitsune AUC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='748▼24% 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='759▼22% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='751▼23% F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='419▼57% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='366▼61% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='402▼58% DeepLog AUC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='716▼27% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='621▼26% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='767▼22% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='653▼34% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='666▼32% F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='513▼47% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='508▼45% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='572▼40% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='628▼34% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='597▼37% H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' AUC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='988▲8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='974▲4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='985▲2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='993▲29% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='988▲13% F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='978▲19% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='927▲42% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='957▲67% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='970▲54% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='960▲36% 1 The results are N/A because Jaqen is designed for detection of volumetric attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2 - means that the average AUC is lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='60, which is nearly the result of random guessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' attacks, which aims to verify the long-run performances of HyperVision (see Appendix B3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, we validate the robustness of HyperVision against evasion attacks with obfus- cation techniques, which can be found in Appendix B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We use five state-of-the-art generic malicious traffic detection methods as baselines: Jaqen (sampling based recording and signature based de- tection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Jaqen [51] uses Sketches to obtain flow statistics and applies the threshold based detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We prototype Jaqen on the testbed, and adjust the signatures for each statistic and each attack to obtain the best accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' FlowLens (sampling based recording and ML based de- tection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' FlowLens [5] uses sampled flow distribution and supervised learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', random forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We use the hyper- parameter setting with the best accuracy used in the paper to retrain the ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Whisper (flow-level features and ML based detection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Whisper [30], [31] extracts the frequency domain features of flows and uses clustering to learn the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We deploy Whisper on the physical testbed without modifications and then retrain the clustering model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Kitsune (packet-level features and DL based detection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Kitsune extracts per-packet features and uses autoencoders to learn the features which is an unsupervised method [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We use its default hyper-parameters and retrain the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' DeepLog (event based recording and DL based detection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' DeepLog is a general log analyzer using LSTN RNN [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We use the logs of connections for detection and its original hyper-parameter setting to achieve the best accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, in the baselines above, we do not include DPI- based encrypted malicious traffic detection because they are unable to investigate encrypted payloads [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Also, we do not compare the task-specific detection methods [3], [76] because they cannot achieve acceptable detection accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Features in FlowLens, Kitsune, and Whisper are similar to them, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', flow features [3], packet header features [2], and time-series [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We mainly use AUC and F1 score because they are most widely used in the literature [8], [20], [30], [35], [56], [75], [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Also, we use other six metrics to validate the improvements of HyperVision, including precision, recall, F2, ACC, FPR, and EER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 9 TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' DETECTION ACCURACY OF HYPERVISION AND THE BASELINES ON TRADITIONAL BRUTE FORCE ATTACKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Method Metric Brute Scanning Amplification Attack Source Spoofing DDoS ICMP NTP SSH SQL DNS HTTP HTTPS NTP DNS CharG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' SSDP RIPv1 Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' CLDAP SYN RST UDP ICMP Jaqen AUC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9706 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9851 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9988 0.' metadata={'source': 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ROC of detecting HTTP scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Precision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Recall Jaqen FlowLens Whisper Kitsune DeepLog H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (c) PRC of detecting NTP DDoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Precision 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Recall Jaqen FlowLens Whisper Kitsune DeepLog H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (d) PRC of detecting SYN DDoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' ROC and PRC of HyperVision and all the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Hyper-parameter Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We conduct four-fold cross val- idation to avoid overfitting and hyper-parameter bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specif- ically, the datasets are equally partitioned into four subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Each subset is used once as a validation set to tune the hyper-parameters via the empirical study and the remaining three subsets are used as testing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Finally, four results are averaged to produce final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, our ablation study shows that the different threshold settings incur at most 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2% accuracy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Therefore, the hyper-parameter selection has limited impacts on the detection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Accuracy Evaluation Table III summarizes the detection accuracy and the im- provements of HyperVision over the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In gen- eral, HyperVision achieves average F1 ranging between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='927 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='978 and average AUC ranging between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='974 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='993 on the 80 datasets, which are 35% and 13% improvements over the best accuracy of the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In 44 datasets, none of the baselines achieves F1 higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='80, which means that they are not effective to detect the attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Due to the page limits, we do not show the failed detection results of these baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Traditional Brute Force Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' First, we measure the performance of the baselines by using the flooding attacks with short flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Although HyperVision is designed for encrypted malicious traffic detection, we find that it can also detect tra- ditional attacks accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The results are shown in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='992 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='999 AUC and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='929 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='999 F1, which achieves at most 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3% improvement of F1 and AUC over the best performance of the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The ROC and PRC results are illustrated in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' According to Figure 10(a) and 10(b), we observe that HyperVision has less false positives while achieving similar accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 10(c) and Figure 10(d) show that the PRC of HyperVision is largely better than the baselines, which means that it has a higher precision when all methods reach the same recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Second, by comparing HyperVision with Jaqen, we can see that HyperVision can realize higher accuracy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', a 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4% F1 improvement) than Jaqen with the best threshold set manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' That is, the unsupervised method allows reducing manual design efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, it has 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3% AUC improvement over the typical supervised ML based method (FlowLens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, we assume that HyperVision cannot acquire labeled datasets for training, which is more realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Also, it outper- forms Whisper with 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6% AUC, which is an unsupervised detection in high-speed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We observe that Kitsune and DeepLog have lower accuracy because they cannot afford high- speed backbone traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Third, we measure the detection accuracy of probing vulnerable applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' As shown in Figure 11, we see that HyperVision can detect the low-rate attacks with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='920 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='994 F1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='916 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='999 AUC under 6 ∼ 268 attackers with 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 ∼ 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 Kpps total bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' It also achieves at most 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8% F1 and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3% AUC improvements over the baselines that have a more significant accuracy decrease than the high-rate attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For example, FlowLens only achieves averagely 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='684 F1, which is only 77% under the high-rate attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Although Jaqen can be deployed on programmable switches, its thresholds are invalided by the low-rate attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' And Whisper is unable to detect the attacks with two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, Kitsune and DeepLog cannot detect the attacks because of the low rate of malicious packets (≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The reason why HyperVision can detect the slow probing while maintaining the similar accuracy to the high-rate attacks is that the graph preserves flow interaction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Although the flows from a single attacker are slow, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', at least 244 pps, HyperVision can record and analyze their interaction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, each flow in the stealthy attack traffic can be represented by an edge in the graph, while the vertices in the graph indicate the addresses generating the traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, the 10 SMTP NetBios Telnet VLC SNMP RDP HTTP DNS ICMP SSH Jaqen FlowLens Whisper Kitsune DeepLog H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9790 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9944 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9791 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9323 (b) F1 of detecting probing vulnerable application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Heatmap of accuracy for probing vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' traffic can be captured by identifying vertices with large out- degrees (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', a large number of edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, the brute force attacks validate that our method is effective to capture the DDoS traffic because it utilizes the short flow aggregation to construct the edge associated with short flows and avoids inspecting each short spoofing flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Besides, the experiment results also show that the critical vertices denote the addresses of major active flows, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', web servers, DNS servers, and scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, we exclude the results of the baselines that cannot detect encrypted traffic with lower rates in the following sections due to the page limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Encrypted Flooding Traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 12 shows the detection accuracy under flooding attacks using encrypted traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Gen- erally, HyperVision achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='856 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='981 F1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='917 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='998 AUC, which are 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7% and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3% accuracy im- provements over the baselines that can detect such attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, as shown in Figure 12(a) and 12(b), we observe that HyperVision can accurately detect the link flooding traffic consists of various encrypted traffic with different parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For instance, it can detect the Crossfire attack using HTTPS web requests generated by different sizes of botnets [41] with at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='939 F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The massive web traffic generated by bots, which is low-rate (≤ 4Kbps) and encrypted, evades the detection of Whisper and FlowLens (F1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' As shown in Figure 14(a), HyperVision can detect the attack efficiently by splitting the botnet clusters into a single connected component to exclude the interference from the similar benign web traffic, where the inner layer denotes botnets and the outer denotes decoy servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, we find that HyperVision can detect low-rate TCP DoS attacks that use burst encrypted video traffic for at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='995 AUC and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='938 F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Although Whisper has slightly better AUC in some cases, we find that it cannot achieve high accuracy on all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' As a result, it has only 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5% AUC in the worse case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, HyperVision can aggregate the short flows in the SSH connection injection attacks and achieves more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='95 F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The attacks exploiting protocol vulnerabilities realize low-rate packet injection and evade the detection of FlowLens (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', AUC ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='774, F1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='513).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 12(c) and 12(d) illustrate that HyperVision can identify slow and persisted password attempts for the channels Size 100 Size 200 Size 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2s Burst 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5s Burst 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0s Burst ACK Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' IPID Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' IPID Port 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 AUC Crossfire Attack Low-rate TCP DoS SSH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Conn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Injection Flowlens Whisper H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (a) AUC of detecting encrypted link-flooding and encrypted channel injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Size 100 Size 200 Size 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2s Burst 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5s Burst 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0s Burst ACK Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' IPID Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' IPID Port 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 F1 Crossfire Attack Low-rate TCP DoS SSH Conn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Injection (b) F1 of detecting encrypted link-flooding and encrypted channel injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 35 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 257 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 486 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 19 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 43 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 83 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Victim 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 AUC SSH Telnet (c) F1 of password cracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 35 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 257 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 486 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 19 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 43 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 83 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Victim 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 F1 SSH Telnet (d) AUC of password cracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Detection accuracy of encrypted flooding traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Padding Oracle XSS [Xsssniper] SSL Vul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [SSLScan] Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [Commix] Code Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [Commix] Agent Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [Commix] CVE- 2014-6271 CVE- 2013-2028 CSRF [Bolt] Crawler [Scrapy] Spam [1 Bot] Spam [50 Bots] Spam [100 Bots] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='00 AUC Whisper Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' AUC H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' AUC Whisper H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (a) AUC of detecting encrypted web attack traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Padding Oracle XSS [Xsssniper] SSL Vul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [SSLScan] Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [Commix] Code Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [Commix] Agent Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [Commix] CVE- 2014-6271 CVE- 2013-2028 CSRF [Bolt] Crawler [Scrapy] Spam [1 Bot] Spam [50 Bots] Spam [100 Bots] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='00 F1 Whisper Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' F1 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' F1 Whisper H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (b) F1 of detecting encrypted web attack traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Accuracy of encrypted web attack traffic detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (a) Crossfire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (b) SSH cracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (c) XSS detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (d) P2P botnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Subgraph with various encrypted malicious traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' with over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='881 F1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='917 AUC, which are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='19 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='28 times improvements over FlowLens and Whisper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The reason is that HyperVision maintains the interaction patterns of attackers using the graph, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', the massive short flows for login attempts shown as red edges in Figure 14(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 11 Magic Trickster Plankton Penetho Zsone CCleaner Feiwo Mobidash Adload WebComp Koler Svpeng Ransombo Wannalocker Dridex BitCoinM TrojanM CoinMiner THBot Emotet Snojan Trickbot Sality Mazarbot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='00 AUC Spyware Adware Ransomeware Miner Botware Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' AUC AUC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='00 F1 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' F1 F1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision can detect various encrypted malware traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 10 15 20 25 30 35 40 45 50 Throughput [Gb/s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='10 PDF Avg: 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 Gb/s Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020 (a) Graph construction throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 10 20 30 40 50 60 70 Maximum Throughput [Gb/s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='02 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (b) Max construction throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 0 50 100 150 200 250 300 Throughput [Gb/s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Throughput of graph construction and detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} 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1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 PDF Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020 Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020 (a) Graph construction latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Flow Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Long Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Short Flow Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Long Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Short 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Latency [s] Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020 Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020 (b) Construct latency composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Latency [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 PDF 99th Percentile Avg: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='82 s Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020 (c) Graph detection latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Total Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Identify Pre Cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Critical Vertex Cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 Ltency [10x s] (d) Detection latency composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Latency of graph construction and detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Encrypted Web Malicious Traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 13 presents the detection accuracy of the encrypted traffic generated by various web vulnerabilities discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='985 average AUC and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='957 average F1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8% and 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2% increase compared to Whisper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The flow based ML detection cannot detect web encrypted malicious traffic because the traf- fic has single-flow patterns that are almost same to benign web access flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision can accurately detect the encrypted web malicious traffic, because, as shown in Figure 14(c), it captures the traffic from the frequent interactions as the edges associated with long flows, and identifies the malicious traffic (denoted by red edges) generated by the attacker (denoted by the green vertex) by clustering the edges associated with benign web traffic that are connected to the same critical vertex (denoted by the red solid vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Encrypted Malware Traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We show the detection accuracy of encrypted malware traffic in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, the 0 10 20 30 40 50 60 70 80 90 100 Time [s] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Memory Usage [GB] Overall Graph (a) Runtime memory usages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Head Suri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Zeek H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Head Suri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Zeek H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Storage Usage [10x MB] Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020 Benign Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020 RST DoS (b) Graph storage usages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Hardware resource usages of HyperVision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' encrypted malware traffic is hard to detect for the baselines because it is slow and persistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' However, HyperVision ac- curately detects the malware campaigns with at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='964 AUC and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='891 F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, it captures the C&C servers of spyware for exfiltration as abnormal critical vertices that are connected by massive infected hosts in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' As a result, it detects the encrypted malicious traffic of the malware with at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='942 F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For example, to detect Sality P2P botnet shown in Figure 14(d), HyperVision collects the interactions among similar P2P bots, aggregates the encrypted short flows as edges, and finally clusters the edges with higher loss than benign interaction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Similarly, it can capture the static servers of adware, malware component delivery servers, the infected miner pools as abnormal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, the low-rate malicious flows (at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='814 pps) are represented as the edges associated with short flows connected to critical vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Meanwhile, the massive long flows with almost 100% encrypted packet proportion are represented as the edges associated with long flows to the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Therefore, a critical vertex connected with the edges indicates the malware campaign that is significantly different from benign vertices with large degrees, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', benign websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Performance Results Throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We truncate the packets to the first 200 bytes on the physical testbed and increase the sending rates until the graph construction module reaches maximum throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 16 shows the throughput of the graph construction and the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 16(a) presents the distribution of average throughput within a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0s time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We observe that HyperVision constructs the graph for 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='21 Gb traffic per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 16(b) presents the maximum throughput in each time window with all the backbone traffic datasets used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision achieves 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='43 ∼ 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='71 peak throughput on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, we measure the throughput of the graph learning module, which inspects flow interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' According to Figure 16(c), we observe that it can analyze 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='14 Gb traffic per second on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, the de- tection throughput is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 times higher than the construction so that the detection can analyze the recorded traffic iteratively to consider the past interaction information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We observe that the average throughput exhibits a bimodal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The peak of low throughput (around 75 Gb/s) is caused by lacking the information on the graph for analyzing during cold start stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 12 Figure 16(d) illustrates the throughput when the performance of the system is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We observe that it achieves 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 ∼ 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 Gb/s throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, the throughput on Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' and Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2020 datasets is lower because of their low original traffic volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We measure the latency caused by graph construction and detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 17(a) presents the PDF of the maximum latency for constructing each edge within a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0s window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We observe that HyperVision has 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='09s ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='04s average construc- tion latency with an upper bound of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='93s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The distribution is a significant bimodal one because the receive side scaling (RSS) on the Intel NIC is unbalanced on the threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The light-load threads have only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='75s latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We analyze the composition of the latency in Figure 17(b) (where the error bar is 10th and 90th percentile) and find that the flow classification, short flow aggregation, and long flow distribution fitting share 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='95%, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='03%, and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0% latency, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We measure the average detection latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 17(c) shows that the learning module has a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='83s latency on average with a 99th percentile of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='48s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We also analyze the latency in each step (see Figure 17(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We see that 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8% of the latency comes from pre-clustering (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='66s on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' However, the pre-clustering step reduces the processing overhead of the subsequent processing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', selecting critical vertex and clustering, for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 × 10−3s (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='64%) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4 × 10−3s (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='40%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Resource Consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 18(a) presents the memory usage of HyperVision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, the DPDK huge pages require 6GB memory and thus we measure the consumption when the usage reaches 6GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We observe that the increasing rate of memory for maintaining the graph is only 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 MB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Finally, HyperVision utilizes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='78 GB memory to maintain the flow interaction patterns extracted from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='82 TB ongoing traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision incurs low memory consumption because the fea- ture distribution fitting for long flow and short flow aggregation make the in-memory graph compact which ensures low-latency detection and long-term recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, the memory consumption of the learning algorithm is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='452 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='619 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision can export the graph to disk for forensic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Figure 18(b) shows the storage used for recording the first 45s traffic of the MAWI dataset by different methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', HyperVision, event based network monitors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', Suricata [74] and Zeek [86]), and raw packet headers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We observe that HyperVision achieves 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='99%, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7%, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1% storage reduction over the baselines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Meanwhile, our analysis shows that HyperVision retains more traffic information than the existing tools (see Section VII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, the graph based analysis is more efficient than these existing tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' RELATED WORK Graph Based Anomaly Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Graph based structures have been used for task-specific traffic detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' These meth- ods heavily rely on DPI and thus cannot be applied to detect encrypted traffic [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' analyzed the download re- lationship graph to identify malware downloading [45], which is similar to WebWitness [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Eshete et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' constructed HTTP interaction graphs to detect malware static resources [24], and Invernizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' used a graph constructed from plain- text traffic to identify malware infrastructures [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Different from these works, HyperVision constructs the interaction graph without parsing specific application layer headers and thus achieves task-agnostic encrypted traffic detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Note that, the provenance graph based attack forensic analysis [83], [87] is orthogonal to our traffic detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' DTMC Based Anomaly Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Discrete-Time Markov Chain (DTMC) has been used to model the behaviors of users/devices [1], [71], [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' These methods aim to predict behaviors of users and devices by utilizing DTMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For in- stance, Peek-a-Boo predicted user activities [1], Aegis pre- dicted user behaviors for abnormal event detection [72], and 6thSense predicted sensor behaviors for detecting sensor-based attacks [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Different to these methods, our work utilizes DTMC to quantify the benefits of building the compact graph for detecting various unknown attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' ML Based Malicious Traffic Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' ML based detec- tion can detect zero-day attacks [12] and achieve higher accuracy than the traditional signature based methods [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For example, Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' leveraged frequency domain features to realize realtime detection [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Barradas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' developed Flowlens to extract flow distribution features on data-plane and detect attacks by applying random forest [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Stealthwatch detected attacks by analyzing flow features extracted from NetFlow [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Mirsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' developed Kitsune to learn the per-packet features by adopting auto-encoders [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For task- specific methods, Nelms et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [60], Invernizzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [38], and Bilge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [8] detected traffic in the different stages of malware campaigns by using statistical ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Bartos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [6] and Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [75] detect malformed HTTP request traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Holland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [35] developed an automatic pipeline for traffic detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' All these methods cannot effectively detect attacks based on encrypted traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Task-Specific Encrypted Traffic Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The existing encrypted traffic detection relies on domain knowledge for short-term flow-level features [2], [16], [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For example, Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' leveraged SDN to achieve crossfire attack de- tection [90], and Xing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' designed the primitives for the programmable switch to detect link flooding attacks [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For encrypted malware traffic, Bilge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' [8] leveraged the traffic history to detect C&C server, and Tegeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' developed supervised learning using time-scale flow features extracted from malware binaries [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' studies the feasibility of detecting malware encrypted communication via malformed TLS headers [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' To the best of our knowledge, our HyperVision is the first system that enables unsupervised detection for the encrypted traffic with unknown patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Encrypted Traffic Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision aims to iden- tify the malicious behaviors according to encrypted traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' It is different from encrypted traffic classifications that decide if the traffic is generated by certain applications or users [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For instance, Rimmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' leveraged DL for web fingerprint, which de-anonymizes Tor traffic by classifying encrypted web traffic [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Siby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' showed that classifying encrypted DNS traffic can jeopardize the user privacy [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Similarly, Bahramali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' classified the encrypted traffic of instant mes- saging applications [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Ede et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' designed semi-supervised learning for mobile applications fingerprinting [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' All these classifications are orthogonal to HyperVision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 13 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' CONCLUSION In this paper, we present HyperVision, an ML based realtime detection system for encrypted malicious traffic with unknown patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision utilizes a compact in-memory graph to retain flow interaction patterns, while not requiring prior knowledge on the traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, HyperVision uses two different strategies to represent the interaction patterns generated by short and long flows and aggregates the informa- tion of these flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We develop an unsupervised graph learning method to detect the traffic by utilizing the connectivity, sparsity, and statistical features in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, we establish an information theory based analysis framework to demonstrate that HyperVision preserves near-optimal informa- tion of flows for effective detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The experiments with 92 real-world attack traffic datasets demonstrate that HyperVision achieves at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='86 F1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='92 AUC with over 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 Gb/s detection throughput and average detection latency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='83s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' In particular, 44 out of the attacks can evade all five state- of-the-art methods, which demonstrate the effectiveness of HyperVision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Acar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} 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here: detecting DPI evasion attacks with context learning,” in CoNEXT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' ACM, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 183–197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Details of Implementations We present the details of the flow classification and short flow aggregation algorithm in Algorithm 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The features used for edge pre-clustering and clustering are shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' And Table VII shows the hyper-parameters used in HyperVision and the recommended values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 15 TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' THE FEATURES OF EDGES USED IN HYPERVISION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Edge Group Data Description Edge Denoting Short Flows structural bool Denoting short flows with the same source address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' bool Denoting short flows with the same source port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' bool Denoting short flows with the same destination address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' bool Denoting show flows with the same destination port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The in-degree of the connected source vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The out-degree of the connected source vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The in-degree of the connected destination vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The out-degree of the connected destination vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' statistical int The number of flows denoted by the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The length of the feature sequence associated with the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The sum of packet lengths in the feature sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The mask of protocols in the feature sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' float The mean of arrival intervals in the feature sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Edge Denoting Long Flows structural int The in-degree of the connected source vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The out-degree of the connected source vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The in-degree of the connected destination vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The out-degree of the connected destination vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' statistical float The flow completion time of the denoted long flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' float The packet rate of the denoted long flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The number of packets in the denoted long flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The maximum bin size for fitting packet length distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The length associated with the maximum bin size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The maximum bin size for fitting protocol distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' int The protocol associated with the maximum bin size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' DETAILS OF MALICIOUS TRAFFIC DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Class Dataset Label Description Att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 Vic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 Enc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Ratio Malware Related Encrypted Traffic Spyware Magic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Magic Hound spyware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2 479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='13% Trickster Encrypted C&C connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2 793 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='63 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0% Plankton Pulling components from CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3 579 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8% Penetho Wifi cracking APK spyware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 516 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='57 100% Zsone Multi-round encrypted uploads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 479 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='98 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0% CCleaner Unwanted software downloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 4 466 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='09% Adware Feiwo Encrypted ad API calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='00K 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 100% Mobidash Periodical statistic ad updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3 624 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='08 100% WebComp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' WebCompanion click tricker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3 281 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='38 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2% Adload Static resources for PPI adware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 280 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='09% Ransom- ware Svpeng Periodical C&C interactions (10s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2 403 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='26% Koler Invalid TLS connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3 333 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='22 100% Ransombo Executable malware downloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 5 369 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7% WannaL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Wannalocker delivers components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2 275 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='49 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3% Dridex Victim locations uploading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 429 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='10 100% Miner BitCoinM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Abnormal encrypted channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='54K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='79 100% TrojanM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Long SSL connections to C&C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='37K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='39 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4% CoinM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Periodical connections to pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='40K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='21 100% Botware THBot Getting C&C server addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 4 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='72 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='71% Emotet Communication to C&C servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='17K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='43 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6% Snojan PPI malware downloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3 326 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='94 100% Trickbot Connecting to alternative C&C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 4 347 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='57 100% Mazarbot Long C&C connections to cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3 409 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='13 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9% Sality A P2P botware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 20 247 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='19 100% Encrypted Flooding Traffic Link Flooding CrossfireS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We use the botnet cluster sizes and the ratio of decony servers (HTTPS) in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 100 313 197 100% CrossfireM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 200 313 278 100% CrossfireL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 500 313 503 100% LrDoS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 We use the traffic of an encrypted video application and the settings in WAN experiments [44] 1 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='57 100% LrDoS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 1 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='25 100% LrDoS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='90 100% SSH Inject ACK Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' SSH injection via ACK rate-limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='78 IPID Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' SSH injection via IPID counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='28 IPID Port Requires of the SSH injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='83 Password Cracking Telnet S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Telnet servers in AS38635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='63 100% Telnet M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Telnet servers in AS2501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='70 100% Telnet L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Telnet servers in AS2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 83 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='76 100% SSH S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' SSH servers in AS9376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='39 100% SSH M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' SSH servers in AS2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 257 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='49 100% SSH L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' SSH servers in AS2501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 486 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='53 100% Encrypted Web Traffic Web Attacks Oracle TLS padding Oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='99 100% XSS Xsssniper XSS detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 100% SSLScan SSL vulnerabilities detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 100% Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Commix parameter injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 100% Cookie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Commix cookie injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 100% Agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Commix agent-based injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 100% WebCVE Exploiting CVE-2013-2028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='30 100% WebShell Exploiting CVE-2014-6271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 100% CSRF Bolt CSRF detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='73 100% Crawl A crawler using scrapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 100% SMTP SSL Spam1 Spam using SMTP-over-SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='2 100% Spam50 Encrypted spam with 50 bots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 50 1 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 100% Spam100 Brute spam using 100 bots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 100 1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9 100% Traditional Brute Force Attack Brute Scanning ICMP We use the brute force scanning rates identified by darknet in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We reproduce the scan using Zmap which targets the peers and customers of AS 2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 211K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='61 NTP 1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3K 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='87 SSH 1 205K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='79 SQL 1 112K 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='04 DNS 1 198K 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='61 HTTP 1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='68 HTTPS 1 209K 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='89 Source Spoof SYN We use the protocol types and the packet rates in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='50K 1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='41 RST 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5K 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='79 UDP 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='50K 1 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 ICMP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='20K 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='13 Amplification Attack NTP We use the packet rates and the vulnerable protocols observed in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' And we use the number of the reflectors in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 650 1 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 DNS 200 1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 CharGen 200 1 175 SSDP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='30K 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='23 RIPv1 500 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='04 Memcache 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='60K 1 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 CLDAP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='30K 1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8 Probing Vulnerable Application Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' SMTP We use the sending rates of vulnerable application discovery disclosed by a darknet [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We estimate the number of scanners by the number of visible active addresses from the vantage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', realword measurements) and the size of the darknet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 11 158K 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='97 Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='NetBios 28 444K 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='Telnet 156 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='23M 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='VLC 22 352K 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='SNMP 6 110K 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='51 Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='RDP 172 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='30M 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='HTTP 94 640K 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='DNS 28 428K 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='ICMP 268 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='82M 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3 Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='SSH 72 994K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='63 1 Att.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' and Vic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' indicate the number of attackers and victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' is short for total bandwidth in the unit of Mb/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' RECOMMENDED HYPER-PARAMETER CONFIGURATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Group Hyper-Parameter Description Value Graph Construction PKT TIMEOUT Flow completion time threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0s FLOW LINE Flow classification threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 15 AGG LINE Flow aggregation threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 20 Graph Pre- Processing ϵ DBSCAN hyper-parameters for clustering components and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 4 × 10−3 minPoint 40 Traffic Detection K K-means hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 10 T Loss threshold for malicious traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0 α Balancing the terms in the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1 β 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 γ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7 Algorithm 1: Secure flow classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Input: Per-packet features: PktInfo, the hash table for flow collecting: FlowHashTable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Output: Classified flows: ShortFlow and LongFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 time now := PktInfo[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='time, last check := time now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2 for pkt in PktInfo do // Aggregate packets into flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3 if Hash(pkt) not in FlowHashTable then 4 FlowHashTable adds an entry for pkt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 5 FlowHashTable[Hash(pkt)] appends pkt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 6 if time now − last check > JUDGE INTERVAL then 7 for flow in FlowHashTable do // Judge the completion of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 8 if time now − flow[−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='time > PKT TIMEOUT then // Classify the flow via the number of packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 9 if flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='size > FLOW LINE then 10 ShortFlow adds flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 11 else 12 LongFlow adds flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 13 FlowHashTable clears the states of flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 14 last check ← time now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' // Record the time of checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 15 time now ← pkt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' // Update the timer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Algorithm 2: Short flow aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Input: Short flows: ShortFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Output: Constructed edges: ShortEdge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 1 Initialize ProtoHashTable as an empty table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' // Select candidate protocols for the aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2 for flow in ShortFlow do // Calculate the protocol mask of a short flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3 flow proto := (flow[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='proto|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|flow[−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='proto).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 4 if Hash(flow proto) not in ProtoHashTable then 5 ProtoHashTable adds an entry for flow proto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 6 Append flow to ProtoHashTable[Hash(flow proto)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' // Perform the source aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 7 for flows in ProtoHashTable with same protocols do 8 SrcAddrTable collects the flows with same sources in flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 9 for sflow in SrcAddrTable do // The flows can be aggregated and denoted by one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 10 if sflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='size > AGG LINE then 11 edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='features := sflow[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 12 edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='source := sflow[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 13 if an unique destination in sflow then // Source and destination aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 14 edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='destination saves the unique destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 15 else // Source aggregation only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 16 Record each destination in sflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 17 Add the constructed edge to ShortEdge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 18 SrcAddrTable evicts sflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 19 DstAddrTable collects flows with same destinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 20 Inspect the flows with the same destinations similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' // Process short flows which cannot be aggregated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 21 ShortEdge adds flows in SrcAddrTable and DstAddrTable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 16 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Details of Experiments 1) Details of Datasets: We present the detailed properties of the 80 newly collected datasets in Table VI, including the number of attackers and victims, the packet rates of attack flows, and the ratios of encrypted traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' All the datasets are collected and labeled using the same method as MAWI datasets [80] and CIC datasets [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2) Detection Accuracy of Other Datasets: We use 12 existing datasets to eliminate the impact of dataset bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Overall, HyperVision achieves 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='8%, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='0%, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='1% F1 im- provements over the best accuracy of the baselines on Kitsune datasets [56], CIC-IDS2017 datasets [15], and CIC-DDoS2019 datasets [14], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' From the Kitsune datasets, we validate the correctness of the deployed baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 3) Long-run Performances: By using the CIC datasets [14], [15], we validate the long-run performances of HyperVision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, the experiments show that HyperVision achieves over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='95 F1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='99 AUC in long-run detection (6∼8 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The results also verify that the accumulation of detection errors cannot interfere with HyperVision, and HyperVision can detect multiple attacks simultaneously even in the presence of attacks with changed addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, the memory consumption of the compact graph is bounded by 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='6 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 4) Robustness Against Obfuscation Techniques: We vali- date our method under evasion attacks with different obfusca- tion techniques according to a recent study [30], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', injecting three kinds of benign traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The results demonstrate that the accuracy decrease incurred by the obfuscation is bounded by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='3% F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Specifically, when benign TLS traffic, UDP video traffic, and normal ICMP traffic is injected into brute force attack traffic, the average F1 decreases by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='7%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='9%, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='4%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The reason why the obfuscation techniques incur negligible accuracy decrease is that they only manipulate patterns of a single flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision can still detect the evasion attacks by learning the interaction patterns among various flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Details of Theoretical Analysis 1) Analysis of Event based Mode: Let random variable IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' indicate if the event based mode records an event for a flow denoted by a random variable sequence, ⟨s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' , sL⟩, L ∼ G(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' And we assume that the mode can merge repetitive events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' First, we obtain the probability distribution of the random variable IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' : P[IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 1] = 1 − P[IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 0], P[IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 0] = ∞ � l=1 P[L = l] · P[IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 0|L = l] = ∞ � l=1 (1 − q)l−1 · q · (1 − ps)l = q(1 − ps) 1 − (1 − q)(1 − ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (21) Then, we obtain the entropy of the random variable IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' : HEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = H[IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='] = −P[IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 0] ln P[IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 0] − P[IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 1] ln P[IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (22) We observe that ∂H[IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='] ∂q ≈ 0 when q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, we use the second-order taylor series of q to approach HEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' : HEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 2q(1 − ps) ln[ (ps−1)q ps(q−1)−q ] ps(q − 1) − q = −2θ ln θ, (23) where θ = ζ η, ζ = q − qps, and η = q − ps(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Similarly, we obtain the expected data scale LEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' and the information density DEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' : LEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = P[IEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 1] = ps ps(1 − q) + q = −ps η , DEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = HEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' LEve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 2ζ ps · ln θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (24) Here, we complete the analysis for the event based mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 2) Analysis of Sampling based Mode: We use XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' to denote the random variable to be recorded as the flow information in the sampling based mode which is the sum of the observed per-packet features denoted by the random variable sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We can obtain the distribution of XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' as follows: XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = L � i=1 si, si ∼ B(s, p) ⇒ XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' ∼ B(Ls, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (25) The amount of the information recorded by the sampling based mode is the Shannon entropy of XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='. We decompose the entropy as conditional entropy and mutual information: HSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = H[XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='] = H[XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L] + I(XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (26) We assume that the mutual information between the se- quence length L and the accumulative statistic XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' is close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' It implies the impossibility of inferring the statistic from the length of the packet sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Then we obtain a lower bound of the entropy as an estimation which is verified to be a tight bound via numerical analysis: � � � HSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = H[XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L] = ∞ � l=1 P[L = l] · H[XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L = l] H[XSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L = l] = 1 2 ln 2πelsp(1 − p), ⇒ HSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 1 2 ln 2πesp(1 − p) + q 2 ∞ � l=1 (1 − q)l−1 ln l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (27) We observed that the second-order taylor series can accu- rately approach the second term of the entropy: HSamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = 1 2 ln 2πesp(1 − p) + ln 2 2 q(1 − q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (28) Finally, we obtain the expected data scale and the informa- tion density similar to the analysis for the event based mode and complete the analysis for the sampling based mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 17 3) Analysis of Graph based Mode in HyperVision: Hyper- Vision applies different recording strategies for short and long flows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', when L > K it extracts the histogram for long flow feature distribution fitting, and when L ≤ K it records detailed per-packet features and aggregates short flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Let XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' denote the random set of the recorded information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' For short flows, all the random variables are collected in XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='. For long flows, XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' collects s counters of the histogram for each state on the state diagram of the DTMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' First, we decompose the entropy of the graph based recording mode as the terms for short and long flows: HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = H[XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L] = ∞ � l=1 P[L = l] · H[XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L = l] = H[X S H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L] + H[X L H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L] (29) � � � � � � � H[X S H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L] = K � l=1 P[L = l] · H[XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L = l] H[X L H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L] = ∞ � l=K+1 P[L = l] · H[XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L = l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Short Flow Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' HyperVision records detailed per- packet feature sequences for short flows which is the same as the brute recording in the idealized mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Thus, the increasing rate of information equals the entropy rate of the DTMC: H[XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L = l] = l · H[G], (30) H[X S H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L] = K � l=1 P[L = l] · l · H[G] = q · H[G] · K � l=1 (1 − q)l−1 · l = 1 − (Kq + 1)(1 − q)K q H[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (31) Long Flow Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' When L > K, the random set collects the counters for distribution fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' When the DTMC has s states, the histogram has s counters υ1, υ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' , υs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=', XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = {υ1, υ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' , υs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' We assume that the counters are independent: υi = L � j=1 δj, δj = � 1, if sj is the ith state 0, else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (32) We observe that ⟨υ1, υ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' , υs⟩ is a binomial process: υi ∼ B(L, P[si = i]) ∼ B(L, Ci spi(1 − p)s−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (33) To obtain the closed-form solution, we use (sp)ie−sp i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' as an estimation of Ci spi(1−p)s−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, the length of the per- packet feature sequence of a long flow is relatively large which implies υi approaches a Poisson distribution: υi ∼π(L · P[si = i]) ∼π(λi), λi = (sp)ie−sp i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (34) Basing on the distribution of the collected counters, we obtain the entropy of the random set: � � � H[υi|L = l] = 1 2 ln 2πel (sp)ie−sp i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' H[X L H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L = l] = s� i=1 H[υi|L = l], (35) H[X L H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L] = ∞ � l=K+1 P[L = l] · H[X L H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L = l] = ∞ � l=K+1 q(1 − q)l−1 · s � i=1 1 2 ln 2πel (sp)ie−sp i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = (1 − q)K 2 [s ln 2πe + s(s + 1) 2 ln sp − sp2 − s � i=1 ln i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='] + qs 2 [ ∞ � l=K+1 (1 − q)l−1 ln l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' The assumption of q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='5 implies Kth order taylor series can accurately approach the last term in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Moreover, we utilize the quadric term of s in the taylor series of �s i=1 ln i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' to approach the entropy of long flows (γ is Euler–Mascheroni constant): H[X L H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L] = 1 4s(1 − q)K[(1 + s) ln ps+ 2 ln 2πe + 2q ln K − 2s(1 + p + γ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' (36) Finally, we take (31) and (36) in (29) and complete the analysis for the entropy of the graph based recording mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Similarly, we obtain the expected data scale by analyzing the conditions of short and long flows separately: LH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = E[LS H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L] + E[LL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='|L] = K � l=1 P[L = l] · L C + ∞ � l=K+1 s · P[L = l] = s(1 − q)K + 1 − (Kq + 1)(1 − q)K Cq , (37) where C is the average number of flows denoted by an edge associated with short flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' Also, we obtain the expected information density by its definition: DH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' = HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='/LH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' and complete the analysis for the graph based recording mode used by HyperVision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNFRT4oBgHgl3EQf9DhY/content/2301.13686v1.pdf'} diff --git a/WdE0T4oBgHgl3EQfmAG-/content/tmp_files/2301.02494v1.pdf.txt b/WdE0T4oBgHgl3EQfmAG-/content/tmp_files/2301.02494v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..86ede0fea6bbc51d12d8f437fafa21966ca4ba9c --- /dev/null +++ b/WdE0T4oBgHgl3EQfmAG-/content/tmp_files/2301.02494v1.pdf.txt @@ -0,0 +1,1627 @@ +Task Aware Feature Extraction Framework for +Sequential Dependence Multi-Task Learning +Xuewen Tao∗ +xuewen.txw@mybank.cn +MYbank, Ant Group +Beijing, China +Mingming Ha∗ +hamingming_0705@foxmail.com +School of Automation and Electrical +Engineering, University of Science +and Technology Beijing; MYbank, Ant +Group +Beijing, China +Xiaobo Guo† +xb_guo@bjtu.edu.cn +Institute of Information Science, +Beijing Jiaotong University, Mybank, +Ant Group, +Beijing, China +Qiongxu Ma +qiongxu.mqx@mybank.cn +MYbank, Ant Group +Shanghai, China +Hongwei Cheng +chw286885@mybank.cn +MYbank, Ant Group +Shanghai, China +Wenfang Lin +moxi.lwf@mybank.cn +MYbank, Ant Group +Hangzhou, Zhejiang, China +Abstract +Multi-task learning (MTL) has been successfully implemented +in many real-world applications, which aims to simultane- +ously solve multiple tasks with a single model. The gen- +eral idea of multi-task learning is designing kinds of global +parameter sharing mechanism and task-specific feature ex- +tractor to improve the performance of all tasks. However, +sequential dependence between tasks are rarely studied but +frequently encountered in e-commence online recommen- +dation, e.g. impression, click and conversion on displayed +product. There is few theoretical work on this problem and +biased optimization object adopted in most MTL methods +deteriorates online performance. Besides, challenge still re- +mains in balancing the trade-off between various tasks and +effectively learn common and specific representation. In this +paper, we first analyze sequential dependence MTL from +rigorous mathematical perspective and design a dependence +task learning loss to provide an unbiased optimizing object. +And we propose a Task Aware Feature Extraction (TAFE) +framework for sequential dependence MTL, which enables to +selectively reconstruct implicit shared representations from +a sample-wise view and extract explicit task-specific infor- +mation in an more efficient way. Extensive experiments on +∗These authors contributed equally to this research. +†Xiaobo Guo is the corresponding author. +Permission to make digital or hard copies of all or part of this work for +personal or classroom use is granted without fee provided that copies are not +made or distributed for profit or commercial advantage and that copies bear +this notice and the full citation on the first page. Copyrights for components +of this work owned by others than ACM must be honored. Abstracting with +credit is permitted. To copy otherwise, or republish, to post on servers or to +redistribute to lists, requires prior specific permission and/or a fee. Request +permissions from permissions@acm.org. +Conference’17, July 2017, Washington, DC, USA +© 2022 Association for Computing Machinery. +ACM ISBN 978-1-4503-XXXX-X/18/06...$15.00 +https://doi.org/XXXXXXX.XXXXXXX +offline datasets and online A/B implementation demonstrate +the effectiveness of our proposed TAFE. +CCS Concepts: • Information systems → Information +systems applications; Computational advertising; • Multi- +task Learning; +Keywords: Recommender System, Sequential Dependency, +Multi-Task Learning, Representation Learning +ACM Reference Format: +Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei +Cheng, and Wenfang Lin. 2022. Task Aware Feature Extraction +Framework for Sequential Dependence Multi-Task Learning. In +Proceedings of ACM Conference (Conference’17). ACM, New York, +NY, USA, 13 pages. https://doi.org/XXXXXXX.XXXXXXX +1 +Introduction +Multi-task learning (MTL) has been successfully applied es- +pecially for online recommendation in various real-world +scenarios such as E-commerce or financial service. MTL aims +to simultaneously learn multiple tasks in a single model and +has been proved to have obvious advantages compared with +single-task learning [4], since it is able to provide comple- +mentary information by implicitly passing the message be- +tween different tasks. [21], [3]. Click-through rate (CTR), +conversion rate (CVR) and click-through & conversion rate +(CTCVR) are the typical optimizing object on the industrial +recommendation indicating the a customer acquisition pro- +cess. These behaviors are strictly sequential dependent on +the previous one especially for fiance service and we illus- +trate a multi-step conversion example in Figure 1. A customer +will convert through stages of Impression → Click → Autho- +rize → Conversion. Conversion behaviors such like applying +loans, make deposit or purchasing investment products are +only permitted after an authorization. Series works from +ESMM [13] to ESCM2 [25] pay more attention on the unbi- +ased CVR estimation problem in a view of causality to correct +arXiv:2301.02494v1 [cs.LG] 6 Jan 2023 + +Conference’17, July 2017, Washington, DC, USA +Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei Cheng, and Wenfang Lin +Conversion +Stages +Impression +Click +Authorization +Loan +Finance +Figure 1. An illustration of a multi-step conversion in Fi- +nance Service. +sample selection bias. The dependent relation between the +tasks like CTR and CVR is implicitly implied via the dis- +tribution of sample space. Recently, [26] captures the task +dependency through the information transfer between dif- +ferent conversion steps and combines a calibrator to further +constrain the dependent relationship. However, the depen- +dency between each steps is still not deeply defined and +discussed in most MTL works from a theoretical format. +Besides, as mentioned above, MTL methods improve pre- +diction result through an information passing mechanism +between tasks, which suggests improper feature sharing will +result in even poorer or imbalanced performance in different +tasks known as a negative transfer phenomenon. Therefore, +general approach in MTL mainly focuses on designing kinds +of information extraction modules (experts) to learn com- +mon and task-specific representations. Such as Cross-Stitch +Network [14] and Sluice Network [17] employ a linear combi- +nation to leverage representations of different tasks but also +require much more training parameters. SOTA method of +Multi-gate Mixture-of-Experts (MMoE) approach [12] adopts +an ensemble of experts submodules and gating network to +model task relationships while consuming less computation. +Progressive Layered Extraction (PLE) [19], separates task- +common and task-specific parameters explicitly which could +further avoid parameter conflicts caused by complex task +correlation. These approaches assign individual parameters +to each task to better exploit task information and improve +model generalization. Nonetheless, feature expressivity with +respect to each task is still limited since task-irrelevant in- +formation passing from the shared structure and more fine- +grained representation learning is necessary. +In this paper, we first provide a formal definition of MTL +on sequential dependence problem, and propose an optimiz- +ing object paradigm for recover the dependent relationship +based on theoretical proof. And we also present a novel MTL +framework called Task Aware Feature Extraction (TAFE) +to selectively and dynamically enhance the representation +learning for respective tasks along with the dependency- +based object. TAFE consists of two main modules: Adaptive +Sample-wise Representation Generator (ASRG) and explicit +Task-Specific Adapter (TSA). ASRG employs a dynamic se- +lection mechanism to learn the hierarchical feature inter- +action from a sample-wise view to further separates the +task-irrelevant information. The implement of explicit TSA +enables fine-grained feature learning by introducing task- +specific indicator vectors. In a summary, main contributions +of this paper are presented as follows: +• The sequential dependence multi-task learning prob- +lem is first formulated, and its connections and differ- +ences with the general multi-task learning problem are +illustrated. Moreover, the distribution dependence re- +lationship between the adjacent task spaces is revealed +from a theoretical perspective. +• We present a multi-task learning framework named +TAFE for selectively fine-grained feature representa- +tion learning from a sample-wise view. ASRG and TSA +modules within TAFE adatively reconstruct the im- +plicit shared representations and extract explicit task- +specific information in an more efficient way. +• Extensive experiments on public and real-world indus- +trial dataset are conducted to evaluate the effectiveness +of TAFE. Experiment results demonstrate that our pro- +posed approach outperforms the state-of-the-art MTL +methods. Furthermore, we explore the boundary of +TAFE in real-world industrial applications to prove its +efficiency for large-scale online recommendations. +2 +Preliminaries +In this section, the multi-task learning with sequential de- +pendence and the data distribution discrepancy of different +tasks are discussed. Then, from the expected loss’s point +of view, the distribution relationship between the adjacent +tasks is revealed. +2.1 +Problem Formulation +Consider a SDMTL problem over an input space X and a set +of task {T𝑖}𝑁 +𝑖=1, where 𝑁 is the number of tasks and the cor- +responding task spaces are denoted as {T1, . . . , T𝑁 }. A large +dataset of data points {𝑥𝑗,𝑜1 +𝑗, . . . ,𝑜𝑁 +𝑗 }𝑀 +𝑗=1 are given, where 𝑀 +is the number of data points and 𝑜 𝑗 +𝑖 ∈ {0, 1} corresponding +to a binary classification problem or 𝑜 𝑗 +𝑖 ∈ R for a regression +problem is the label of the 𝑖-th task for the 𝑗-th data point. +Differing from the general MTL problem, for the SDMTL +problem, there exists the sequential dependence relationship +between tasks in the sense that the current task T𝑖 depends +on the previous task T𝑖−1, i.e., T𝑖−1 → T𝑖. Let 𝑋 and 𝑇𝑖 be +the random variables over the input space X and task out- +put space T𝑖, respectively. In this paper, for convenience of +analysis, each task is set as a binary classification task. As +mentioned in literature [15] and shown in Fig. 1, for sequen- +tial dependence, one of core properties is that if the event +𝑇𝑖−1 is not triggered, then the event 𝑇𝑖 must not occur, i.e., + +夫Task Aware Feature Extraction Framework for Sequential Dependence Multi-Task Learning +Conference’17, July 2017, Washington, DC, USA +𝑃(𝑇𝑖 = 1, 𝑃𝑖−1 = 0|𝑋) = 0, where 𝑃(·|·) denotes the condi- +tional probability. Therefore, according to this property, the +random variables 𝑇𝑖 satisfies +𝑃(𝑇𝑖 = 1|𝑋) = +∑︁ +𝑡𝑖−1,...𝑡1∈{0,1} +𝑃(𝑇𝑖 = 1,𝑇𝑖−1 = 𝑡𝑖−1, . . . ,𝑇1 = 𝑡1|𝑋) += 𝑃(𝑇𝑖 = 1,𝑇𝑖−1 = 1, . . . ,𝑇1 = 1|𝑋), +𝑃(𝑇𝑖 = 0|𝑋) = +∑︁ +𝑡𝑖−1,...𝑡1∈{0,1} +𝑃(𝑇𝑖 = 0,𝑇𝑖−1 = 𝑡𝑖−1, . . . ,𝑇1 = 𝑡1|𝑋), +(1) +which implies that the positive samples of𝑇𝑖 are derived from +the positive samples of the task𝑇𝑖 while the negative samples +consist of the negative samples of tasks 𝑇𝑖,𝑇𝑖−1, . . . ,𝑇1 due to +the sequential dependence. +In addition, the sequential dependence relationship is also +embodied in the constraints with respect to the conversion +probabilities of the adjacent tasks. In [26], the sequential +dependence relationship is formalized as +𝑃(𝑇1 = 1|𝑋) ≥𝑃(𝑇2 = 1,𝑇1 = 1|𝑋) +· · · +≥𝑃(𝑇𝑖−1 = 1, . . . ,𝑇1 = 1|𝑋) +≥𝑃(𝑇𝑖 = 1,𝑇𝑖−1 = 1, . . . ,𝑇1 = 1|𝑋). +(2) +Then, a behavioral expectation calibrator is introduced into +AITM [26] to guarantee the sequential dependence relation- +ship (2). When the outputs of the model violate this condi- +tion, the designed loss will output a positive penalty term. +However, the condition (2) cannot completely reflect the +dependence relationship between tasks. Reconsidering the +dependence relationship between 𝑃(𝑇𝑖−1 = 1, . . . ,𝑇1 = 1|𝑋) +and 𝑃(𝑇𝑖 = 1, . . . ,𝑇1 = 1|𝑋), it leads to +𝑃(𝑇𝑖−1 = 1|𝑋) − 𝑃(𝑇𝑖 = 1|𝑋) += 𝑃(𝑇𝑖−1 = 1, . . . ,𝑇1 = 1|𝑋) − 𝑃(𝑇𝑖 = 1, . . . ,𝑇1 = 1|𝑋) += 𝑃(𝑇𝑖−1 = 1, . . . ,𝑇1 = 1|𝑋) +× +� +1 − 𝑃(𝑇𝑖 = 1|𝑇𝑖−1 = 1, . . . ,𝑇1 = 1,𝑋) +� += 𝑃(𝑇𝑖−1 = 1, . . . ,𝑇1 = 1|𝑋)𝑃(𝑇𝑖 = 0|𝑇𝑖−1 = 1, . . . ,𝑇1 = 1,𝑋) += 𝑃(𝑇𝑖 = 0,𝑇𝑖−1 = 1, . . . ,𝑇1 = 1|𝑋). +(3) +Therefore, the dependence relationship between the adjacent +tasks needs to satisfy the equality constraints (24). +Define a parametric hypothesis class per task as 𝑓𝑖 (𝑥;𝜃𝑠,𝜃𝑖) : +X → T𝑖, where 𝜃𝑠 and 𝜃𝑖 are shared parameters and task- +specific parameters of the task 𝑖. Also, the task-specific loss +function is defined as 𝐿𝑖 (·, ·) : T𝑖 × T𝑖 → R+. Similar to the +general MTL problem, the objective of SDMTL is to minimize +the following expected loss: +min +𝜃𝑠,𝜃1,...,𝜃𝑖 +𝑁 +∑︁ +𝑖=1 +𝐸𝑋,𝑇1,...,𝑇𝑁 ∼O[𝑤𝑖𝐿𝑖 (𝑓𝑖 (𝑋;𝜃𝑠,𝜃𝑖),𝑇𝑖)] +s.t. 𝑓𝑖 (𝑋;𝜃𝑠,𝜃𝑖) − 𝑓𝑖−1(𝑋;𝜃𝑠,𝜃𝑖) += 𝑃(𝑇𝑖 = 0,𝑇𝑖−1 = 1, . . . ,𝑇1 = 1|𝑋) +(4) +where O is the distribution with domain X × T1 × · · · × T𝑁 , +and 𝑤𝑖 is the static or dynamically computed weight per task. +In this case, the sequential dependence is embodied in the +expression 𝑃(𝑇𝑖 = 1|𝑇𝑖−1 = 0,𝑋) = 0. +1 +2 +3 +4 + +Figure 2. Distribution discrepancy of different task spaces +in SDMTL. The curved surface represents the distribution +O with domain X × T1 × · · · × T𝑁 . The colored circles from +the outside to the inside denote the domains of tasks T1, T2, +T3, and T4, respectively. +Considering (24), we can regarded the relationship be- +tween 𝑇𝑖 and 𝑇𝑖−1 as a new sequential dependence task. The +task-specific outputs 𝑓𝑖 (𝑥𝑗;𝜃𝑠,𝜃𝑖) of model also need to sat- +isfy this dependence relationship. +Since there exists the dependence relationship between the +previous and current tasks, i.e., T1 → T2 → . . . → T𝑁 , the +sample space of the current task depends on the previous task. +In general, the sample space of the previous task contains +the sample space of the current one as shown in Fig. 2, which +leads to the data distribution discrepancy between these two +sample spaces. Consider the general CTR, CVR and CTCVR +estimation tasks, i.e., impression→click→conversion. We +use the random variables 𝑌 ∈ {0, 1} and 𝑍 ∈ {0, 1} to denote +the click event and the conversion event, respectively. Then, +CTR, CVR and CTCVR with feature input 𝑋 are defined as +𝑃(𝑌 |𝑋), 𝑃(𝑍 |𝑌 = 1,𝑋) and 𝑃(𝑍,𝑌 = 1|𝑋), which satisfy +𝑃(𝑍 = 1,𝑌 = 1|𝑋) = 𝑃(𝑌 = 1|𝑋)𝑃(𝑍 = 1|𝑌 = 1,𝑋), +(5) +where 𝑃(·|·) denotes the conditional probability. In this case, +the training space of the traditional CVR estimation task +is generally determined by the samples with 𝑌 = 1 in the +CTR estimation task. However, for a new user, there are no +impression and click records. The conversion rate estimation +task is actually to estimate 𝑃(𝑍 = 1|𝑋). Considering 𝑃(𝑍 = +1,𝑌 = 0|𝑋) = 0 [15], we can obtain +𝑃(𝑍 = 1|𝑋) =𝑃(𝑍 = 1,𝑌 = 0|𝑋) + 𝑃(𝑍 = 1,𝑌 = 1|𝑋), +=𝑃(𝑍 = 1,𝑌 = 1|𝑋), +𝑃(𝑍 = 0|𝑋) =𝑃(𝑍 = 0,𝑌 = 0|𝑋) + 𝑃(𝑍 = 0,𝑌 = 1|𝑋). +(6) +According to (6), it is observed that the negative samples of +the conversion event 𝑍 are derived from the entire space of +the click event 𝑌. If the negative data points derived from +the space with 𝑌 = 0 is used to predict 𝑃(𝑍 = 1|𝑋), then the + +Conference’17, July 2017, Washington, DC, USA +Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei Cheng, and Wenfang Lin +data distribution discrepancy between training space and +inference space leads to inaccurate predictions. However, a +model directly learning 𝑃(𝑍 = 1|𝑋) will ignore the sequen- +tial dependence information between the click event and the +conversion event. +2.2 +Distribution Dependence Relationship Between +Inference Space and Local Space +In this subsection, the random variables 𝑇𝑖−1 and 𝑇𝑖 of ad- +jacent tasks are denoted as 𝑌 and 𝑍, respectively. In what +follows, the relationship of expected losses between domains +of adjacent tasks Y and Z is established, where Y → Z. +Considering the sequential dependence relationship between +Y and Z, the input random variable 𝑋 over the input space +X, and random variables 𝑌 ∈ {0, 1} and 𝑍 ∈ {0, 1} over the +task output spaces Y and Z, we can obtain +𝑃(𝑍 |𝑋) = +∑︁ +𝑦∈{0,1} +𝑃(𝑍,𝑌 = 𝑦|𝑋). +(7) +In tasks Y and Z, the sample space with data points {𝑥𝑗 ∈ +X,𝑜𝑌 +𝑗 +∈ {0, 1},𝑜𝑍 +𝑗 +∈ {0, 1}} is called ieference space, i.e., +entire space for Y and Z, and the sample space with data +points {𝑥𝑗 ∈ X,𝑜𝑌 +𝑗 ∈ {1},𝑜𝑍 +𝑗 ∈ {0, 1}} is called local space, +also called training space in some traditional CVR estimation +methods [13]. The distributions of reference and local spaces +are denoted as D and C, respectively. +Therefore, the objective of these two tasks Y and Z with +sequential dependence in inference space is to minimize the +following expected loss: +𝐸𝑋,𝑌,𝑍∼D [𝐿(𝑓𝑌 (𝑋),𝑌) + 𝐿(𝑓𝑍 (𝑋),𝑍)] +=𝐸𝑋,𝑌∼D [𝐿(𝑓𝑌 (𝑋),𝑌)] + 𝐸𝑋,𝑍∼D [𝐿(𝑓𝑍 (𝑋),𝑍)] += +∫ +D +𝐿(𝑓𝑌 (𝑥),𝑦)𝑃D(𝑥,𝑦)d𝑥d𝑦 ++ +∫ +D +𝐿(𝑓𝑍 (𝑥),𝑧)𝑃D(𝑥,𝑧)d𝑥d𝑧, +(8) +where 𝑃D(·, ·) is the joint distribution in inference space. On +the other hand, if the model is trained in the local space C, +then the task Y determines the sample distribution of the +task Z. With this operation, the expected loss becomes the +following form: +𝐸𝑋,𝑌∼D [𝐿(𝑓𝑌 (𝑋),𝑌)] + 𝐸𝑋,𝑍∼C[𝐿(𝑓𝑍 (𝑋),𝑍)] += +∫ +D +𝐿(𝑓𝑌 (𝑥),𝑦)𝑃D(𝑥,𝑦)d𝑥d𝑦 ++ +∫ +C +𝐿(𝑓𝑍 (𝑥),𝑧)𝑃C(𝑥,𝑧)d𝑥d𝑧, +(9) +where 𝑃C(·, ·) is the joint distribution in local space. Next, the +relationship between expected loss in (8) and (9) is revealed. +Theorem 2.1. If the expected losses in the reference and local +spaces are defined as in (8) and (9), then, for any loss function +𝐿(·, ·), they satisfy +𝐸𝑋,𝑌,𝑍∼D [𝐿(𝑓𝑌 (𝑋),𝑌) + 𝐿(𝑓𝑍 (𝑋),𝑍)] +=𝐸𝑋,𝑌∼D [𝐿(𝑓𝑌 (𝑋),𝑌)] ++ 𝐸𝑋,𝑍∼C +� +𝑃D(𝑌 = 1) +𝑃D(𝑍 |𝑋) +𝑃D(𝑍,𝑌 = 1|𝑥) 𝐿(𝑓𝑍 (𝑋),𝑍) +� +. (10) +Proof. Considering the definitions of the reference and local +spaces, and their corresponding expected losses given in (8) +and (9), we can obtain +𝑃C(𝑋,𝑍) = 𝑃D(𝑋,𝑍 |𝑌 = 1) +(11) +in the sense that the joint distribution of 𝑋 and 𝑍 in C is +equivalent to, under 𝑌 = 1, the joint distribution of 𝑋 and 𝑍 +in D. +According to (11) and the definition of 𝐸𝑋,𝑍∼D [𝐿(𝑓𝑍 (𝑋),𝑍)], +the second term in the right-hand side of (10) satisfies +𝐸𝑋,𝑍∼C +� +𝑃D(𝑌 = 1) +𝑃D(𝑍 |𝑋) +𝑃D(𝑍,𝑌 = 1|𝑋) 𝐿(𝑓𝑍 (𝑋),𝑍) +� +=𝐸𝑋,𝑍∼C +� +𝑃D(𝑌 = 1) +𝑃D(𝑌 = 1|𝑋,𝑍) 𝐿(𝑓𝑍 (𝑋),𝑍) +� += +∫ +C +𝑃D(𝑌 = 1) +𝑃D(𝑌 = 1|𝑥,𝑧) 𝐿(𝑓𝑍 (𝑥),𝑧)𝑃C(𝑥,𝑧)d𝑥d𝑧 += +∫ +D +𝐿(𝑓𝑍 (𝑥),𝑧) +𝑃D(𝑌 = 1) +𝑃D(𝑌 = 1|𝑥,𝑧) 𝑃D(𝑥,𝑧|𝑌 = 1)d𝑥d𝑧 += +∫ +D +𝐿(𝑓𝑍 (𝑥),𝑧) +𝑃D(𝑌 = 1) +𝑃D(𝑌 = 1|𝑥,𝑧) +𝑃D(𝑥,𝑧,𝑌 = 1) +𝑃D(𝑌 = 1) +d𝑥d𝑧 += +∫ +D +𝐿(𝑓𝑍 (𝑥),𝑧)𝑃D(𝑥,𝑧)d𝑥d𝑧 +=𝐸𝑋,𝑍∼D [𝐿(𝑓𝑍 (𝑋),𝑍)]. +(12) +Therefore, equations (8), (9) and (12) imply that the relation- +ship (10) holds. +□ +Obviously, the distribution shift also exists in CTR, CVR +and CTCVR estimations when they are trained in different +spaces. +3 +The Task Aware feature Extraction +Framework +The whole architecture of proposed TAFE for sequential +dependence multi-task learning is illustrated in Figure 3. +TAFE consists of two representation learning modules ASRG +and TSA to dynamically extract implicit and explicit feature +information from a sample-wise view, and a sequential de- +pendence task learning loss to reconstruct an unbiased task +relationship on a global training space. Adaptive Sample- +wise Representation Generator (ASRG) is responsible for hi- +erarchical shared-representation learning, adopting inducing +points to interact with different feature field corresponding +to each input. Task Specific Adapter (TSA) module coop- +erates with ASRG but works as a task-ware information +extractor through designed task indicator and is with an in- +dependent message passing structure to better solve the task + +Task Aware Feature Extraction Framework for Sequential Dependence Multi-Task Learning +Conference’17, July 2017, Washington, DC, USA +conflict. Besides those two, a sequence dependency learning +loss between tasks is proposed and theoretical proved, which +is able to describe the conditional dependent probability for +sequential based multi-task learning from the whole training +space and consequently improve the prediction result by +precisely capturing the task relationship. We will elaborate +ASRG and TSA in section 3.1 and 3.2, and lastly discuss the +relationship between sequence dependence tasks in section +3.3. +Task Tower 𝑖-1 +Task Tower 𝑖 +CONCAT +Sequential dependency +Learning +… +Embedding Feas +Filed 1 +Filed 2 +Filed n +… +Adaptive Sample-wise +Representation Generator (ASRG) +High-Order +Inducing Points +Task Indicator 𝑖 +Task Indicator 𝑖-1 +N +ℒ!"#$%& +ℒ'"($%& +) +~𝑃(y)|𝑥)) +ℒ'"($%& +))* +~𝑃(y)"*|𝑥)"*) +𝑦! +y!"# +ℒ!"#$%&~𝑃 ∆y) 𝑥)"*, 𝑥) , ∆y)= y)"* − y) +Task Specific +Adapter(TSA) +Task Specific +Adapter(TSA) +Figure 3. An illustration of the overall architecture of TAFE. +3.1 +Adaptive Sample-wise Representation +Generator +Fine-grained feature information extraction corresponding +to different tasks is crucial in multi-task learning and signifi- +cantly affects model performance. But feature generalization +also needs to be included to balance the trade-off between +tasks in terms of shared information. Based on these consid- +erations, we propose a novel representation learning mod- +ule, named Adaptive Sample-wise Representation Generator +(ASRG). Besides learning generalized shared-information, +we design a dynamic selector to learn the feature interac- +tion from a sample-wise view to further separates the task- +irrelevant info. The structure of ASRG is shown in Figure 4, +which mainly consists of an dynamic activation layer and a +feature interaction learning layer. +Dynamic Activation Layer. In recommendation scenario, +input field usually contains kinds of user and item features. +Adaptive Sample-wise +Representation Generator +(ASRG) +Multi-Head Attention +Add & Norm +Feature Embeddingi +Inducing Points ++ +Element-wise +Q +V +K +0.8 0.1 0.6 0.3 +0.7 0.5 0.2 0.9 +0.1 0.6 0.7 0.3 +1.0 +0.0 +0.0 +0.0 +1.0 +0.0 +0.0 +1.0 +0.0 +0.0 +1.0 +0.0 +Dynamic Activation Layer +Feature Embeddingo +𝒙 +𝒇(𝒙) +𝜸=𝟏 𝟎!𝟒 +𝒇(𝒙) +𝒙 +𝜸=𝟏 +𝒇 𝒙 = +𝟎, 𝒙 ≤ 𝜸 +𝟐 +− 𝟐 +𝜸𝟐 𝒙𝟑 + 𝟑 +𝟐𝜸 𝒙 + 𝟏 +𝟐 , − 𝜸 +𝟐 < 𝒙 < 𝜸 +𝟐 +𝟏, 𝒙 ≥ 𝜸 +𝟐 +Dynamic Selector +Dynamic Selector +Transformation Layer +Figure 4. The detail structure of Adaptive Sample-wise Rep- +resentation Generator +Given an input x from 𝐹 different feature fields, we denote x +as the concatenation of all feature fields: +x = [𝑥1,𝑥2, . . . ,𝑥𝐹], +(13) +where 𝑥𝑖 represents the value of the 𝑖-th feature. As a com- +monly data preprocssing for online recommendation sce- +nario with better generalization, we discretize numerical +features 𝑥𝑖 through a Log-round operation to get an unique +value, and randomly initialize it with a vector of 𝑑𝑓 dimen- +sion. Thus, we obtain the input embedding for each feature +field as 𝐻 = [ℎ1,ℎ2, . . . ,ℎ𝐹]T, where 𝐻 ∈ R𝐹×𝑑𝑓 . +A transformation Layer is first applied to project the input +embeddings into a 𝐾 dimension vector. The transformation +layer can be any type of deep neural network structure and +here we chose a standard MLP layer just for simplicity. The +output 𝑧𝐾 is defined as a dynamic selector: +𝑧𝐾 = MLP(𝐻) +(14) +where 𝑧𝐾 ∈ R𝐾. Then, we implement a dynamic activation +function 𝑓𝐷 inspired by [7] to get a sparser representation +of 𝑧𝐾, whose formulation is as follows: +𝑧𝐾 = 𝑓𝐷 (𝑧𝐾) +(15) + +XConference’17, July 2017, Washington, DC, USA +Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei Cheng, and Wenfang Lin +where 𝑓𝐷 is formulated as: +𝑓𝐷 (𝑧) = + + +0, +𝑧 ≤ −𝛾 +2 +− 2 +𝛾3𝑧3 + 3 +2𝛾 𝑧 + 1 +2, +−𝛾 +2 < 𝑧 < 𝛾 +2 +1, +𝑧 ≥ 𝛾 +2 +(16) +where 𝛾 = 𝑀𝑎𝑥{10 − 2𝑒-4 · 𝑠𝑡𝑒𝑝, 1𝑒-3} and maximum 𝑠𝑡𝑒𝑝 +during the training process is around 1𝑒6. Dynamic selector +𝑧𝐾 works as a information filter which selectively interacts +with input from the sample-wise view due to the Transfor- +mation Layer. As visualized in Figure 5, the output shape of +𝑓𝐷 becomes steeper with the increase of training step. By +utilizing 𝑓𝐷, 𝑧𝐾 creates a 𝐾 dimension sparse vector only con- +tains values of 0 and 1 corresponding to each input sample. +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +z +0.00 +0.25 +0.50 +0.75 +1.00 +fD(z) +Step=1 +Step=25000 +Step=100000 +Figure 5. Output of the dynamic activation function 𝑓𝐷 with +the increase of training step. +Feature Interaction Learning Layer. Attention mecha- +nism for learning hierarchical feature interaction is gen- +erally adopted but requires quadratic time complexity in +standard self-attention structure. Here we design a learnable +matrix called inducing points 𝐼, enlightened from Set Trans- +former [10] to reduce the computational complexity from +quadratic to linear. We define the inducing points 𝐼 ∈ R𝐾×𝑑𝑓 , +where 𝐾 is the same as in dynamic selector 𝑧𝐾. After a +element-wise operation, we get a modified query ˆ𝑄 as: +ˆ𝑄 = 𝐼 ⊙ 𝑧𝐾 +(17) +Then, we calculate the output 𝑂 𝑗 from the attention opera- +tion according to the following formulation: +𝑂 𝑗 = Attention( ˆ𝑄 𝑗, 𝐾𝑗,𝑉𝑗; 𝜆) +(18) +where ˆ𝑄 𝑗 = 𝐼𝑗 ⊙ 𝑧𝐾, 𝐾𝑗 = 𝐻𝑊 𝐾 +𝑗 ,𝑉𝑗 = 𝐻𝑊 𝑉 +𝑗 with trainable +parameter 𝜆 = +� +𝐼𝑗,𝑊 𝐾 +𝑗 ,𝑊 𝑉 +𝑗 +�𝑚 +𝑗=1 and 𝑚 represents the num- +ber of multi-head. Then we get output 𝑂 from the multi-head +attention with parameter 𝑊 𝑂 as: +𝑂 = concat (𝑂1, . . . ,𝑂ℎ)𝑊 𝑂 +(19) +Finally, the adaptive representation𝑌𝐴𝑆𝑅𝐺 learned from ASRG +can be formulated in a way of residual network: +𝑌𝐴𝑆𝑅𝐺 = LayerNorm(𝑂, 𝐻) +(20) +The time complexity of feature interaction learning layer +reduces from 𝑂(𝐹 2) to 𝑂(𝐾 × 𝐹) by introducing 𝐼. As sug- +gested in [16], 𝐾, the reduced dimension of 𝐼 could be viewed +as 𝐾 independent memory cells interacting with each fea- +ture field, which is further automatically selected by 𝑧𝑘 to +distinguish feature information explicitly from a sample- +wise view. Compared with traditional shared-representation +learning structure in most MTL methods, ASRG learns more +distinctive info in terms of a dynamic activation layer and +feature interaction learning layer, which is mainly attributed +to the former one combining a transformation layer and +dynamic activation function to generate an adaptive mask +corresponding to each input sample. +3.2 +Explicit Task-Specific Adapter +Besides effective shared-representation generated by ASRG, +specific feature learning according to each task will strongly +affect the model performance since it directly enhances the +task-relevant information. In most MTL works, task-targeted +feature extractors, such as the task-specific experts proposed +by PLE are deliberately designed to learn the representa- +tion for each task. However, mutual interference between +different tasks still exists since the shared and task-specific +components are not completely separated in these cases. +In order to learn the task-aware information among differ- +ent tasks with a more independent and thoroughly separated +structure, we introduce a module named explicit Task Spe- +cific Adapters (TSAs) as detail plotted in Figure 6. TSA uti- +lizes parameterized task indicator vector to interact with pre- +vious sample-wise common shared info from ASRG, which +is able to extract task-specific representation by directly op- +timizing respective task object. The approach is similarly +adopted in PAL [18] and K-adapter [23]. +Task Specific Adapter(TSA) +Attention +Task Indicator +Q +K +V +Feature Embedding o +Add & Norm +Task Aware Embedding +Feature Embeddingi +Figure 6. The detail structure of Task-Specific Adapter. +As illustrated in Figure 6, we take the output 𝑌𝐴𝑆𝑅𝐺 ∈ +R𝐾×𝑑𝑓 from the Adaptive Sample-wise Representation Gen- +erator to interact with a learnable task indicator vector 𝛼𝑖 +corresponding to each task 𝑖. The 𝐹𝑖 is the output calculated +through an attention operation between 𝑌𝐴𝑆𝑅𝐺 and 𝛼𝑖 as: +𝐹𝑖 = Attention(𝛼𝑖,𝑌𝐴𝑆𝑅𝐺,𝑌𝐴𝑆𝑅𝐺), +(21) +where 𝛼𝑖 ∈ R1×𝑑𝑓 is the task indicator vector and 𝐹𝑖 ∈ R1×𝑑𝑓 +denotes the middle output of task-aware representation for + +Task Aware Feature Extraction Framework for Sequential Dependence Multi-Task Learning +Conference’17, July 2017, Washington, DC, USA +each task 𝑖 correspondingly. Here, Attention is the same +attention calculation operation as in formula (18). Conse- +quently, for each task𝑖, we refer the task-specific information +generated from the 𝑘-th TSA layer as 𝑇 𝑘 +𝑖 and we calculate it +also through a residual network and layer normalization for +training efficiency as in (20): +𝑇 𝑘 +𝑖 = LayerNorm(𝑇 𝑘−1 +𝑖 ++ 𝐹𝑘 +𝑖 ), +(22) +where𝑇 𝑘−1 +𝑖 +∈ R1×𝑑𝑓 means the output from the previous TSA +layer for task 𝑖, and 𝐹𝑘 +𝑖 ∈ R1×𝑑𝑓 is the task-aware embedding +learned by the interaction between the task indicator for the +𝑘-th layer with task 𝑖 and shared-common embedding. Note, +𝑇 0 +𝑖 is ignored at the first iteration. +It can be observed in Figure 6, task aware embedding𝑇 ob- +tained from Task Specific Adapters is trained independently +and whose message doesn’t pass into ASRG module among +different layers. The proposed structure keeps the implicit +(from ASRG) and explicit (from TSA) representation learn- +ing modules more separated, which not only isolates the +negative interference between tasks more thoroughly but +also provides a extendable multi-task learning framework +especially necessary in industrial implementation. +3.3 +Loss Function Design Towards Sequential +Dependence Multi-Task Learning +For the multi-task learning without sequential dependence, +the loss function is generally designed as the following form: +L(𝜃𝑠,𝜃1, . . . ,𝜃𝑁 ) = 1 +𝑀 +𝑁 +∑︁ +𝑖=1 +𝑀 +∑︁ +𝑗=1 +𝑤𝑖𝐿 +� +𝑓𝑖 (𝑥𝑗;𝜃𝑠,𝜃𝑖),𝑜𝑖 +𝑗 +� +. (23) +From the loss function (23), it can be observed that this loss +function cannot learn the sequential dependence relation- +ship. As mentioned in subsection 2.1, the corresponding loss +function for SDMTL is designed as +L(𝜃𝑠,𝜃1, . . . ,𝜃𝑁 ) +=L𝑀−𝑇𝑎𝑠𝑘 + L𝐷−𝑇𝑎𝑠𝑘 +(24) += 1 +𝑀 +𝑁 +∑︁ +𝑖=1 +𝑀 +∑︁ +𝑗=1 +𝑤𝑖𝐿 +� +𝑓𝑖 (𝑥𝑗;𝜃𝑠,𝜃𝑖),𝑜𝑖 +𝑗 +� ++ 1 +𝑀 +𝑁 +∑︁ +𝑖=2 +𝑀 +∑︁ +𝑗=1 +𝐿 +� +𝑓𝑖−1(𝑥𝑗;𝜃𝑠,𝜃𝑖−1) − 𝑓𝑖 (𝑥𝑗;𝜃𝑠,𝜃𝑖),𝑜𝑖−1 +𝑗 +− 𝑜𝑖 +𝑗 +� +, +(25) +where L𝑀−𝑇𝑎𝑠𝑘 and L𝐷−𝑇𝑎𝑠𝑘 are the loss functions of the +main tasks, i.e., T𝑖, and the loss functions of the sequen- +tial dependence relationship, respectively. With this oper- +ation, each task and their corresponding dependence rela- +tionship can be trained separately. The loss functions of the +dependence relationship can be regarded as a regularization +term. Therefore, the SDMTL problem can also be considered +as a general MTL with the constraints 𝑓𝑖−1(𝑥𝑗;𝜃𝑠,𝜃𝑖−1) − +𝑓𝑖 (𝑥𝑗;𝜃𝑠,𝜃𝑖) = 𝑜𝑖−1 +𝑗 +− 𝑜𝑖 +𝑗. Note that the selection of negative +samples determines the training space. Therefore, the pos- +itive samples of the task T𝑖 are derived from the current +task while the negative samples of T𝑖 are derived from differ- +ent tasks T𝑖−1, . . . , T1. Similar to the subsection 3.2, expected +losses of the dependence relationship derived from the entire +space D and local space C are discussed as follows. +Theorem 3.1. If the dependence relationship is learned in the +entire space D and the local space C, respectively, and the cor- +responding expected losses are denoted as 𝐸𝑋,𝑌,𝑍∼D [𝐿(𝑓𝑌 (𝑋)− +𝑓𝑍 (𝑋),𝑌 −𝑍)] and 𝐸𝑋,𝑌,𝑍∼C[𝐿(𝑓𝑌 (𝑋) − 𝑓𝑍 (𝑋),𝑌 −𝑍)], then +these two expected losses satisfy +𝐸𝑋,𝑌,𝑍∼D [𝐿(𝑓𝑌 (𝑋) − 𝑓𝑍 (𝑋),𝑌 − 𝑍)] += 𝐸𝑋,𝑌,𝑍∼C +� +𝑃D(𝑌 = 1)𝑃D(𝑌 − 𝑍 |𝑥) +𝑃D(𝑌 − 𝑍,𝑌 = 1|𝑋) +× 𝐿(𝑓𝑌 (𝑋) − 𝑓𝑍 (𝑋),𝑌 − 𝑍) +� +(26) +Proof. According to Bayes’ theorem, we can obtain the fol- +lowing equality: +𝑃D(𝑌 = 1)𝑃D(𝑌 − 𝑍 |𝑥) +𝑃D(𝑌 − 𝑍,𝑌 = 1|𝑋) += +𝑃D(𝑌 = 1) +𝑃D(𝑌 = 1|𝑋,𝑌 − 𝑍) . +(27) +Considering the right-hand side of (26) and (27), it leads to +𝐸𝑋,𝑌,𝑍∼C +� +𝑃D(𝑌 = 1) +𝑃D(𝑌 = 1|𝑋,𝑌 − 𝑍) 𝐿(𝑓𝑌 (𝑋) − 𝑓𝑍 (𝑋),𝑌 − 𝑍) +� += +∫ +C +� +𝑃D(𝑌 = 1) +𝑃D(𝑌 = 1|𝑥,𝑦 − 𝑧) 𝐿(𝑓𝑌 (𝑥) − 𝑓𝑍 (𝑥),𝑦 − 𝑧) +𝑃C(𝑥,𝑦 − 𝑧) +� +d𝑥d𝑦d𝑧 += +∫ +D +� +𝑃D(𝑌 = 1) +𝑃D(𝑌 = 1|𝑥,𝑦 − 𝑧) 𝑃D(𝑥,𝑦 − 𝑧|𝑌 = 1) +× 𝐿(𝑓𝑌 (𝑥) − 𝑓𝑍 (𝑥),𝑦 − 𝑧) +� +d𝑥d𝑦d𝑧 += +∫ +D +𝐿(𝑓𝑌 (𝑥) − 𝑓𝑍 (𝑥),𝑦 − 𝑧)𝑃D(𝑥,𝑦 − 𝑧)d𝑥d𝑦d𝑧 += 𝐸𝑋,𝑌,𝑍∼D [𝐿(𝑓𝑌 (𝑋) − 𝑓𝑍 (𝑋),𝑌 − 𝑍)], +(28) +which implies that (26) holds. The proof is completed. +□ +4 +Experiments +In this section, we describe the experiments to evaluate the +performance of the proposed TAFE framework, which are +conducted on both public benchmark dataset and real-world +industrial dataset in financial service. We also analyze the +contribution of each modules consisting of TAFE to further +understand the working mechanism and demonstrate the +effectiveness of proposed method for sequential dependence +multi-task learning. + +Conference’17, July 2017, Washington, DC, USA +Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei Cheng, and Wenfang Lin +4.1 +Experimental Setup +4.1.1 +Datasets. Experiments are conducted on two dataset: +the public benchmark Ali-CCP and an industrial dataset from +the financial scenario. +• Ali-CCP dataset 1 contains 84 million samples from +an online E-commence recommendation platform in +TaoBao. 5 million users’ clicking and conversion be- +haviors are sampled from this dataset. We consider +CTR and CVR as two tasks. +• Industrial Dataset is collected from a real-world fi- +nancial service scenario based on our industrial on- +line recommendation platform, which describes users’ +preferences in financial products. There are 73 million +samples for this dataset containing 30 million users’ +records. Prediction of users’ clicking and conversion +behaviors on financial product like credit loan are two +tasks for evaluation. +4.1.2 +Baseline Methods. To validate the effectiveness of +TAFE, we conduct our experiments on the following repre- +sentative methods for comparison, which are SOTA multi- +task learning approaches or recent sequence dependency +learning method: +• Single-Task is a three-layer MLP network with hid- +den layer size of [256,128,64] for single-task optimiza- +tion. +• Shared-Bottom constructs a shared bottom layer to +learn the common representation across all tasks and +introduces separated task tower for the object opti- +mization respectively. +• MMOE is inspired by the classic MoE method which +adopts a group of shared bottom subnetworks as ex- +perts and introduces gating network assigning differ- +ent tasks with distinctive weights. +• PLE generalizes CGC method and employs a progres- +sive routing mechanism to extract and separate deeper +semantic knowledge. +• AITM is a shared-bottom structure with adaptive in- +formation transfer for modeling sequential dependency +among multi-step conversions. +• TAFE is our proposed approach which adopts adaptive +sample-wise representation generator and the explicit +task-specific adapter, with dependency learning loss +for the sequence dependence multi-task learning. +4.1.3 +Implementation of L𝐷−𝑇𝑎𝑠𝑘. As discussed in sec- +tion 3.3, L𝐷−𝑇𝑎𝑠𝑘 can be regarded as a regularization, which +constraints the relationship between sequential dependence +tasks during training process. In this paper, we constructs +a MSE (Mean-Squared Loss) as an implementation for 𝐿 in +formula (24): +1https://tianchi.aliyun.com/dataset/dataDetail?dataId=408 +Table 1. Detailed hyper-parameters settings for each dataset. +Dataset +Hyper-parameters Settings +Ali-CCP +𝐵 = 1024,𝑑𝑓 = 18, 𝑀 = 2, 𝐾 = 64, 𝐿 = 4, 𝜆 = 10−3 +Industrial dataset +𝐵 = 1024,𝑑𝑓 = 18, 𝑀 = 2, 𝐾 = 64, 𝐿 = 4, 𝜆 = 10−3 +𝐿𝑀𝑆𝐸 = 1 +𝑀 +𝑀 +∑︁ +𝑗=1 +𝑤 · (𝑦𝑗 − ˆ𝑦𝑗)2 +(29) +where 𝑦𝑗 is label from 𝑜𝑖−1 +𝑗 +− 𝑜𝑖 +𝑗 and ˆ𝑦𝑗 is the output from +𝑓𝑖−1(𝑥𝑗;𝜃𝑠,𝜃𝑖−1) − 𝑓𝑖 (𝑥𝑗;𝜃𝑠,𝜃𝑖) for input 𝑗. Each sample is +equally treated with 𝑤 = 1. +4.1.4 +Training Setup. In the experiments, we implement +all models through the Pytorch framework. We randomly +divide each dataset into the training set and test set chrono- +logically, accounting for 90% and 10% respectively. Each ex- +periment is repeated 5 times, the average performance and +the p value are both reported. We select the optimal hyper- +parameters for each model in terms of grid search [11] for +fair comparison. The batch size 𝐵 on each datasets is set +as 1024 respectively during the training process. Adam op- +timizer [9] is applied with a learning rate 𝜆 of 0.001. The +dimension 𝑑𝑓 of input embedding layer is 18. The number +of the stacked layers 𝐿, the number of the attention heads +𝑀 and the number of the inducing points 𝐾 is illustrated +in Table 1. The activation function of MLP in single-task +modeling is ReLU. +4.2 +Performance Comparison +The experimental results for all comparison methods with +the evaluation metric AUC for each task are presented in +Table 3. The best performance on different datasets are high- +lighted in boldface. As can be observed, TAFE outperforms +most baseline models for each task on both datasets respec- +tively. +The average performance on Ali-CCP dataset is poor both +on CTR and CVR targets on all compared methods, which +probably implies the input features are not qualified enough +to express the targets or the irrelevant information affects +significantly. For the latter case, task-specific feature extrac- +tion will play a key role to the prediction results in terms of +filtering negative interference. As observed, TAFE achieves +0.6203 and 0.6456 of AUC for CTR and CVR tasks respectively, +with gains of 1.16% and 7.31% compared to the Singel-Task +method. The improvements of CTR is significant with com- +parison to other methods but slightly poorer than PLE in +the object of CVR. The result seems to be attributed to the +trade-off between tasks considered by TAFE and TAFE. The +difference improvements between CTR and CVR is smaller +in TAFE and suggests a more balanced optimization among +tasks. The performance on the Industrial dataset of TAFE + +Task Aware Feature Extraction Framework for Sequential Dependence Multi-Task Learning +Conference’17, July 2017, Washington, DC, USA +Table 2. The performance (AUC) comparison with baselines.The Gain means the mean AUC improvement compared with +Single-Task method. ** indicates that the improvement of the proposed TAFE is statistically significant compared with the best +baseline at a p-value < 0.01 over paired samples t-test. +Models +Ali-CPP +Industrial Dataset +CTR +CVR +𝐺𝑎𝑖𝑛𝐶𝑇𝑅 +𝐺𝑎𝑖𝑛𝐶𝑉 𝑅 +CTR +CVR +𝐺𝑎𝑖𝑛𝐶𝑇𝑅 +𝐺𝑎𝑖𝑛𝐶𝑉 𝑅 +Single-Task +0.6089 +0.6011 +– +– +0.7081 +0.7616 +– +– +Shared-Bottom +0.6098 +0.6225 +0.15% +3.56% +0.7050 +0.7614 +-0.37% +0.11% +MMOE +0.6177 +0.6223 +1.45% +3.53% +0.7134 +0.7673 +0.82% +0.90% +PLE +0.6195 +0.6355 +1.74% +5.72% +0.7140 +0.7675 +0.90% +0.92% +AITM +0.6133 +0.6391 +0.72% +6.32% +0.7110 +0.7667 +0.47% +0.81% +TAFE +0.6198 +0.6436** +1.79% +7.07% +0.7167** +0.7714** +1.29% +1.43% +Table 3. The performance (AUC) comparison of ablation study. +Models +Ali-CPP +Industrial Dataset +CTR +CVR +𝐺𝑎𝑖𝑛𝐶𝑇𝑅 +𝐺𝑎𝑖𝑛𝐶𝑉 𝑅 +CTR +CVR +𝐺𝑎𝑖𝑛𝐶𝑇𝑅 +𝐺𝑎𝑖𝑛𝐶𝑉 𝑅 +TAFE w/o ASRG +0.6178 +0.6379 +-0.32% +-0.89% +0.7131 +0.7670 +-0.50% +-0.57% +TAFE w/o TSA +0.6158 +0.6382 +-0.65% +-0.84% +0.7141 +0.7695 +-0.36% +-0.25% +TAFE w/o L𝐷−𝑇𝑎𝑠𝑘 +0.6199 +0.6319 +0.02% +-1.82% +0.7160 +0.7695 +-0.10% +-0.25% +TAFE +0.6198 +0.6436 +– +– +0.7167 +0.7714 +– +– +obtains an considerable gain of 1.29% and 1.43% for both +targets and significantly outperforms other methods. Com- +pared with PLE, which achieves a second best result, the +proposed model still gets an increase of the gain by 43% and +55% and further demonstrates its effectiveness. +4.3 +Ablation Study +We conduct ablation study on different submodules in TAFE +in order to provide a detail analysis of its function and ef- +ficiency. The variant models of TAFE consists of following +structures and the notation is just for simplicity: +• TAFE without ASRG: removing the dynamic activa- +tion layer in ASRG and replacing with a standard self +attention operation. +• TAFE without TSA: removing task indicator in TSA +layers for all corresponding tasks. +• TAFE without L𝐷−𝑇𝑎𝑠𝑘: removing the sequence de- +pendence learning loss L𝐷−𝑇𝑎𝑠𝑘 as denoted as in (24). +• TAFE: complete structure of TAFE. +The results of ablation study are presented in Table ?? with +an evaluation metric AUC on both datasets for CTR and +CVR tasks. As observed, the complete structure of TAFE +outperforms all other TAFE-variants and we can draw the +following conclusions for each submodule: +(1). Adaptive Sample-wise Representation Generator con- +tributes to learn fine-grained and generalized shared repre- +sentation for both tasks. In which, dynamic selector enables +to select essential information for each sample which en- +hance the knowledge learning. Without the dynamic selec- +tion layer, model performance drops most for both CTR and +CVR target as -0.5% and -0.57% in Industrial dataset. Fully +interaction learning via a standard multi-head self-attention +can’t provide enough shared info. We believe that ASRG re- +construct the necessary information in an adaptive manner +which not only learns the feature field interaction but filter +the noise by utilizing a group-level attention from a sample- +wise view. The contribution in Ali-CPP is still obvious since +whose features seems less expressive as discussed in section +4.2 and is greatly benefited by ASRG. +(2). Explicit Task-Specific Adapters works as a task-sensitive +feature extractor, which is quite crucial in the multi-task +learning to precisely extract task-relevant information for +each task. It can be evidently observed that without the +task attention mechanism (proposed task indicator), model +performance drops dramatically and is slightly better than +without ASRG in Industrial dataset but worse in Ali-CPP. +It is suggested that a vanilla task-specific tower structure +doesn’t generate enough information during task optimizing +process. +(3). The proposed sequence dependence learning loss L𝐷−𝑇𝑎𝑠𝑘 +based on theoretical proof contributes to the model perfor- +mance in terms of the additional information passing among +related tasks. Although it seems less significant compared to +other submodules in Industrial dataset but contributes most +in the CVR task in Ali-CPP. CVR task probably depends on +CTR task heavily and L𝐷−𝑇𝑎𝑠𝑘 modifies the biased object +of original definition, which further optimizes the param- +eters by recovering their complete probability-dependent +relationship. +4.4 +Analysis of Dynamic Selector +Dynamic selector 𝑧𝐾 defined in formula (15) functions as a +sparse mask generated based on the input sample, which +cooperates with Inducing points 𝐼 interacting with feature + +Conference’17, July 2017, Washington, DC, USA +Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei Cheng, and Wenfang Lin +fields selectively. We conduct several case studies of 𝑧𝐾 to +provide an intuitive analysis as visualized in Figure 7. +0.50 +0.55 +0.60 +0.65 +0.70 +(a) +0.0% +10.0% +20.0% +30.0% +40.0% +6 +4 +2 +0 +2 +4 +(b) +4 +2 +0 +2 +4 +Low +Mid +High +Figure +7. +An +illustration +of +dynamic +selector 𝑧𝐾. +(a).Distribution of the selection rate for different samples. +(b). Plot of sample embeddings with high, middle and +low selection rate is colored in green, yellow and blue +respectively. +As noted in section 3.1 , 𝑧𝐾 is a 𝐾 dimension vector only +with values of 0 and 1, where 1 means interacting with im- +plicit field group in terms of 𝐼 correspondingly and vice versa. +We first plot the distribution of the non-zero rate (selection +rate) of 𝑧𝐾 on the test samples of industrial dataset in Figure 7 +(a). It can be observed for most samples, the selection rate is +between 55% and 60% and indicates that more than half the +interaction groups are required for information extraction. +For specific cases, some needs just less interaction groups +and some needs more. We could regard this as a multi-view +representations for each sample, such as different numbers +of perspectives qualified enough to describe a customer’s +interest specially on online recommendation scenario. We +further randomly plots sample embeddings with high (top +1%), mid (around 58%) and low (bottom 1%) selection rates in +Figure 7 (b). As illustrated, samples with different interaction +degrees show significant difference in embedding space and +probably implying distinctive intentions. . +4.5 +Efficiency Evaluation +In this section, we evaluate time and storage efficiency of our +proposed method. We record the time cost during the train- +ing (per epoch) and inference process for TAFE and other +baseline models in Figure 8 (a), and their respective memory +cost in Figure 8 (b). As illustrated in (a), TAFE requires 1177 +Table 4. The offline performances (AUC) comparison for +two real-world financial scenes. +Models +Scene 1 +Scene 2 +CTR +CVR +CTR +CVR +MMOE +0.8102 +0.8034 +0.8110 +0.8719 +TAFE +0.8102 +0.8072 +0.8123 +0.8773 +Gain +- +0.47% +0.16% +0.62% +seconds for training an epoch on the Ali-CCP dataset with 38 +millions samples, which is less efficient than other methods +(295s for the best results from Shared-Bottom) but similar to +the PLE (1195s). Since we generally train the model in an of- +fline manner especially for large-scale data, relatively higher +training efficiency is within tolerance. On the inference time, +the essential factor considered in the online industrial appli- +cation, TAFE spends 47 seconds for forward propagation on +the test data with 4.2 millions samples. Its deviation from top +performance models like AITM and Shared-Bottom is 12 sec- +onds and it is acceptable considering most industrial online +inference situation’ QPS threshold. Besides, TAFE has the +least parameters with 89 Mb (178 Mb for the largest model +of Single-Task) as plotted in Figure 8 (b), which makes it +easily deployed and portable. In a conclusion, TAFE achieves +an significant improvement with an appropriate computa- +tional time and advantageous storage capacity compared +with other approved MTL methods. +PLE +TAFE +MMOE +Single-Task +AITM +Shared-Bottom +(a) +0 +250 +500 +750 +1000 +1250 +1500 +train cost time(s) +train time(s) +0 +20 +40 +60 +80 +inference cost time(s) +inference time(s) +Single-Task +PLE +MMOE +AITM +Shared-Bottom +TAFE +(b) +0 +50 +100 +150 +Size(Mb) +Figure 8. Efficiency comparison among models. (a). Training +and Inference time (b). Memory Cost + +Task Aware Feature Extraction Framework for Sequential Dependence Multi-Task Learning +Conference’17, July 2017, Washington, DC, USA +4.6 +Online A/B Performance +We implement an online A/B test between TAFE and SOTA +multi-task learning method of MMOE for one week. Two +models are deployed on two real-world financial advertising +scenarios with the objective of maximizing the CVR for the +financial products as investment and credit loan. The offline +comparison is presented in Table 4, TAFE gets an improve- +ment of 0.16% for CTR in Scene 2, 0.47% and 0.62% for CVR +on each scene correspondingly. In Figure 9, we can observe +the online performances of TAFE compared with MMOE. As +shown in the plot, TAFE achieves significant and consistent +improvements for both scenarios during the whole period, +average increase of 9.22 % in scenario (a) and 19.87% in sce- +nario (b) on the CVR task. The experiment result proves the +efficiency and the stability of proposed mehtod, which is +qualified enough for large-scale industrial application. +Day1 +Day2 +Day3 +Day4 +Day5 +Day6 +(a) +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +0.014 +CVR on Scene 1 +TAFE +MMoE +Day1 +Day2 +Day3 +Day4 +Day5 +Day6 +(b) +0.01 +0.02 +0.03 +0.04 +0.05 +CVR on Scene 2 +TAFE +MMoE +Figure 9. Online A/B results on two real-world scenarios (a) +and (b). +5 +Related Work +Multi-task Learning. Multi-task Learning (MTL) is pro- +posed to learn the shared information among tasks to im- +prove the model generalization and performance [2]. How- +ever, multi-task learning scenario usually suffers from the +performance deterioration as negative transfer because of the +complex relationship between different tasks [1, 20]. There- +fore, much feature learning works in structure designing +are proposed for necessary information extraction accord- +ing to specific task and balancing the performances among +all tasks. Cross-Stitch Network [14] use a linear combina- +tion of shared representations to learn the task-specific em- +beddings for each task. Based on the idea of Cross-Stitch, +Sluice Network [17] is a generalized meta-architecture with +more task-specific parameters by dividing each layer into +task-specific and shared subspaces and achieves better per- +formance specially for less correlated tasks. However, these +approaches could not capture the sample dependence and +require more training data and less efficient for large-scale +application. Inspired by the MoE [8] structure, multi-gate +Mixture-of-Experts (MMoE) [12] employs an ensemble of +experts submodules and gating network to combine the rep- +resentation of the bottom experts to learn the task relation- +ship while consuming less computation. Similarly, Multiple +Relational Attention Network (MRAN) [28] models multiple +relationships by three attention-based learning mechanism. +Compared with MMoE, Progressive Layered Extraction (PLE) +method [19] propose a novel MTL framework as show in +Figure ?? (a), which separates task-common and task-specific +parameters more explicitly and adopts a progressive separa- +tion routing mechanism to better alleviate parameter con- +flicts caused by complex task correlation. +Sequential Dependence Multi-task Learning. The most +classical applications of the sequential dependence MTL +(SDMTL) are the multi-step conversion process of customer +acquisition in e-commerce, display advertising or finance +systems. In general, the multi-step conversion process in- +volves impression→click→ · · · →conversion, which cor- +responds to several estimation tasks like post-view click- +through rate (CTR), post-click conversion rate (CVR) and +post-view click-through & conversion rate (CTCVR) estima- +tions and so fourth. Differing from the general MTL, there +exist dependence relationships between the adjacent tasks +in the SDMTL problem. For the CVR estimation problem, +Entire Space Multi-task Model (ESMM) is proposed in [13] +to overcome Sample Selection Bias (SSB) and Data Sparsity +(DS) issues by introducing two auxiliary tasks of predicting +the post-view click-through rate (CTR) and post-view click +through & conversion rate (CTCVR). With this operation, +the performance of the CVR estimation will depend heavily +on the performance auxiliary tasks. As the number of steps +increases in multi-step conversion path, the accumulation +of performance errors becomes intolerable. Aimed at the DS +problem of the CVR estimation, in [25], a novel user sequen- +tial behavior graph is established to achieve post-click be- +havior decomposition by inserting disjoint purchase-related +deterministic action and other action into between click and +conversion. Considering micro behaviors (user’s interactions +with items) and macro behaviors (user’s interactions with +specific components on the item detail page) of users, Wen +et al. [24] propose a Hierarchically Modeling both Micro +and Macro behaviors for CVR prediction to address SSB and +DS issues by using the abundant supervisory labels from +micro and macro behaviors. To models the sequential depen- +dence among multi-step conversions, Adaptive Information + +Conference’17, July 2017, Washington, DC, USA +Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei Cheng, and Wenfang Lin +Transfer Multi-task (AITM) framework with adaptive in- +formation transfer module is developed in [26] to directly +predict the end-to-end conversion probabilities of each step. +Besides, causal approaches have also been applied to achieve +the debiasing post-click conversion rate estimation lately +[5, 6, 22, 27]. However, for the sequential dependence multi- +task learning problem, there is rare literature to develop a +formalization description. +6 +Conclusions +In this paper, we propose a sequence dependence multi-task +learning framework named as Task Aware Feature Extrac- +tion (TAFE), which could selectively reconstruct implicit +shared representations from a sample-wise view and extract +explicit task-specific information in an more efficient way +compared with common task-aware tower structure. We +accomplish this by involving an Adaptive Sample-wise Rep- +resentation Generator and a Task-Specific Adapter. For the +multi-task learning with dependency generally encountered +in E-commence online recommendation, we provide a detail +theoretical proof about the dependent relationship from rig- +orous mathematical perspective. Based on our analysis, we +design a dependence task learning loss to complete optimiz- +ing object in an unbiased format. The performance gains of +TAFE compared to several SOTA multi-task approaches on +both public and real-world industrial datasets demonstrates +its effectiveness and generalization characteristics. Besides, +we carefully conduct ablation study, case study, efficiency +evaluation and online A/B test to further analyze the con- +tributions from different modules and its applicability for +large-scale industrial scenarios. +References +[1] Jonathan Baxter. 1997. A Bayesian/information theoretic model of +learning to learn via multiple task sampling. Machine learning 28, 1 +(1997), 7–39. +[2] Rich Caruana. 1997. Multitask learning. Machine learning 28, 1 (1997), +41–75. +[3] Ling Chen, Donghui Chen, Fan Yang, and Jianling Sun. 2021. Neural +episodic control. In A deep multi-task representation learning method +for time series classification and retrieval. Information Sciences, 17–32. +[4] Michael Crawshaw. 2020. 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Revisiting multi-task learning in the +deep learning era. arXiv preprint arXiv:2004.13379 2 (2020). +[22] Hao Wang, Tai-Wei Chang, Tianqiao Liu, Jianmin Huang, Zhichao +Chen, Chao Yu, Ruopeng Li, and Wei Chu. 2022. ESCM2: Entire Space +Counterfactual Multi-Task Model for Post-Click Conversion Rate Esti- +mation. arXiv preprint arXiv:2204.05125 (2022). +[23] Ruize Wang, Duyu Tang, Nan Duan, Zhongyu Wei, Xuanjing Huang, +Guihong Cao, Daxin Jiang, Ming Zhou, et al. 2020. K-adapter: Infusing +knowledge into pre-trained models with adapters. arXiv preprint +arXiv:2002.01808 (2020). +[24] Hong Wen, Jing Zhang, Fuyu Lv, Wentian Bao, Tianyi Wang, and Zu- +long Chen. 2021. Hierarchically modeling micro and macro behaviors +via multi-task learning for conversion rate prediction. 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In Proceedings of the 25th ACM SIGKDD International +Conference on Knowledge Discovery & Data Mining. 1123–1131. + diff --git a/WdE0T4oBgHgl3EQfmAG-/content/tmp_files/load_file.txt b/WdE0T4oBgHgl3EQfmAG-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d2dd5f6b620a89adcdcacd8fe3b5340c6cd4b19f --- /dev/null +++ b/WdE0T4oBgHgl3EQfmAG-/content/tmp_files/load_file.txt @@ -0,0 +1,809 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf,len=808 +page_content='Task Aware Feature Extraction Framework for Sequential Dependence Multi-Task Learning Xuewen Tao∗ xuewen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='txw@mybank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='cn MYbank, Ant Group Beijing, China Mingming Ha∗ hamingming_0705@foxmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='com School of Automation and Electrical Engineering, University of Science and Technology Beijing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' MYbank, Ant Group Beijing, China Xiaobo Guo† xb_guo@bjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='cn Institute of Information Science, Beijing Jiaotong University, Mybank, Ant Group, Beijing, China Qiongxu Ma qiongxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='mqx@mybank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='cn MYbank, Ant Group Shanghai, China Hongwei Cheng chw286885@mybank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='cn MYbank, Ant Group Shanghai, China Wenfang Lin moxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='lwf@mybank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='cn MYbank, Ant Group Hangzhou, Zhejiang, China Abstract Multi-task learning (MTL) has been successfully implemented in many real-world applications, which aims to simultane- ously solve multiple tasks with a single model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The gen- eral idea of multi-task learning is designing kinds of global parameter sharing mechanism and task-specific feature ex- tractor to improve the performance of all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' However, sequential dependence between tasks are rarely studied but frequently encountered in e-commence online recommen- dation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' impression, click and conversion on displayed product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' There is few theoretical work on this problem and biased optimization object adopted in most MTL methods deteriorates online performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Besides, challenge still re- mains in balancing the trade-off between various tasks and effectively learn common and specific representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In this paper, we first analyze sequential dependence MTL from rigorous mathematical perspective and design a dependence task learning loss to provide an unbiased optimizing object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' And we propose a Task Aware Feature Extraction (TAFE) framework for sequential dependence MTL, which enables to selectively reconstruct implicit shared representations from a sample-wise view and extract explicit task-specific infor- mation in an more efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Extensive experiments on ∗These authors contributed equally to this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' †Xiaobo Guo is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='XXXXXXX offline datasets and online A/B implementation demonstrate the effectiveness of our proposed TAFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' CCS Concepts: • Information systems → Information systems applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Computational advertising;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' • Multi- task Learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Keywords: Recommender System, Sequential Dependency, Multi-Task Learning, Representation Learning ACM Reference Format: Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei Cheng, and Wenfang Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Task Aware Feature Extraction Framework for Sequential Dependence Multi-Task Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In Proceedings of ACM Conference (Conference’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ACM, New York, NY, USA, 13 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='XXXXXXX 1 Introduction Multi-task learning (MTL) has been successfully applied es- pecially for online recommendation in various real-world scenarios such as E-commerce or financial service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' MTL aims to simultaneously learn multiple tasks in a single model and has been proved to have obvious advantages compared with single-task learning [4], since it is able to provide comple- mentary information by implicitly passing the message be- tween different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' [21], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Click-through rate (CTR), conversion rate (CVR) and click-through & conversion rate (CTCVR) are the typical optimizing object on the industrial recommendation indicating the a customer acquisition pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' These behaviors are strictly sequential dependent on the previous one especially for fiance service and we illus- trate a multi-step conversion example in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' A customer will convert through stages of Impression → Click → Autho- rize → Conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Conversion behaviors such like applying loans, make deposit or purchasing investment products are only permitted after an authorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Series works from ESMM [13] to ESCM2 [25] pay more attention on the unbi- ased CVR estimation problem in a view of causality to correct arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='02494v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='LG] 6 Jan 2023 Conference’17, July 2017, Washington, DC, USA Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei Cheng, and Wenfang Lin Conversion Stages Impression Click Authorization Loan Finance Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' An illustration of a multi-step conversion in Fi- nance Service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' sample selection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The dependent relation between the tasks like CTR and CVR is implicitly implied via the dis- tribution of sample space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Recently, [26] captures the task dependency through the information transfer between dif- ferent conversion steps and combines a calibrator to further constrain the dependent relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' However, the depen- dency between each steps is still not deeply defined and discussed in most MTL works from a theoretical format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Besides, as mentioned above, MTL methods improve pre- diction result through an information passing mechanism between tasks, which suggests improper feature sharing will result in even poorer or imbalanced performance in different tasks known as a negative transfer phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Therefore, general approach in MTL mainly focuses on designing kinds of information extraction modules (experts) to learn com- mon and task-specific representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Such as Cross-Stitch Network [14] and Sluice Network [17] employ a linear combi- nation to leverage representations of different tasks but also require much more training parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' SOTA method of Multi-gate Mixture-of-Experts (MMoE) approach [12] adopts an ensemble of experts submodules and gating network to model task relationships while consuming less computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Progressive Layered Extraction (PLE) [19], separates task- common and task-specific parameters explicitly which could further avoid parameter conflicts caused by complex task correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' These approaches assign individual parameters to each task to better exploit task information and improve model generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Nonetheless, feature expressivity with respect to each task is still limited since task-irrelevant in- formation passing from the shared structure and more fine- grained representation learning is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In this paper, we first provide a formal definition of MTL on sequential dependence problem, and propose an optimiz- ing object paradigm for recover the dependent relationship based on theoretical proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' And we also present a novel MTL framework called Task Aware Feature Extraction (TAFE) to selectively and dynamically enhance the representation learning for respective tasks along with the dependency- based object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' TAFE consists of two main modules: Adaptive Sample-wise Representation Generator (ASRG) and explicit Task-Specific Adapter (TSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ASRG employs a dynamic se- lection mechanism to learn the hierarchical feature inter- action from a sample-wise view to further separates the task-irrelevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The implement of explicit TSA enables fine-grained feature learning by introducing task- specific indicator vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In a summary, main contributions of this paper are presented as follows: The sequential dependence multi-task learning prob- lem is first formulated, and its connections and differ- ences with the general multi-task learning problem are illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Moreover, the distribution dependence re- lationship between the adjacent task spaces is revealed from a theoretical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We present a multi-task learning framework named TAFE for selectively fine-grained feature representa- tion learning from a sample-wise view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ASRG and TSA modules within TAFE adatively reconstruct the im- plicit shared representations and extract explicit task- specific information in an more efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Extensive experiments on public and real-world indus- trial dataset are conducted to evaluate the effectiveness of TAFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Experiment results demonstrate that our pro- posed approach outperforms the state-of-the-art MTL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Furthermore, we explore the boundary of TAFE in real-world industrial applications to prove its efficiency for large-scale online recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 2 Preliminaries In this section, the multi-task learning with sequential de- pendence and the data distribution discrepancy of different tasks are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Then, from the expected loss’s point of view, the distribution relationship between the adjacent tasks is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1 Problem Formulation Consider a SDMTL problem over an input space X and a set of task {T𝑖}𝑁 𝑖=1, where 𝑁 is the number of tasks and the cor- responding task spaces are denoted as {T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' , T𝑁 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' A large dataset of data points {𝑥𝑗,𝑜1 𝑗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑜𝑁 𝑗 }𝑀 𝑗=1 are given, where 𝑀 is the number of data points and 𝑜 𝑗 𝑖 ∈ {0, 1} corresponding to a binary classification problem or 𝑜 𝑗 𝑖 ∈ R for a regression problem is the label of the 𝑖-th task for the 𝑗-th data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Differing from the general MTL problem, for the SDMTL problem, there exists the sequential dependence relationship between tasks in the sense that the current task T𝑖 depends on the previous task T𝑖−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=', T𝑖−1 → T𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Let 𝑋 and 𝑇𝑖 be the random variables over the input space X and task out- put space T𝑖, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In this paper, for convenience of analysis, each task is set as a binary classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As mentioned in literature [15] and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 1, for sequen- tial dependence, one of core properties is that if the event 𝑇𝑖−1 is not triggered, then the event 𝑇𝑖 must not occur, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=', 夫Task Aware Feature Extraction Framework for Sequential Dependence Multi-Task Learning Conference’17, July 2017, Washington, DC, USA 𝑃(𝑇𝑖 = 1, 𝑃𝑖−1 = 0|𝑋) = 0, where 𝑃(·|·) denotes the condi- tional probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Therefore, according to this property, the random variables 𝑇𝑖 satisfies 𝑃(𝑇𝑖 = 1|𝑋) = ∑︁ 𝑡𝑖−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝑡1∈{0,1} 𝑃(𝑇𝑖 = 1,𝑇𝑖−1 = 𝑡𝑖−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 𝑡1|𝑋) = 𝑃(𝑇𝑖 = 1,𝑇𝑖−1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1|𝑋), 𝑃(𝑇𝑖 = 0|𝑋) = ∑︁ 𝑡𝑖−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝑡1∈{0,1} 𝑃(𝑇𝑖 = 0,𝑇𝑖−1 = 𝑡𝑖−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 𝑡1|𝑋), (1) which implies that the positive samples of𝑇𝑖 are derived from the positive samples of the task𝑇𝑖 while the negative samples consist of the negative samples of tasks 𝑇𝑖,𝑇𝑖−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 due to the sequential dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In addition, the sequential dependence relationship is also embodied in the constraints with respect to the conversion probabilities of the adjacent tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In [26], the sequential dependence relationship is formalized as 𝑃(𝑇1 = 1|𝑋) ≥𝑃(𝑇2 = 1,𝑇1 = 1|𝑋) · · ≥𝑃(𝑇𝑖−1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1|𝑋) ≥𝑃(𝑇𝑖 = 1,𝑇𝑖−1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1|𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (2) Then, a behavioral expectation calibrator is introduced into AITM [26] to guarantee the sequential dependence relation- ship (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' When the outputs of the model violate this condi- tion, the designed loss will output a positive penalty term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' However, the condition (2) cannot completely reflect the dependence relationship between tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Reconsidering the dependence relationship between 𝑃(𝑇𝑖−1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1|𝑋) and 𝑃(𝑇𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1|𝑋), it leads to 𝑃(𝑇𝑖−1 = 1|𝑋) − 𝑃(𝑇𝑖 = 1|𝑋) = 𝑃(𝑇𝑖−1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1|𝑋) − 𝑃(𝑇𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1|𝑋) = 𝑃(𝑇𝑖−1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1|𝑋) × � 1 − 𝑃(𝑇𝑖 = 1|𝑇𝑖−1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1,𝑋) � = 𝑃(𝑇𝑖−1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1|𝑋)𝑃(𝑇𝑖 = 0|𝑇𝑖−1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1,𝑋) = 𝑃(𝑇𝑖 = 0,𝑇𝑖−1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1|𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (3) Therefore, the dependence relationship between the adjacent tasks needs to satisfy the equality constraints (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Define a parametric hypothesis class per task as 𝑓𝑖 (𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖) : X → T𝑖, where 𝜃𝑠 and 𝜃𝑖 are shared parameters and task- specific parameters of the task 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Also, the task-specific loss function is defined as 𝐿𝑖 (·, ·) : T𝑖 × T𝑖 → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Similar to the general MTL problem, the objective of SDMTL is to minimize the following expected loss: min 𝜃𝑠,𝜃1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=',𝜃𝑖 𝑁 ∑︁ 𝑖=1 𝐸𝑋,𝑇1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=',𝑇𝑁 ∼O[𝑤𝑖𝐿𝑖 (𝑓𝑖 (𝑋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖),𝑇𝑖)] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 𝑓𝑖 (𝑋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖) − 𝑓𝑖−1(𝑋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖) = 𝑃(𝑇𝑖 = 0,𝑇𝑖−1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑇1 = 1|𝑋) (4) where O is the distribution with domain X × T1 × · · · × T𝑁 , and 𝑤𝑖 is the static or dynamically computed weight per task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In this case, the sequential dependence is embodied in the expression 𝑃(𝑇𝑖 = 1|𝑇𝑖−1 = 0,𝑋) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' \ue2401 \ue2402 \ue2403 \ue2404 \ue23b Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Distribution discrepancy of different task spaces in SDMTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The curved surface represents the distribution O with domain X × T1 × · · · × T𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The colored circles from the outside to the inside denote the domains of tasks T1, T2, T3, and T4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Considering (24), we can regarded the relationship be- tween 𝑇𝑖 and 𝑇𝑖−1 as a new sequential dependence task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The task-specific outputs 𝑓𝑖 (𝑥𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖) of model also need to sat- isfy this dependence relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Since there exists the dependence relationship between the previous and current tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=', T1 → T2 → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' → T𝑁 , the sample space of the current task depends on the previous task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In general, the sample space of the previous task contains the sample space of the current one as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 2, which leads to the data distribution discrepancy between these two sample spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Consider the general CTR, CVR and CTCVR estimation tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=', impression→click→conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We use the random variables 𝑌 ∈ {0, 1} and 𝑍 ∈ {0, 1} to denote the click event and the conversion event, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Then, CTR, CVR and CTCVR with feature input 𝑋 are defined as 𝑃(𝑌 |𝑋), 𝑃(𝑍 |𝑌 = 1,𝑋) and 𝑃(𝑍,𝑌 = 1|𝑋), which satisfy 𝑃(𝑍 = 1,𝑌 = 1|𝑋) = 𝑃(𝑌 = 1|𝑋)𝑃(𝑍 = 1|𝑌 = 1,𝑋), (5) where 𝑃(·|·) denotes the conditional probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In this case, the training space of the traditional CVR estimation task is generally determined by the samples with 𝑌 = 1 in the CTR estimation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' However, for a new user, there are no impression and click records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The conversion rate estimation task is actually to estimate 𝑃(𝑍 = 1|𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Considering 𝑃(𝑍 = 1,𝑌 = 0|𝑋) = 0 [15], we can obtain 𝑃(𝑍 = 1|𝑋) =𝑃(𝑍 = 1,𝑌 = 0|𝑋) + 𝑃(𝑍 = 1,𝑌 = 1|𝑋), =𝑃(𝑍 = 1,𝑌 = 1|𝑋), 𝑃(𝑍 = 0|𝑋) =𝑃(𝑍 = 0,𝑌 = 0|𝑋) + 𝑃(𝑍 = 0,𝑌 = 1|𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (6) According to (6), it is observed that the negative samples of the conversion event 𝑍 are derived from the entire space of the click event 𝑌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' If the negative data points derived from the space with 𝑌 = 0 is used to predict 𝑃(𝑍 = 1|𝑋), then the Conference’17, July 2017, Washington, DC, USA Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei Cheng, and Wenfang Lin data distribution discrepancy between training space and inference space leads to inaccurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' However, a model directly learning 𝑃(𝑍 = 1|𝑋) will ignore the sequen- tial dependence information between the click event and the conversion event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='2 Distribution Dependence Relationship Between Inference Space and Local Space In this subsection, the random variables 𝑇𝑖−1 and 𝑇𝑖 of ad- jacent tasks are denoted as 𝑌 and 𝑍, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In what follows, the relationship of expected losses between domains of adjacent tasks Y and Z is established, where Y → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Considering the sequential dependence relationship between Y and Z, the input random variable 𝑋 over the input space X, and random variables 𝑌 ∈ {0, 1} and 𝑍 ∈ {0, 1} over the task output spaces Y and Z, we can obtain 𝑃(𝑍 |𝑋) = ∑︁ 𝑦∈{0,1} 𝑃(𝑍,𝑌 = 𝑦|𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (7) In tasks Y and Z, the sample space with data points {𝑥𝑗 ∈ X,𝑜𝑌 𝑗 ∈ {0, 1},𝑜𝑍 𝑗 ∈ {0, 1}} is called ieference space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=', entire space for Y and Z, and the sample space with data points {𝑥𝑗 ∈ X,𝑜𝑌 𝑗 ∈ {1},𝑜𝑍 𝑗 ∈ {0, 1}} is called local space, also called training space in some traditional CVR estimation methods [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The distributions of reference and local spaces are denoted as D and C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Therefore, the objective of these two tasks Y and Z with sequential dependence in inference space is to minimize the following expected loss: 𝐸𝑋,𝑌,𝑍∼D [𝐿(𝑓𝑌 (𝑋),𝑌) + 𝐿(𝑓𝑍 (𝑋),𝑍)] =𝐸𝑋,𝑌∼D [𝐿(𝑓𝑌 (𝑋),𝑌)] + 𝐸𝑋,𝑍∼D [𝐿(𝑓𝑍 (𝑋),𝑍)] = ∫ D 𝐿(𝑓𝑌 (𝑥),𝑦)𝑃D(𝑥,𝑦)d𝑥d𝑦 + ∫ D 𝐿(𝑓𝑍 (𝑥),𝑧)𝑃D(𝑥,𝑧)d𝑥d𝑧, (8) where 𝑃D(·, ·) is the joint distribution in inference space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' On the other hand, if the model is trained in the local space C, then the task Y determines the sample distribution of the task Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' With this operation, the expected loss becomes the following form: 𝐸𝑋,𝑌∼D [𝐿(𝑓𝑌 (𝑋),𝑌)] + 𝐸𝑋,𝑍∼C[𝐿(𝑓𝑍 (𝑋),𝑍)] = ∫ D 𝐿(𝑓𝑌 (𝑥),𝑦)𝑃D(𝑥,𝑦)d𝑥d𝑦 + ∫ C 𝐿(𝑓𝑍 (𝑥),𝑧)𝑃C(𝑥,𝑧)d𝑥d𝑧, (9) where 𝑃C(·, ·) is the joint distribution in local space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Next, the relationship between expected loss in (8) and (9) is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' If the expected losses in the reference and local spaces are defined as in (8) and (9), then, for any loss function 𝐿(·, ·), they satisfy 𝐸𝑋,𝑌,𝑍∼D [𝐿(𝑓𝑌 (𝑋),𝑌) + 𝐿(𝑓𝑍 (𝑋),𝑍)] =𝐸𝑋,𝑌∼D [𝐿(𝑓𝑌 (𝑋),𝑌)] + 𝐸𝑋,𝑍∼C � 𝑃D(𝑌 = 1) 𝑃D(𝑍 |𝑋) 𝑃D(𝑍,𝑌 = 1|𝑥) 𝐿(𝑓𝑍 (𝑋),𝑍) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Considering the definitions of the reference and local spaces, and their corresponding expected losses given in (8) and (9), we can obtain 𝑃C(𝑋,𝑍) = 𝑃D(𝑋,𝑍 |𝑌 = 1) (11) in the sense that the joint distribution of 𝑋 and 𝑍 in C is equivalent to, under 𝑌 = 1, the joint distribution of 𝑋 and 𝑍 in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' According to (11) and the definition of 𝐸𝑋,𝑍∼D [𝐿(𝑓𝑍 (𝑋),𝑍)], the second term in the right-hand side of (10) satisfies 𝐸𝑋,𝑍∼C � 𝑃D(𝑌 = 1) 𝑃D(𝑍 |𝑋) 𝑃D(𝑍,𝑌 = 1|𝑋) 𝐿(𝑓𝑍 (𝑋),𝑍) � =𝐸𝑋,𝑍∼C � 𝑃D(𝑌 = 1) 𝑃D(𝑌 = 1|𝑋,𝑍) 𝐿(𝑓𝑍 (𝑋),𝑍) � = ∫ C 𝑃D(𝑌 = 1) 𝑃D(𝑌 = 1|𝑥,𝑧) 𝐿(𝑓𝑍 (𝑥),𝑧)𝑃C(𝑥,𝑧)d𝑥d𝑧 = ∫ D 𝐿(𝑓𝑍 (𝑥),𝑧) 𝑃D(𝑌 = 1) 𝑃D(𝑌 = 1|𝑥,𝑧) 𝑃D(𝑥,𝑧|𝑌 = 1)d𝑥d𝑧 = ∫ D 𝐿(𝑓𝑍 (𝑥),𝑧) 𝑃D(𝑌 = 1) 𝑃D(𝑌 = 1|𝑥,𝑧) 𝑃D(𝑥,𝑧,𝑌 = 1) 𝑃D(𝑌 = 1) d𝑥d𝑧 = ∫ D 𝐿(𝑓𝑍 (𝑥),𝑧)𝑃D(𝑥,𝑧)d𝑥d𝑧 =𝐸𝑋,𝑍∼D [𝐿(𝑓𝑍 (𝑋),𝑍)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (12) Therefore, equations (8), (9) and (12) imply that the relation- ship (10) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' □ Obviously, the distribution shift also exists in CTR, CVR and CTCVR estimations when they are trained in different spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 3 The Task Aware feature Extraction Framework The whole architecture of proposed TAFE for sequential dependence multi-task learning is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' TAFE consists of two representation learning modules ASRG and TSA to dynamically extract implicit and explicit feature information from a sample-wise view, and a sequential de- pendence task learning loss to reconstruct an unbiased task relationship on a global training space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Adaptive Sample- wise Representation Generator (ASRG) is responsible for hi- erarchical shared-representation learning, adopting inducing points to interact with different feature field corresponding to each input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Task Specific Adapter (TSA) module coop- erates with ASRG but works as a task-ware information extractor through designed task indicator and is with an in- dependent message passing structure to better solve the task Task Aware Feature Extraction Framework for Sequential Dependence Multi-Task Learning Conference’17, July 2017, Washington, DC, USA conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Besides those two, a sequence dependency learning loss between tasks is proposed and theoretical proved, which is able to describe the conditional dependent probability for sequential based multi-task learning from the whole training space and consequently improve the prediction result by precisely capturing the task relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We will elaborate ASRG and TSA in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='2, and lastly discuss the relationship between sequence dependence tasks in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Task Tower 𝑖-1 Task Tower 𝑖 CONCAT Sequential dependency Learning … Embedding Feas Filed 1 Filed 2 Filed n … Adaptive Sample-wise Representation Generator (ASRG) High-Order Inducing Points Task Indicator 𝑖 Task Indicator 𝑖-1 N ℒ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' "#$%& ℒ\'"($%& ) ~𝑃(y)|𝑥)) ℒ\'"($%& ))* ~𝑃(y)"*|𝑥)"*) 𝑦!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' y!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' "# ℒ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' "#$%&~𝑃 ∆y) 𝑥)"*, 𝑥) , ∆y)= y)"* − y) Task Specific Adapter(TSA) Task Specific Adapter(TSA) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' An illustration of the overall architecture of TAFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1 Adaptive Sample-wise Representation Generator Fine-grained feature information extraction corresponding to different tasks is crucial in multi-task learning and signifi- cantly affects model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' But feature generalization also needs to be included to balance the trade-off between tasks in terms of shared information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Based on these consid- erations, we propose a novel representation learning mod- ule, named Adaptive Sample-wise Representation Generator (ASRG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Besides learning generalized shared-information, we design a dynamic selector to learn the feature interac- tion from a sample-wise view to further separates the task- irrelevant info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The structure of ASRG is shown in Figure 4, which mainly consists of an dynamic activation layer and a feature interaction learning layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Dynamic Activation Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In recommendation scenario, input field usually contains kinds of user and item features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Adaptive Sample-wise Representation Generator (ASRG) Multi-Head Attention Add & Norm Feature Embeddingi Inducing Points + Element-wise Q V K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 Dynamic Activation Layer Feature Embeddingo 𝒙 𝒇(𝒙) 𝜸=𝟏 𝟎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝟒 𝒇(𝒙) 𝒙 𝜸=𝟏 𝒇 𝒙 = 𝟎, 𝒙 ≤ 𝜸 𝟐 − 𝟐 𝜸𝟐 𝒙𝟑 + 𝟑 𝟐𝜸 𝒙 + 𝟏 𝟐 , − 𝜸 𝟐 < 𝒙 < 𝜸 𝟐 𝟏, 𝒙 ≥ 𝜸 𝟐 Dynamic Selector Dynamic Selector Transformation Layer Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The detail structure of Adaptive Sample-wise Rep- resentation Generator Given an input x from 𝐹 different feature fields, we denote x as the concatenation of all feature fields: x = [𝑥1,𝑥2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑥𝐹], (13) where 𝑥𝑖 represents the value of the 𝑖-th feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As a com- monly data preprocssing for online recommendation sce- nario with better generalization, we discretize numerical features 𝑥𝑖 through a Log-round operation to get an unique value, and randomly initialize it with a vector of 𝑑𝑓 dimen- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Thus, we obtain the input embedding for each feature field as 𝐻 = [ℎ1,ℎ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,ℎ𝐹]T, where 𝐻 ∈ R𝐹×𝑑𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' A transformation Layer is first applied to project the input embeddings into a 𝐾 dimension vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The transformation layer can be any type of deep neural network structure and here we chose a standard MLP layer just for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The output 𝑧𝐾 is defined as a dynamic selector: 𝑧𝐾 = MLP(𝐻) (14) where 𝑧𝐾 ∈ R𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' we implement a dynamic activation function 𝑓𝐷 inspired by [7] to get a sparser representation of 𝑧𝐾,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' whose formulation is as follows: 𝑧𝐾 = 𝑓𝐷 (𝑧𝐾) (15) XConference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' July 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' USA Xuewen Tao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Mingming Ha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Xiaobo Guo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Qiongxu Ma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Hongwei Cheng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' and Wenfang Lin where 𝑓𝐷 is formulated as: 𝑓𝐷 (𝑧) = \uf8f1\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f3 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 𝑧 ≤ −𝛾 2 − 2 𝛾3𝑧3 + 3 2𝛾 𝑧 + 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' −𝛾 2 < 𝑧 < 𝛾 2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 𝑧 ≥ 𝛾 2 (16) where 𝛾 = 𝑀𝑎𝑥{10 − 2𝑒-4 · 𝑠𝑡𝑒𝑝,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 1𝑒-3} and maximum 𝑠𝑡𝑒𝑝 during the training process is around 1𝑒6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Dynamic selector 𝑧𝐾 works as a information filter which selectively interacts with input from the sample-wise view due to the Transfor- mation Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As visualized in Figure 5, the output shape of 𝑓𝐷 becomes steeper with the increase of training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' By utilizing 𝑓𝐷, 𝑧𝐾 creates a 𝐾 dimension sparse vector only con- tains values of 0 and 1 corresponding to each input sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='00 fD(z) Step=1 Step=25000 Step=100000 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Output of the dynamic activation function 𝑓𝐷 with the increase of training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Feature Interaction Learning Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Attention mecha- nism for learning hierarchical feature interaction is gen- erally adopted but requires quadratic time complexity in standard self-attention structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Here we design a learnable matrix called inducing points 𝐼, enlightened from Set Trans- former [10] to reduce the computational complexity from quadratic to linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We define the inducing points 𝐼 ∈ R𝐾×𝑑𝑓 , where 𝐾 is the same as in dynamic selector 𝑧𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' After a element-wise operation, we get a modified query ˆ𝑄 as: ˆ𝑄 = 𝐼 ⊙ 𝑧𝐾 (17) Then, we calculate the output 𝑂 𝑗 from the attention opera- tion according to the following formulation: 𝑂 𝑗 = Attention( ˆ𝑄 𝑗, 𝐾𝑗,𝑉𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 𝜆) (18) where ˆ𝑄 𝑗 = 𝐼𝑗 ⊙ 𝑧𝐾, 𝐾𝑗 = 𝐻𝑊 𝐾 𝑗 ,𝑉𝑗 = 𝐻𝑊 𝑉 𝑗 with trainable parameter 𝜆 = � 𝐼𝑗,𝑊 𝐾 𝑗 ,𝑊 𝑉 𝑗 �𝑚 𝑗=1 and 𝑚 represents the num- ber of multi-head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Then we get output 𝑂 from the multi-head attention with parameter 𝑊 𝑂 as: 𝑂 = concat (𝑂1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝑂ℎ)𝑊 𝑂 (19) Finally, the adaptive representation𝑌𝐴𝑆𝑅𝐺 learned from ASRG can be formulated in a way of residual network: 𝑌𝐴𝑆𝑅𝐺 = LayerNorm(𝑂, 𝐻) (20) The time complexity of feature interaction learning layer reduces from 𝑂(𝐹 2) to 𝑂(𝐾 × 𝐹) by introducing 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As sug- gested in [16], 𝐾, the reduced dimension of 𝐼 could be viewed as 𝐾 independent memory cells interacting with each fea- ture field, which is further automatically selected by 𝑧𝑘 to distinguish feature information explicitly from a sample- wise view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Compared with traditional shared-representation learning structure in most MTL methods, ASRG learns more distinctive info in terms of a dynamic activation layer and feature interaction learning layer, which is mainly attributed to the former one combining a transformation layer and dynamic activation function to generate an adaptive mask corresponding to each input sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='2 Explicit Task-Specific Adapter Besides effective shared-representation generated by ASRG, specific feature learning according to each task will strongly affect the model performance since it directly enhances the task-relevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In most MTL works, task-targeted feature extractors, such as the task-specific experts proposed by PLE are deliberately designed to learn the representa- tion for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' However, mutual interference between different tasks still exists since the shared and task-specific components are not completely separated in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In order to learn the task-aware information among differ- ent tasks with a more independent and thoroughly separated structure, we introduce a module named explicit Task Spe- cific Adapters (TSAs) as detail plotted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' TSA uti- lizes parameterized task indicator vector to interact with pre- vious sample-wise common shared info from ASRG, which is able to extract task-specific representation by directly op- timizing respective task object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The approach is similarly adopted in PAL [18] and K-adapter [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Task Specific Adapter(TSA) Attention Task Indicator Q K V Feature Embedding o Add & Norm Task Aware Embedding Feature Embeddingi Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The detail structure of Task-Specific Adapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As illustrated in Figure 6, we take the output 𝑌𝐴𝑆𝑅𝐺 ∈ R𝐾×𝑑𝑓 from the Adaptive Sample-wise Representation Gen- erator to interact with a learnable task indicator vector 𝛼𝑖 corresponding to each task 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The 𝐹𝑖 is the output calculated through an attention operation between 𝑌𝐴𝑆𝑅𝐺 and 𝛼𝑖 as: 𝐹𝑖 = Attention(𝛼𝑖,𝑌𝐴𝑆𝑅𝐺,𝑌𝐴𝑆𝑅𝐺), (21) where 𝛼𝑖 ∈ R1×𝑑𝑓 is the task indicator vector and 𝐹𝑖 ∈ R1×𝑑𝑓 denotes the middle output of task-aware representation for Task Aware Feature Extraction Framework for Sequential Dependence Multi-Task Learning Conference’17, July 2017, Washington, DC, USA each task 𝑖 correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Here, Attention is the same attention calculation operation as in formula (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Conse- quently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' for each task𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' we refer the task-specific information generated from the 𝑘-th TSA layer as 𝑇 𝑘 𝑖 and we calculate it also through a residual network and layer normalization for training efficiency as in (20): 𝑇 𝑘 𝑖 = LayerNorm(𝑇 𝑘−1 𝑖 + 𝐹𝑘 𝑖 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (22) where𝑇 𝑘−1 𝑖 ∈ R1×𝑑𝑓 means the output from the previous TSA layer for task 𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' and 𝐹𝑘 𝑖 ∈ R1×𝑑𝑓 is the task-aware embedding learned by the interaction between the task indicator for the 𝑘-th layer with task 𝑖 and shared-common embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Note, 𝑇 0 𝑖 is ignored at the first iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' It can be observed in Figure 6, task aware embedding𝑇 ob- tained from Task Specific Adapters is trained independently and whose message doesn’t pass into ASRG module among different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The proposed structure keeps the implicit (from ASRG) and explicit (from TSA) representation learn- ing modules more separated, which not only isolates the negative interference between tasks more thoroughly but also provides a extendable multi-task learning framework especially necessary in industrial implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='3 Loss Function Design Towards Sequential Dependence Multi-Task Learning For the multi-task learning without sequential dependence, the loss function is generally designed as the following form: L(𝜃𝑠,𝜃1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝜃𝑁 ) = 1 𝑀 𝑁 ∑︁ 𝑖=1 𝑀 ∑︁ 𝑗=1 𝑤𝑖𝐿 � 𝑓𝑖 (𝑥𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖),𝑜𝑖 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (23) From the loss function (23), it can be observed that this loss function cannot learn the sequential dependence relation- ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As mentioned in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1, the corresponding loss function for SDMTL is designed as L(𝜃𝑠,𝜃1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ,𝜃𝑁 ) =L𝑀−𝑇𝑎𝑠𝑘 + L𝐷−𝑇𝑎𝑠𝑘 (24) = 1 𝑀 𝑁 ∑︁ 𝑖=1 𝑀 ∑︁ 𝑗=1 𝑤𝑖𝐿 � 𝑓𝑖 (𝑥𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖),𝑜𝑖 𝑗 � + 1 𝑀 𝑁 ∑︁ 𝑖=2 𝑀 ∑︁ 𝑗=1 𝐿 � 𝑓𝑖−1(𝑥𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖−1) − 𝑓𝑖 (𝑥𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖),𝑜𝑖−1 𝑗 − 𝑜𝑖 𝑗 � , (25) where L𝑀−𝑇𝑎𝑠𝑘 and L𝐷−𝑇𝑎𝑠𝑘 are the loss functions of the main tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=', T𝑖, and the loss functions of the sequen- tial dependence relationship, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' With this oper- ation, each task and their corresponding dependence rela- tionship can be trained separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The loss functions of the dependence relationship can be regarded as a regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Therefore, the SDMTL problem can also be considered as a general MTL with the constraints 𝑓𝑖−1(𝑥𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖−1) − 𝑓𝑖 (𝑥𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖) = 𝑜𝑖−1 𝑗 − 𝑜𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Note that the selection of negative samples determines the training space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Therefore, the pos- itive samples of the task T𝑖 are derived from the current task while the negative samples of T𝑖 are derived from differ- ent tasks T𝑖−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' , T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Similar to the subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='2, expected losses of the dependence relationship derived from the entire space D and local space C are discussed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' If the dependence relationship is learned in the entire space D and the local space C, respectively, and the cor- responding expected losses are denoted as 𝐸𝑋,𝑌,𝑍∼D [𝐿(𝑓𝑌 (𝑋)− 𝑓𝑍 (𝑋),𝑌 −𝑍)] and 𝐸𝑋,𝑌,𝑍∼C[𝐿(𝑓𝑌 (𝑋) − 𝑓𝑍 (𝑋),𝑌 −𝑍)], then these two expected losses satisfy 𝐸𝑋,𝑌,𝑍∼D [𝐿(𝑓𝑌 (𝑋) − 𝑓𝑍 (𝑋),𝑌 − 𝑍)] = 𝐸𝑋,𝑌,𝑍∼C � 𝑃D(𝑌 = 1)𝑃D(𝑌 − 𝑍 |𝑥) 𝑃D(𝑌 − 𝑍,𝑌 = 1|𝑋) × 𝐿(𝑓𝑌 (𝑋) − 𝑓𝑍 (𝑋),𝑌 − 𝑍) � (26) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' According to Bayes’ theorem, we can obtain the fol- lowing equality: 𝑃D(𝑌 = 1)𝑃D(𝑌 − 𝑍 |𝑥) 𝑃D(𝑌 − 𝑍,𝑌 = 1|𝑋) = 𝑃D(𝑌 = 1) 𝑃D(𝑌 = 1|𝑋,𝑌 − 𝑍) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (27) Considering the right-hand side of (26) and (27), it leads to 𝐸𝑋,𝑌,𝑍∼C � 𝑃D(𝑌 = 1) 𝑃D(𝑌 = 1|𝑋,𝑌 − 𝑍) 𝐿(𝑓𝑌 (𝑋) − 𝑓𝑍 (𝑋),𝑌 − 𝑍) � = ∫ C � 𝑃D(𝑌 = 1) 𝑃D(𝑌 = 1|𝑥,𝑦 − 𝑧) 𝐿(𝑓𝑌 (𝑥) − 𝑓𝑍 (𝑥),𝑦 − 𝑧) 𝑃C(𝑥,𝑦 − 𝑧) � d𝑥d𝑦d𝑧 = ∫ D � 𝑃D(𝑌 = 1) 𝑃D(𝑌 = 1|𝑥,𝑦 − 𝑧) 𝑃D(𝑥,𝑦 − 𝑧|𝑌 = 1) × 𝐿(𝑓𝑌 (𝑥) − 𝑓𝑍 (𝑥),𝑦 − 𝑧) � d𝑥d𝑦d𝑧 = ∫ D 𝐿(𝑓𝑌 (𝑥) − 𝑓𝑍 (𝑥),𝑦 − 𝑧)𝑃D(𝑥,𝑦 − 𝑧)d𝑥d𝑦d𝑧 = 𝐸𝑋,𝑌,𝑍∼D [𝐿(𝑓𝑌 (𝑋) − 𝑓𝑍 (𝑋),𝑌 − 𝑍)], (28) which implies that (26) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' □ 4 Experiments In this section, we describe the experiments to evaluate the performance of the proposed TAFE framework, which are conducted on both public benchmark dataset and real-world industrial dataset in financial service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We also analyze the contribution of each modules consisting of TAFE to further understand the working mechanism and demonstrate the effectiveness of proposed method for sequential dependence multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei Cheng, and Wenfang Lin 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1 Experimental Setup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1 Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Experiments are conducted on two dataset: the public benchmark Ali-CCP and an industrial dataset from the financial scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Ali-CCP dataset 1 contains 84 million samples from an online E-commence recommendation platform in TaoBao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 5 million users’ clicking and conversion be- haviors are sampled from this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We consider CTR and CVR as two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Industrial Dataset is collected from a real-world fi- nancial service scenario based on our industrial on- line recommendation platform, which describes users’ preferences in financial products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' There are 73 million samples for this dataset containing 30 million users’ records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Prediction of users’ clicking and conversion behaviors on financial product like credit loan are two tasks for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='2 Baseline Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' To validate the effectiveness of TAFE, we conduct our experiments on the following repre- sentative methods for comparison, which are SOTA multi- task learning approaches or recent sequence dependency learning method: Single-Task is a three-layer MLP network with hid- den layer size of [256,128,64] for single-task optimiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Shared-Bottom constructs a shared bottom layer to learn the common representation across all tasks and introduces separated task tower for the object opti- mization respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' MMOE is inspired by the classic MoE method which adopts a group of shared bottom subnetworks as ex- perts and introduces gating network assigning differ- ent tasks with distinctive weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' PLE generalizes CGC method and employs a progres- sive routing mechanism to extract and separate deeper semantic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' AITM is a shared-bottom structure with adaptive in- formation transfer for modeling sequential dependency among multi-step conversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' TAFE is our proposed approach which adopts adaptive sample-wise representation generator and the explicit task-specific adapter, with dependency learning loss for the sequence dependence multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='3 Implementation of L𝐷−𝑇𝑎𝑠𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As discussed in sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='3, L𝐷−𝑇𝑎𝑠𝑘 can be regarded as a regularization, which constraints the relationship between sequential dependence tasks during training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In this paper, we constructs a MSE (Mean-Squared Loss) as an implementation for 𝐿 in formula (24): 1https://tianchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='aliyun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='com/dataset/dataDetail?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='dataId=408 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Detailed hyper-parameters settings for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Dataset Hyper-parameters Settings Ali-CCP 𝐵 = 1024,𝑑𝑓 = 18, 𝑀 = 2, 𝐾 = 64, 𝐿 = 4, 𝜆 = 10−3 Industrial dataset 𝐵 = 1024,𝑑𝑓 = 18, 𝑀 = 2, 𝐾 = 64, 𝐿 = 4, 𝜆 = 10−3 𝐿𝑀𝑆𝐸 = 1 𝑀 𝑀 ∑︁ 𝑗=1 𝑤 · (𝑦𝑗 − ˆ𝑦𝑗)2 (29) where 𝑦𝑗 is label from 𝑜𝑖−1 𝑗 − 𝑜𝑖 𝑗 and ˆ𝑦𝑗 is the output from 𝑓𝑖−1(𝑥𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖−1) − 𝑓𝑖 (𝑥𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='𝜃𝑠,𝜃𝑖) for input 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Each sample is equally treated with 𝑤 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='4 Training Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In the experiments, we implement all models through the Pytorch framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We randomly divide each dataset into the training set and test set chrono- logically, accounting for 90% and 10% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Each ex- periment is repeated 5 times, the average performance and the p value are both reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We select the optimal hyper- parameters for each model in terms of grid search [11] for fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The batch size 𝐵 on each datasets is set as 1024 respectively during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Adam op- timizer [9] is applied with a learning rate 𝜆 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The dimension 𝑑𝑓 of input embedding layer is 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The number of the stacked layers 𝐿, the number of the attention heads 𝑀 and the number of the inducing points 𝐾 is illustrated in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The activation function of MLP in single-task modeling is ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='2 Performance Comparison The experimental results for all comparison methods with the evaluation metric AUC for each task are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The best performance on different datasets are high- lighted in boldface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As can be observed, TAFE outperforms most baseline models for each task on both datasets respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The average performance on Ali-CCP dataset is poor both on CTR and CVR targets on all compared methods, which probably implies the input features are not qualified enough to express the targets or the irrelevant information affects significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' For the latter case, task-specific feature extrac- tion will play a key role to the prediction results in terms of filtering negative interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As observed, TAFE achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6203 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6456 of AUC for CTR and CVR tasks respectively, with gains of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='16% and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='31% compared to the Singel-Task method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The improvements of CTR is significant with com- parison to other methods but slightly poorer than PLE in the object of CVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The result seems to be attributed to the trade-off between tasks considered by TAFE and TAFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The difference improvements between CTR and CVR is smaller in TAFE and suggests a more balanced optimization among tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The performance on the Industrial dataset of TAFE Task Aware Feature Extraction Framework for Sequential Dependence Multi-Task Learning Conference’17, July 2017, Washington, DC, USA Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The performance (AUC) comparison with baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='The Gain means the mean AUC improvement compared with Single-Task method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ** indicates that the improvement of the proposed TAFE is statistically significant compared with the best baseline at a p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='01 over paired samples t-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Models Ali-CPP Industrial Dataset CTR CVR 𝐺𝑎𝑖𝑛𝐶𝑇𝑅 𝐺𝑎𝑖𝑛𝐶𝑉 𝑅 CTR CVR 𝐺𝑎𝑖𝑛𝐶𝑇𝑅 𝐺𝑎𝑖𝑛𝐶𝑉 𝑅 Single-Task 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6011 – – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7616 – – Shared-Bottom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6225 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='92% AITM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6391 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='72% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='32% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='47% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='81% TAFE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6436** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='79% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='07% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7167** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7714** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='29% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='43% Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The performance (AUC) comparison of ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Models Ali-CPP Industrial Dataset CTR CVR 𝐺𝑎𝑖𝑛𝐶𝑇𝑅 𝐺𝑎𝑖𝑛𝐶𝑉 𝑅 CTR CVR 𝐺𝑎𝑖𝑛𝐶𝑇𝑅 𝐺𝑎𝑖𝑛𝐶𝑉 𝑅 TAFE w/o ASRG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='32% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='89% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='50% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='57% TAFE w/o TSA 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='25% TAFE w/o L𝐷−𝑇𝑎𝑠𝑘 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='02% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='82% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='10% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='25% TAFE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6436 – – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='7714 – – obtains an considerable gain of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='29% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='43% for both targets and significantly outperforms other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Com- pared with PLE, which achieves a second best result, the proposed model still gets an increase of the gain by 43% and 55% and further demonstrates its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='3 Ablation Study We conduct ablation study on different submodules in TAFE in order to provide a detail analysis of its function and ef- ficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The variant models of TAFE consists of following structures and the notation is just for simplicity: TAFE without ASRG: removing the dynamic activa- tion layer in ASRG and replacing with a standard self attention operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' TAFE without TSA: removing task indicator in TSA layers for all corresponding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' TAFE without L𝐷−𝑇𝑎𝑠𝑘: removing the sequence de- pendence learning loss L𝐷−𝑇𝑎𝑠𝑘 as denoted as in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' TAFE: complete structure of TAFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The results of ablation study are presented in Table ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' with an evaluation metric AUC on both datasets for CTR and CVR tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As observed, the complete structure of TAFE outperforms all other TAFE-variants and we can draw the following conclusions for each submodule: (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Adaptive Sample-wise Representation Generator con- tributes to learn fine-grained and generalized shared repre- sentation for both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In which, dynamic selector enables to select essential information for each sample which en- hance the knowledge learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Without the dynamic selec- tion layer, model performance drops most for both CTR and CVR target as -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='5% and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='57% in Industrial dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Fully interaction learning via a standard multi-head self-attention can’t provide enough shared info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We believe that ASRG re- construct the necessary information in an adaptive manner which not only learns the feature field interaction but filter the noise by utilizing a group-level attention from a sample- wise view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The contribution in Ali-CPP is still obvious since whose features seems less expressive as discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='2 and is greatly benefited by ASRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Explicit Task-Specific Adapters works as a task-sensitive feature extractor, which is quite crucial in the multi-task learning to precisely extract task-relevant information for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' It can be evidently observed that without the task attention mechanism (proposed task indicator), model performance drops dramatically and is slightly better than without ASRG in Industrial dataset but worse in Ali-CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' It is suggested that a vanilla task-specific tower structure doesn’t generate enough information during task optimizing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The proposed sequence dependence learning loss L𝐷−𝑇𝑎𝑠𝑘 based on theoretical proof contributes to the model perfor- mance in terms of the additional information passing among related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Although it seems less significant compared to other submodules in Industrial dataset but contributes most in the CVR task in Ali-CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' CVR task probably depends on CTR task heavily and L𝐷−𝑇𝑎𝑠𝑘 modifies the biased object of original definition, which further optimizes the param- eters by recovering their complete probability-dependent relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='4 Analysis of Dynamic Selector Dynamic selector 𝑧𝐾 defined in formula (15) functions as a sparse mask generated based on the input sample, which cooperates with Inducing points 𝐼 interacting with feature Conference’17, July 2017, Washington, DC, USA Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei Cheng, and Wenfang Lin fields selectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We conduct several case studies of 𝑧𝐾 to provide an intuitive analysis as visualized in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='70 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='0% 6 4 2 0 2 4 (b) 4 2 0 2 4 Low Mid High Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' An illustration of dynamic selector 𝑧𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='Distribution of the selection rate for different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Plot of sample embeddings with high, middle and low selection rate is colored in green, yellow and blue respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As noted in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='1 , 𝑧𝐾 is a 𝐾 dimension vector only with values of 0 and 1, where 1 means interacting with im- plicit field group in terms of 𝐼 correspondingly and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We first plot the distribution of the non-zero rate (selection rate) of 𝑧𝐾 on the test samples of industrial dataset in Figure 7 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' It can be observed for most samples, the selection rate is between 55% and 60% and indicates that more than half the interaction groups are required for information extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' For specific cases, some needs just less interaction groups and some needs more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We could regard this as a multi-view representations for each sample, such as different numbers of perspectives qualified enough to describe a customer’s interest specially on online recommendation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We further randomly plots sample embeddings with high (top 1%), mid (around 58%) and low (bottom 1%) selection rates in Figure 7 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As illustrated, samples with different interaction degrees show significant difference in embedding space and probably implying distinctive intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='5 Efficiency Evaluation In this section, we evaluate time and storage efficiency of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We record the time cost during the train- ing (per epoch) and inference process for TAFE and other baseline models in Figure 8 (a), and their respective memory cost in Figure 8 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As illustrated in (a), TAFE requires 1177 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The offline performances (AUC) comparison for two real-world financial scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Models Scene 1 Scene 2 CTR CVR CTR CVR MMOE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='8102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='8034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='8110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='8719 TAFE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='8102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='8072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='8123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='8773 Gain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='47% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='16% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='62% seconds for training an epoch on the Ali-CCP dataset with 38 millions samples, which is less efficient than other methods (295s for the best results from Shared-Bottom) but similar to the PLE (1195s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Since we generally train the model in an of- fline manner especially for large-scale data, relatively higher training efficiency is within tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' On the inference time, the essential factor considered in the online industrial appli- cation, TAFE spends 47 seconds for forward propagation on the test data with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='2 millions samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Its deviation from top performance models like AITM and Shared-Bottom is 12 sec- onds and it is acceptable considering most industrial online inference situation’ QPS threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Besides, TAFE has the least parameters with 89 Mb (178 Mb for the largest model of Single-Task) as plotted in Figure 8 (b), which makes it easily deployed and portable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In a conclusion, TAFE achieves an significant improvement with an appropriate computa- tional time and advantageous storage capacity compared with other approved MTL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' PLE TAFE MMOE Single-Task AITM Shared-Bottom (a) 0 250 500 750 1000 1250 1500 train cost time(s) train time(s) 0 20 40 60 80 inference cost time(s) inference time(s) Single-Task PLE MMOE AITM Shared-Bottom TAFE (b) 0 50 100 150 Size(Mb) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Efficiency comparison among models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Training and Inference time (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Memory Cost Task Aware Feature Extraction Framework for Sequential Dependence Multi-Task Learning Conference’17, July 2017, Washington, DC, USA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='6 Online A/B Performance We implement an online A/B test between TAFE and SOTA multi-task learning method of MMOE for one week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Two models are deployed on two real-world financial advertising scenarios with the objective of maximizing the CVR for the financial products as investment and credit loan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The offline comparison is presented in Table 4, TAFE gets an improve- ment of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='16% for CTR in Scene 2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='47% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='62% for CVR on each scene correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In Figure 9, we can observe the online performances of TAFE compared with MMOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As shown in the plot, TAFE achieves significant and consistent improvements for both scenarios during the whole period, average increase of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='22 % in scenario (a) and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='87% in sce- nario (b) on the CVR task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The experiment result proves the efficiency and the stability of proposed mehtod, which is qualified enough for large-scale industrial application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Day1 Day2 Day3 Day4 Day5 Day6 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='014 CVR on Scene 1 TAFE MMoE Day1 Day2 Day3 Day4 Day5 Day6 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content='05 CVR on Scene 2 TAFE MMoE Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Online A/B results on two real-world scenarios (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 5 Related Work Multi-task Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Multi-task Learning (MTL) is pro- posed to learn the shared information among tasks to im- prove the model generalization and performance [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' How- ever, multi-task learning scenario usually suffers from the performance deterioration as negative transfer because of the complex relationship between different tasks [1, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' There- fore, much feature learning works in structure designing are proposed for necessary information extraction accord- ing to specific task and balancing the performances among all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Cross-Stitch Network [14] use a linear combina- tion of shared representations to learn the task-specific em- beddings for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Based on the idea of Cross-Stitch, Sluice Network [17] is a generalized meta-architecture with more task-specific parameters by dividing each layer into task-specific and shared subspaces and achieves better per- formance specially for less correlated tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' However, these approaches could not capture the sample dependence and require more training data and less efficient for large-scale application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Inspired by the MoE [8] structure, multi-gate Mixture-of-Experts (MMoE) [12] employs an ensemble of experts submodules and gating network to combine the rep- resentation of the bottom experts to learn the task relation- ship while consuming less computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Similarly, Multiple Relational Attention Network (MRAN) [28] models multiple relationships by three attention-based learning mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Compared with MMoE, Progressive Layered Extraction (PLE) method [19] propose a novel MTL framework as show in Figure ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' (a), which separates task-common and task-specific parameters more explicitly and adopts a progressive separa- tion routing mechanism to better alleviate parameter con- flicts caused by complex task correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Sequential Dependence Multi-task Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The most classical applications of the sequential dependence MTL (SDMTL) are the multi-step conversion process of customer acquisition in e-commerce, display advertising or finance systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' In general, the multi-step conversion process in- volves impression→click→ · · · →conversion, which cor- responds to several estimation tasks like post-view click- through rate (CTR), post-click conversion rate (CVR) and post-view click-through & conversion rate (CTCVR) estima- tions and so fourth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Differing from the general MTL, there exist dependence relationships between the adjacent tasks in the SDMTL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' For the CVR estimation problem, Entire Space Multi-task Model (ESMM) is proposed in [13] to overcome Sample Selection Bias (SSB) and Data Sparsity (DS) issues by introducing two auxiliary tasks of predicting the post-view click-through rate (CTR) and post-view click through & conversion rate (CTCVR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' With this operation, the performance of the CVR estimation will depend heavily on the performance auxiliary tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' As the number of steps increases in multi-step conversion path, the accumulation of performance errors becomes intolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Aimed at the DS problem of the CVR estimation, in [25], a novel user sequen- tial behavior graph is established to achieve post-click be- havior decomposition by inserting disjoint purchase-related deterministic action and other action into between click and conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Considering micro behaviors (user’s interactions with items) and macro behaviors (user’s interactions with specific components on the item detail page) of users, Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' [24] propose a Hierarchically Modeling both Micro and Macro behaviors for CVR prediction to address SSB and DS issues by using the abundant supervisory labels from micro and macro behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' To models the sequential depen- dence among multi-step conversions, Adaptive Information Conference’17, July 2017, Washington, DC, USA Xuewen Tao, Mingming Ha, Xiaobo Guo, Qiongxu Ma, Hongwei Cheng, and Wenfang Lin Transfer Multi-task (AITM) framework with adaptive in- formation transfer module is developed in [26] to directly predict the end-to-end conversion probabilities of each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Besides, causal approaches have also been applied to achieve the debiasing post-click conversion rate estimation lately [5, 6, 22, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' However, for the sequential dependence multi- task learning problem, there is rare literature to develop a formalization description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' 6 Conclusions In this paper, we propose a sequence dependence multi-task learning framework named as Task Aware Feature Extrac- tion (TAFE), which could selectively reconstruct implicit shared representations from a sample-wise view and extract explicit task-specific information in an more efficient way compared with common task-aware tower structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' We accomplish this by involving an Adaptive Sample-wise Rep- resentation Generator and a Task-Specific Adapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' For the multi-task learning with dependency generally encountered in E-commence online recommendation, we provide a detail theoretical proof about the dependent relationship from rig- orous mathematical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Based on our analysis, we design a dependence task learning loss to complete optimiz- ing object in an unbiased format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' The performance gains of TAFE compared to several SOTA multi-task approaches on both public and real-world industrial datasets demonstrates its effectiveness and generalization characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE0T4oBgHgl3EQfmAG-/content/2301.02494v1.pdf'} +page_content=' Besides, we carefully conduct ablation study, case study, efficiency evaluation and online A/B test to further 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Malomed3,4, and Yongyao Li1,5∗ +1School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528225, China +2Physics Department and Solid-State Institute, Technion, Haifa 32000, Israel +3Department of Physical Electronics, School of Electrical Engineering, +Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel +4Instituto de Alta Investigaci´on, Universidad de Tarapac´a, Casilla 7D, Arica, Chile +5Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano +Optoelectronic Technology, Foshan University, Foshan 528225, China +Creation of stable intrinsically anisotropic self-bound states with embedded vorticity is a chal- +lenging issue. Previously, no such states in Bose-Einstein condensates (BECs) or other physical +settings were known. Dipolar BEC suggests a unique possibility to predict stable anisotropic vor- +tex quantum droplets (AVQDs). We demonstrate that they can be created with the vortex’ axis +oriented perpendicular to the polarization of dipoles. The stability area and characteristics of the +AVQDs in the parameter space are revealed by means of analytical and numerical methods. Further, +the rotation of the polarizing magnetic field is considered, and the largest angular velocities, up to +which spinning AVQDs can follow the rotation in clockwise and anti-clockwise directions, are found. +Collisions between moving AVQDs are studied too, demonstrating formation of bound states with a +vortex-antivortex-vortex structure. A stability domain for such stationary bound states is identified. +A possibility of the creation of AVQDs in a two-component dipolar BEC is briefly considered too. +Nonlocal nonlinearities underlie remarkable phenom- +ena in diverse fields. In particular, while two and three- +dimensional (2D and 3D) self-trapped modes, supported +by the ubiquitous local cubic self-attraction, are unstable +due to the occurrence of the critical and supercritical col- +lapse in the same settings [1–3], the nonlocal nonlinear- +ity arrests the onset of the collapse, thus stabilizing the +2D and 3D solitons [4, 5]. In Bose-Einstein condensates +(BECs), stable 2D and 3D matter-wave solitons were pre- +dicted in free space, making use of long-range van der +Waals interactions between Rydberg atoms [6, 7] or laser- +induced artificial gravity [8], microwave-coupled binary +condensates [9], and dipole-dipole interactions (DDIs) +[10–13]. +Unlike other nonlocal interactions, DDIs fea- +ture strong anisotropy in 3D, while in the 2D geometry +DDIs are isotropic or anisotropic if the dipoles are polar- +ized, respectively, perpendicular to the system’s plane or +making an angle < 90o with it. +The stability of 2D and 3D self-trapped modes with +embedded vorticity is another challenging problem. Soli- +tary vortex modes are often subject to the azimuthal +modulational instability that develops faster than the +collapse, splitting the vortices into fragments [5]. The +nonlocality can help to suppress the splitting instabil- +ity. In BECs, stable vortex solitons supported by non- +local interactions were reported for Rydberg atoms in +3D [14], microwave-coupled binary BECs [9], and dipo- +lar BECs with specially arranged isotropic DDIs in 2D +[15]. All these solitons featuring isotropic shapes, an open +question is whether anisotropic solitons with embedded +vorticity (topological charge) may be made stable in the +free space. Anisotropic DDIs offer a possibility to con- +∗Electronic address: yongyaoli@gmail.com +struct them. +In particular, spin-orbit coupling (SOC) +has helped to predict stable anisotropic solitons mixing +fundamental (zero-vorticity) and vortex components in +the spinor-dipolar BEC [16, 17]. However, vortex com- +ponents play a subordinate role in the SOC system, car- +rying a small part of the soliton’s norm. +Recently, 3D self-bound states in dipolar BECs were +observed in the form of quantum droplets (QDs), which +are stabilized with the help of the beyond-mean-field +(MF) effect, which is represented by the Lee-Huang-Yang +(LHY) term in the respective Gross-Pitaevskii equation +(GPE) [18, 19]. The experimentally demonstrated QDs +feature a strong anisotropy in their density profile in the +free space, but they do not carry vorticity. Their counter- +parts, in the form of isotropic QDs, were experimentally +created in quasi-2D [20] and 3D [21, 22] forms in binary +BECs with attractive inter-component interactions, as +predicted by Petrov [23]. +The stability and shapes of +these self-trapped quantum-fluid states are determined +by the competition between the MF and LHY nonlinear- +ities [24–26]. +This setting is favorable for stabilizing self-bound vor- +tex modes. Isotropic 2D and 3D vortex QDs in the free- +space binary BEC have been predicted to be stable with +topological charges S ≤ 5 [27] and S ≤ 2 [28], respec- +tively. Stable semi-discrete vortex QDs, also with S ≤ 5, +were predicted in an array of tunnel-coupled quasi-1D +potential traps [29]. +For the dipolar QDs, isotropic vortex solutions with +the dipoles polarized parallel to the vortical pivot were +constructed and found to be completely unstable [30]. +No previous work addressed a possibility to construct +strongly anisotropic vortex QD solutions with crossed +dipole polarization and vortical axis (pivot). The main +objective of the present work is to address this option, +and analyze stability of such states. + +2 +We consider anisotropic vortex quantum droplets +(AVQDs) in the 2D geometry, with the dipoles polar- +ized parallel to the plane in which the vortex structure is +built, and the vortex’ axis being directed perpendicular +to the plane and the polarization. Dynamics of this sys- +tem is governed by the scaled form of the 2D GPE with +the LHY correction as +i ∂ +∂tψ = +� +−1 +2∇2 + Φdd(r) + g|ψ|2 + γ|ψ|3 +� +ψ, +(1) +where g > 0 and +γ = 4g5/2 +3π2 +� +1 + 8π2 +3g2 +� +> 0 +(2) +[31] are, respectively, strengths of the local MF and +LHY self-repulsion. The DDI is represented by the term +Φdd(r) = +� +R(r − r′)|ψ(r′)|2dr′, where the kernel of the +long-range interaction is +R(r − r′) = +1 − 3 cos2 Θ +[b2 + (r − r′)2]3/2 , +(3) +with a cutoff scale b [32, 33]. This kernel implies that all +the dipoles are polarized along the x-direction in the 2D +plane, hence cos2 Θ = (x − x′) /|r−r′|2. In this case, the +anisotropic DDIs are chiefly attractive. +Stationary solutions are looked for in the usual form, +ψ(r, t) = φ(r)e−iµt, with wave function φ(r) and real +chemical potential µ. Dynamical invariants of the system +are the total norm and momentum +N = +� +|φ(r)|2dr, +P = i +� +ψ∇ψ∗dr, +(4) +where N is proportional to the number of atoms in the +dipolar BEC, and its energy, +E = 1 +2 +� +dr +� +|∇ψ|2 + g|ψ|4 + Φdd(r)|ψ|2 + 4 +5γ|ψ|5 +� +. (5) +AVQD solutions with integer vorticity S are produced +in the numerical form by means of the imaginary-time +method (ITM), initiated by an anisotropic ansatz, +φ(0)(x, y) = A˜rS exp +� +−α˜r2 + iS˜θ +� +, +(6) +where A and α are positive real constants, and +� +˜r, ˜θ +� +≡ +�� +x2 + β2y2, arctan(βy/x) +� +with an anisotropy factor +β > 1. In this work, we set b = 1 in and β = 2, using N +and g as control parameters. +A typical example of the numerically found stable +AVQD with S = 1 is produced in Figs. 1(a1,a2). The +stability of the AVQDs was tested by direct simulations +of the perturbed evolution for a sufficiently long time [34]. +The stability area for them in the (N, g) plane is shown in +Fig. 1(b). In particular, the stable AVQDs are found at +N > Nmin, where Nmin is a gradually increasing function +FIG. 1: +(a1,a2) +A typical example of +stable +AVQDs +(anisotropic +vortex +quantum +droplets) +with +(N, g) += +(1000, 0.25). +The panels (a1,a2) display, severally, density +and phase patterns of the droplets. (b) In the plane of (N, g), +stable AVQDs with S = 1 and fundamental QDs with S = 0 +coexist in the orange bistability area. In the green area, only +the fundamental QDs are stable. +of g. At N < Nmin, an unstable dipole mode is produced, +instead of the vortex with S = 1, which decays to the +fundamental QD in direct simulations. In the horizontal +direction, stable AVQDs are found at g > gmin, which is +a function of N, e.g., gmin(N = 500) ≈ 0.2. At values +of g slightly smaller than gmin, the AVQDs start spon- +taneous drift, keeping their topological charge. Deeper +into the region of g < gmin, the input given by Eq. (6) +generates an unstable dipole solution, which decays into +a fundamental QD in direct simulations, similar to what +is said above concerning the case of N < Nmin. +An analytical consideration can be developed as fol- +lows. +For stationary QDs with a large norm, one can +apply the Thomas-Fermi (TF) approximation, neglect- +ing the kinetic-energy term in Eq. (1). In this case, the +equilibrium density in the QD, ne, determines the total +energy (5) as +E = 1 +2 +� +εn2 +e + gn2 +e + 4 +5γn5/2 +e +� +Ae, +(7) +where Ae = N/ne is the equilibrium area of the QDs, and +ε = +� +drR(r) ≈ −3.23 represents the nonlocality effect. +Since the equilibrium values provide an energy minimum, +solving dE/dne = 0 and dE/dAe = 0 yield the values of +ne and Ae as +√ne = − 5 +6γ (ε + g), +Ae = 36 +25γ2 +N +(ε + g)2 . +(8) +An obvious condition, √ne > 0, applied to Eq. (8), leads +to g < gmax ≡ −ε ≈ 3.23. At g > gmax, the strong lo- +cal repulsion overcomes the effective nonlocal attraction, +hence no self-bound state can be formed. However, at +g > 2, ne becomes very small and the size of the AVQD +very large, which makes it difficult to reach g = gmax +in the numerical simulations. Furthermore, the chemical +potential of the equilibrium state is found as +µe = (ε + g)ne + γn3/2 +e +. +(9) +These analytical predictions are compared to numerical +findings below in Figs. 2(a1-a3,b1-b3). + +2.0 +10 +(al) +1.5 +(b) +0 +1 +1.5 +0.5 +-10 +60 1.0 +10 +(a2) +2 +0.5 +0 +0 +-10 +-2 +0.0 +0 +500 +1000 1500 2000 +-40 +-20 +0 +20 +40 +N +x3 +FIG. 2: The peak density (IP), chemical potential (µ), effec- +tive area (Aeff), ellipticity (El), and total orbital momentum +(¯Lz), see Eq. (10) versus N (a1-a5) and g (b1-b5). In panels +(a1-a5), g = 0.25 is fixed, while in panels (b1-b5) N = 1000 is +fixed. Red dashed curves in panels (a1-a3,b1-b3) represent the +analytical approximation given by Eqs. (8,9), respectively. +To study the AVQD families systematically, we define +their effective area, ellipticity, and angular momentum: +Aeff = +�� |φ|2dr +�2 +� +|φ|4dr +, El = Wy +Wx +, ¯Lz = +� φ∗ ˆLzφ +N +dr, +(10) +where Wy ≡ +�� |φ(x = 0, y)|2dy +�2 / � |φ(x = 0, y)|4dy, +Wx ≡ +�� +|φ(x, y = 0)|2dx +�2 / +� +|φ(x, y = 0)|4dx, and +ˆLz = −i(x∂y − y∂x). +Figure 2 displays these quanti- +ties (along with IP = |φ|2 +max and µ) versus N and g in +the stability area. +In panel 2(a1), the peak value saturates at (IP)sat ≈ +1.938 if N is sufficiently large, as is expected for the +incompressible quantum fluid. +According to Eq. +(8), +the analytical prediction of the equilibrium density is +ne ≈ 1.942, in close agreement with (IP)sat. Panel 2(a2) +shows that the chemical potential satisfies the Vakhitov- +Kolokolov (VK) criterion, dµ/dN < 0, which is the +well-known necessary stability condition for self-trapped +modes [1–3, 35]. For large N the chemical potential sat- +urates at µ ≈ −0.889. The analytical prediction given by +Eq. (9) is µe ≈ −0.965, which is also close to the numer- +ical result. In panels 2(b1,b2), both IP and µ decay to +zero at g → 2, which agrees well with the analytical pre- +dictions provided by Eqs. (8, 9). In panels 2(a3,b3), the +effective area closely matches the analytical result (8). +In panels 2(a4,b4) the ellipticity remains smaller than +0.25, which indicates that the AVQDs manifest strong +anisotropy with the elongation along the x-direction. Fi- +nally, when the AVQDs features a sufficiently anisotropic +profile (namely, at El ≤ 0.2), the total momentum and +ellipticity roughly satisfy ¯Lz ≈ 2El. +We also find stationary AVQD solutions with higher +vorticities, S ≥ 2, in the numerical form. However, sim- +ulations demonstrate that they are fully unstable [36]. +It is known that zero-vorticity solitons in the dipolar +BECs can rotate, adiabatically following slow in-plane +rotation of the magnetic field which polarizes the mag- +netic moments of the condensates [13]. The rotation can +be introduced in Eq. (3) by replacing Θ → Θ+ωt. When +the rotation is sufficiently slow, viz., ω < ωcr, the AVQD +is able to follow it, in the state of spinning motion. Fig- +ures 3(a-d) show an example of the steady rotation of +an AVQD. Numerical simulations demonstrate that ωcr +decreases with the increase of the size of the soliton. In- +deed, the large size of the QD makes it difficult to syn- +chronize the rotational motion of its core area and remote +edges. The intrinsic vorticity of the AVQD makes it more +tolerant to the rotation in the direction of the inner vor- +ticity than in the opposite one, therefore the simulations +demonstrate two different critical values, ω(±) +cr , as shown +in Figs. 3(e,f). +FIG. 3: (a-d) Steady spinning of an AVQD with (N, g) = +(1000, 0.25), which follows the rotation of the polarizing mag- +netic field with angular velocity ω = 0.25π ×10−3. The shape +of the AVQD is displayed at t = 0 (a), 1000 (b) 2000 (c), 3000 +(d). Panels (e) and (f): the largest angular velocity +���ω(±) +cr +���, +which admits the stable spinning of the AVQD in either of +the two opposite directions, vs. g (at N = 1000) and N (at +g = 0.25), respectively. +FIG. 4: The collision of two AVQDs with identical vorticities, +initiated, at t = 0, by input φ(x−x0, y)e−iηx+φ(x+x0, y)eiηx +with x0 = 64, η = 0.025, g = 0.25, and norm of each AVQD +N = 1000. (a-d) Density patterns at t = 550 (a), 635 (b), 760 +(c), and 900 (d). +Equation (1) being invariant with respect to the +Galilean boost, stable AVQDs can be set in motion by +opposite kicks ±η applied along the x or y-direction. Ac- +cordingly, it is possible to simulate collisions between +AVQDs moving in opposite directions. Results demon- +strate a drastic difference from the usual scenario of colli- +sions in local non-integrable systems, where the increase +of η leads to a transition from inelastic collisions be- +tween slow solitons to quasi-elastic outcomes for fast ones +[37]. In the present setting, elastic collisions are only ob- +served between AVQDs moving in the y-direction if kick +η is relatively small. If η, applied in the y-direction, is +larger, or the head-on collision happens in other direc- + +20 +V0 +-20 +-50 +0 +50 +-50 +0 +50 +-50 +0 +50 +-50 +0 +50 +x +x +x +x +(a) +(b) +(c) +(d)40 +t =0 +t=1000 +5 +3 +) +(-) +c1 +G +cr +(+) +(+) +0 +Wcr +4 +L +-40 +(a) +(b) +2 +3 +40 +t=2000 +t =3000 +2 +- +1 f(e) +(f) +-40 +(c) +(d) +0.3 +0.4 +0.5 500 +10001500 2000 +-40 +0 +40-40 +0 +40 +g +N +x +x2 +1.89 +P +1.62 +1 +(b1) +(a1) +1.35 +0 +-0.6 +0.0 +-0.8 +(a2) +-0.4 +(b2) +-1.0 +-0.8 +1200 +20000 +eff +600 +10000 +A +(a3) +(b3) +0 +0.21 +0.25 +3 +0.18 +0.20 +(a4) +0.15 +(b4) +0.15 +0.36 +0.5 +0.33 +(a5) +0.4 +0.30 +(b5) +0.3 +500 +1000 +1500 +2000 +0.5 +1.0 +1.5 +2.0 +N +g4 +tions in the (x, y) plane, the outcome is inelastic, leading +to merger of AVQDs with identical (S1 = S2 = 1) or op- +posite (S1 = −S2 = 1) vorticities into localized breathing +modes. A noteworthy result is produced by the collision +between the AVQDs with S1 = S2 = 1 traveling in the x- +direction (in which the dipoles are polarized): formation +of a transient state in the form of a breathing vortex- +antivortex-vortex structure with three respective pivots. +Eventually, this long-lived state transforms into a zero- +vorticity breathing one (the present anisotropic system +does not conserve the angular momentum, therefore the +total vorticity is not conserved either). A typical exam- +ple of such a collision is displayed in Fig. 4, where the +central pivot, which represents the antivortex, emerges in +the beginning of the merger of the two colliding vortices +[see panels (b) in the figure]. +FIG. 5: (a) Density and phase patterns of the stable vortex- +antivortex-vortex bound state with (N, g) = (1000, 0.25), +x0 = 13.6 and µ = −0.8482. +(b,c) The chemical potential +of the states of this type versus g (at N = 1000) and N +(at g = 0.25), respectively. +Solid and dashed parts of the +curves represent, severally, stable and unstable states. (d1- +d4) Steady spinning of a vortex-antivortex-vortex bound state +with (N, g) = (1000, 0.25), and the angular velocity of the ro- +tational polarizing magnetic field is ω = 0.2π × 10−3. The +shape of the bound state is displayed at t = 0 (d1), 1250 (d2) +2500 (d3), 3750 (d4). +The production of the above-mentioned long-lived +vortex-antivortex-vortex breather by collisions along the +x direction suggests that the system may support truly +stationary bound states with a similar structure. They +can indeed be produced by means of ITM, starting from +input +φ(0) = +� +± +A±˜r± · e−α±˜r2 +±+i˜θ± + A˜re−α˜r2−i˜θ, +(11) +where A± > 0 and α± > 0 are real constants, ˜r± ≡ +� +(x ± x0)2 + β2y2, ˜θ± ≡ arctan[βy/(x ± x0)], and x0 is +an appropriately chosen separation. A typical example +of such a stable bound state is displayed in Fig. 5(a). +The family of stable bound states is characterized by +dependences of the chemical potential on g and N, as +shown in Figs. 5(b,c). In the latter panels, stable bound +state of the vortex-antivortex-vortex type populate ar- +eas g < 0.35 and 900 < N < 2000. Note also that the +µ(N) relation satisfies the aforementioned VK criterion, +dµ/dN < 0. Similar to what is presented in Fig. 3, it is +possible to apply a rotating magnetic field to the bound +states of the present type, and test a possibility of their +steady spinning motion, see Figs. 5(d1-d4). +In the underlying equation (1), which applies to the +single-component dipolar BEC, nonlinearity constants g +and γ are related by Eq. (2). Following works [23, 38], +it is possible to extend the model for a two-component +system, (ψ1, ψ2). Limiting it to the symmetric set, with +ψ1 = ψ2 = ψ/ +√ +2, one arrives at the equation similar to +Eq. (1), but with independent coefficients in front of the +cubic and quartic terms [39], +i∂tψ = +� +−1 +2∇2 + Φdd(r) + δg|ψ|2 + γ|ψ|3 +� +ψ. +(12) +Here δg ≡ g12 + √g11g22, with coefficients g11 = g22 ≡ +g > 0 and g12 < 0 representing the strength of the +self-repulsion and cross-attraction of the two compo- +nents. Typical examples of a stable AVQD and vortex- +antivortex-vortex bound state produced by Eq. (12) are +shown in Fig. 6 for δg = 0 (i.e., in the case when the MF +interaction is exactly cancelled [40, 41]). These results +indicate that robust AVQD and vortex-antivortex-vortex +bound state exist in the binary condensate as well. +FIG. 6: +Examples of a stable AVQD (left) and vortex- +antivortex-vortex bound state (right) produced by Eq. (12) +with N = 1000, γ = 2.5, and δg = 0. The upper and lower +panels display density and phase patterns of the droplets. +Conclusion We have constructed solutions for stable +AVQDs in the effectively two-dimensional dipolar BEC. +The anisotropy and stability are stipulated by the choice +of the polarization of atomic dipoles parallel to the sys- +tem’s plane and perpendicular to the vortex’ axis. The +stability area of the AVQDs is identified in the system’s +parameter space. Characteristic features of the AVQDs, +such as the peak density, chemical potential, effective +area, ellipticity, and total angular momentum, are pre- +sented. Spinning AVQDs can stably follow rotation of +the polarizing magnetic field, provided that the rotation +is not too fast. +Collisions between slow or fast mov- +ing AVQDs are elastic or inelastic, respectively. In the +latter case, the colliding AVQDs merge into breathers. +In particular, these may be bound states of the vortex- +antivortex-vortex type, which are also found as stable +stationary states. + +(al) +(b1) +10 +0.8 +0.8 +0.6 +0 +0.4 +-10 +0.2 +(a2) +10 +2 +(b2) +2 +0 +0 +0 +-10 +-2 +-2 +-50 +0 +50 +-50 +0 +50 +x +x-0.82 +10 +(al) +1.5 +-0.5 +(b) +(c) +>0 +1 +-0.84 +0.5 +-0.6 +-10 +-0.86 +10 +(a2) +2 +-0.7 +0 +0 +-0.88 +-0.8 +-10 +-2 +-0.90 +0.25 +0.30 +0.35 +0.40 +1000 1500 2000 +-40 +-20 +0 +20 +40 +N +g +x +40 +20 +0 +-20 +(dl +(d4) +(d2 +(d3) +-40 +-40 -20 +0 +20 +40-40-20 +0 +20 +40-40 -20 +0 +20 40-40 -20 +20 +40 +x +x +x +x5 +The present analysis can be extended further. First, it +will be interesting to apply initial torque to an elongated +AVQD mode, and simulate ensuing dynamics, which is +expected to feature oscillations of the droplet’s orienta- +tion around the original elongated direction. Further, it +may also be relevant to simulate motion of a spinning +AVQD, driven by the rotating magnetic field, under the +action of a kick applied to the AVQD. Another relevant +possibility is to construct QD modes with hidden vortic- +ity in the two-component system, i.e., bound states with +vorticities ±1 in the two components with identical den- +sity profiles, cf. Ref. [27]. Finally, a challenging option +is to seek for stable AVQDs in the full 3D setting. +Acknowledgments +This work was supported by NNSFC (China) through +Grants No. 12274077, 11874112, 11905032, by the Nat- +ural Science Foundation of Guangdong province through +Grant No. 2021A1515010214, and 2021A1515111015, the +Key Research Projects of General Colleges in Guangdong +Province through grant No. 2019KZDXM001, the Re- +search Fund of Guangdong-Hong Kong-Macao Joint Lab- +oratory for Intelligent Micro-Nano Optoelectronic Tech- +nology through grant No.2020B1212030010. 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The corresponding simulation times are ∼ +18000 and 32000, respectively. +[35] N. G. Vakhitov and A. A. Kolokolov, Stationary solu- +tions of the wave equation in a medium with nonlinearity +saturation, Radiophys. Quantum Electron. 16, 783-789 +(1973). +[36] AVQDs with S ≥ 2 can be generated by initiating the +ITM with ansatz (6). The result is splitting into an ar- +ray of S unitary vortices set along the stretched axis of +the AVQD. However, these self-bound composite vortex +states are completely unstable. . +[37] Yu. S. Kivshar and B. A, Malomed, Dynamics of solitons +in nearly integrable systems, Rev. Mod. Phys. 61, 763- +915 (1989). +[38] D. S. Petrov and G. E. Astrakharchik, Ultradilute Low- +Dimensional Liquids, Phys. Rev. Lett. 117, 100401 +(2016). +[39] A. Boudjemˆaa, Fluctuations and quantum self-bound +droplets in a dipolar Bose-Bose mixture, Phys. Rev. A +98, 033612 (2018). +[40] N. B. Jørgensen, G. M. Bruun, and J. J. 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A 106, 063315 (2022). + diff --git a/WtE3T4oBgHgl3EQfFgks/content/tmp_files/load_file.txt b/WtE3T4oBgHgl3EQfFgks/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e5879c96a143a3c5093b35e4dbaa25e443bf281d --- /dev/null +++ b/WtE3T4oBgHgl3EQfFgks/content/tmp_files/load_file.txt @@ -0,0 +1,620 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf,len=619 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='04305v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='quant-gas] 11 Jan 2023 Anisotropic vortex quantum droplets in dipolar Bose-Einstein condensates Guilong Li1, Xunda Jiang1, Bin Liu1, Zhaopin Chen2, Boris A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Malomed3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' and Yongyao Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5∗ 1School of Physics and Optoelectronic Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Foshan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Foshan 528225,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' China 2Physics Department and Solid-State Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Technion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Haifa 32000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Israel 3Department of Physical Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' School of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Faculty of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Tel Aviv University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Tel Aviv 69978,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Israel 4Instituto de Alta Investigaci´on,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Universidad de Tarapac´a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Casilla 7D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Arica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Chile 5Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Foshan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Foshan 528225,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' China Creation of stable intrinsically anisotropic self-bound states with embedded vorticity is a chal- lenging issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Previously, no such states in Bose-Einstein condensates (BECs) or other physical settings were known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Dipolar BEC suggests a unique possibility to predict stable anisotropic vor- tex quantum droplets (AVQDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' We demonstrate that they can be created with the vortex’ axis oriented perpendicular to the polarization of dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The stability area and characteristics of the AVQDs in the parameter space are revealed by means of analytical and numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Further, the rotation of the polarizing magnetic field is considered, and the largest angular velocities, up to which spinning AVQDs can follow the rotation in clockwise and anti-clockwise directions, are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Collisions between moving AVQDs are studied too, demonstrating formation of bound states with a vortex-antivortex-vortex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' A stability domain for such stationary bound states is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' A possibility of the creation of AVQDs in a two-component dipolar BEC is briefly considered too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Nonlocal nonlinearities underlie remarkable phenom- ena in diverse fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In particular, while two and three- dimensional (2D and 3D) self-trapped modes, supported by the ubiquitous local cubic self-attraction, are unstable due to the occurrence of the critical and supercritical col- lapse in the same settings [1–3], the nonlocal nonlinear- ity arrests the onset of the collapse, thus stabilizing the 2D and 3D solitons [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In Bose-Einstein condensates (BECs), stable 2D and 3D matter-wave solitons were pre- dicted in free space, making use of long-range van der Waals interactions between Rydberg atoms [6, 7] or laser- induced artificial gravity [8], microwave-coupled binary condensates [9], and dipole-dipole interactions (DDIs) [10–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Unlike other nonlocal interactions, DDIs fea- ture strong anisotropy in 3D, while in the 2D geometry DDIs are isotropic or anisotropic if the dipoles are polar- ized, respectively, perpendicular to the system’s plane or making an angle < 90o with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The stability of 2D and 3D self-trapped modes with embedded vorticity is another challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Soli- tary vortex modes are often subject to the azimuthal modulational instability that develops faster than the collapse, splitting the vortices into fragments [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The nonlocality can help to suppress the splitting instabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In BECs, stable vortex solitons supported by non- local interactions were reported for Rydberg atoms in 3D [14], microwave-coupled binary BECs [9], and dipo- lar BECs with specially arranged isotropic DDIs in 2D [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' All these solitons featuring isotropic shapes, an open question is whether anisotropic solitons with embedded vorticity (topological charge) may be made stable in the free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Anisotropic DDIs offer a possibility to con- ∗Electronic address: yongyaoli@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='com struct them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In particular, spin-orbit coupling (SOC) has helped to predict stable anisotropic solitons mixing fundamental (zero-vorticity) and vortex components in the spinor-dipolar BEC [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' However, vortex com- ponents play a subordinate role in the SOC system, car- rying a small part of the soliton’s norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Recently, 3D self-bound states in dipolar BECs were observed in the form of quantum droplets (QDs), which are stabilized with the help of the beyond-mean-field (MF) effect, which is represented by the Lee-Huang-Yang (LHY) term in the respective Gross-Pitaevskii equation (GPE) [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The experimentally demonstrated QDs feature a strong anisotropy in their density profile in the free space, but they do not carry vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Their counter- parts, in the form of isotropic QDs, were experimentally created in quasi-2D [20] and 3D [21, 22] forms in binary BECs with attractive inter-component interactions, as predicted by Petrov [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The stability and shapes of these self-trapped quantum-fluid states are determined by the competition between the MF and LHY nonlinear- ities [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' This setting is favorable for stabilizing self-bound vor- tex modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Isotropic 2D and 3D vortex QDs in the free- space binary BEC have been predicted to be stable with topological charges S ≤ 5 [27] and S ≤ 2 [28], respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Stable semi-discrete vortex QDs, also with S ≤ 5, were predicted in an array of tunnel-coupled quasi-1D potential traps [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' For the dipolar QDs, isotropic vortex solutions with the dipoles polarized parallel to the vortical pivot were constructed and found to be completely unstable [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' No previous work addressed a possibility to construct strongly anisotropic vortex QD solutions with crossed dipole polarization and vortical axis (pivot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The main objective of the present work is to address this option, and analyze stability of such states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 2 We consider anisotropic vortex quantum droplets (AVQDs) in the 2D geometry, with the dipoles polar- ized parallel to the plane in which the vortex structure is built, and the vortex’ axis being directed perpendicular to the plane and the polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Dynamics of this sys- tem is governed by the scaled form of the 2D GPE with the LHY correction as i ∂ ∂tψ = � −1 2∇2 + Φdd(r) + g|ψ|2 + γ|ψ|3 � ψ, (1) where g > 0 and γ = 4g5/2 3π2 � 1 + 8π2 3g2 � > 0 (2) [31] are, respectively, strengths of the local MF and LHY self-repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The DDI is represented by the term Φdd(r) = � R(r − r′)|ψ(r′)|2dr′, where the kernel of the long-range interaction is R(r − r′) = 1 − 3 cos2 Θ [b2 + (r − r′)2]3/2 , (3) with a cutoff scale b [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' This kernel implies that all the dipoles are polarized along the x-direction in the 2D plane, hence cos2 Θ = (x − x′) /|r−r′|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In this case, the anisotropic DDIs are chiefly attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Stationary solutions are looked for in the usual form, ψ(r, t) = φ(r)e−iµt, with wave function φ(r) and real chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Dynamical invariants of the system are the total norm and momentum N = � |φ(r)|2dr, P = i � ψ∇ψ∗dr, (4) where N is proportional to the number of atoms in the dipolar BEC, and its energy, E = 1 2 � dr � |∇ψ|2 + g|ψ|4 + Φdd(r)|ψ|2 + 4 5γ|ψ|5 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (5) AVQD solutions with integer vorticity S are produced in the numerical form by means of the imaginary-time method (ITM), initiated by an anisotropic ansatz, φ(0)(x, y) = A˜rS exp � −α˜r2 + iS˜θ � , (6) where A and α are positive real constants, and � ˜r, ˜θ � ≡ �� x2 + β2y2, arctan(βy/x) � with an anisotropy factor β > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In this work, we set b = 1 in and β = 2, using N and g as control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' A typical example of the numerically found stable AVQD with S = 1 is produced in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 1(a1,a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The stability of the AVQDs was tested by direct simulations of the perturbed evolution for a sufficiently long time [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The stability area for them in the (N, g) plane is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In particular, the stable AVQDs are found at N > Nmin, where Nmin is a gradually increasing function FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 1: (a1,a2) A typical example of stable AVQDs (anisotropic vortex quantum droplets) with (N, g) = (1000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The panels (a1,a2) display, severally, density and phase patterns of the droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (b) In the plane of (N, g), stable AVQDs with S = 1 and fundamental QDs with S = 0 coexist in the orange bistability area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In the green area, only the fundamental QDs are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' At N < Nmin, an unstable dipole mode is produced, instead of the vortex with S = 1, which decays to the fundamental QD in direct simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In the horizontal direction, stable AVQDs are found at g > gmin, which is a function of N, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=', gmin(N = 500) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' At values of g slightly smaller than gmin, the AVQDs start spon- taneous drift, keeping their topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Deeper into the region of g < gmin, the input given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (6) generates an unstable dipole solution, which decays into a fundamental QD in direct simulations, similar to what is said above concerning the case of N < Nmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' An analytical consideration can be developed as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' For stationary QDs with a large norm, one can apply the Thomas-Fermi (TF) approximation, neglect- ing the kinetic-energy term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In this case, the equilibrium density in the QD, ne, determines the total energy (5) as E = 1 2 � εn2 e + gn2 e + 4 5γn5/2 e � Ae, (7) where Ae = N/ne is the equilibrium area of the QDs, and ε = � drR(r) ≈ −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='23 represents the nonlocality effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Since the equilibrium values provide an energy minimum, solving dE/dne = 0 and dE/dAe = 0 yield the values of ne and Ae as √ne = − 5 6γ (ε + g), Ae = 36 25γ2 N (ε + g)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (8) An obvious condition, √ne > 0, applied to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (8), leads to g < gmax ≡ −ε ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' At g > gmax, the strong lo- cal repulsion overcomes the effective nonlocal attraction, hence no self-bound state can be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' However, at g > 2, ne becomes very small and the size of the AVQD very large, which makes it difficult to reach g = gmax in the numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Furthermore, the chemical potential of the equilibrium state is found as µe = (ε + g)ne + γn3/2 e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (9) These analytical predictions are compared to numerical findings below in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 2(a1-a3,b1-b3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='0 10 (al) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5 (b) 0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5 10 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='0 10 (a2) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5 0 0 10 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='0 0 500 1000 1500 2000 40 20 0 20 40 N x3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 2: The peak density (IP), chemical potential (µ), effec- tive area (Aeff), ellipticity (El), and total orbital momentum (¯Lz), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (10) versus N (a1-a5) and g (b1-b5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In panels (a1-a5), g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='25 is fixed, while in panels (b1-b5) N = 1000 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Red dashed curves in panels (a1-a3,b1-b3) represent the analytical approximation given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (8,9), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' To study the AVQD families systematically, we define their effective area, ellipticity, and angular momentum: Aeff = �� |φ|2dr �2 � |φ|4dr , El = Wy Wx , ¯Lz = � φ∗ ˆLzφ N dr, (10) where Wy ≡ �� |φ(x = 0, y)|2dy �2 / � |φ(x = 0, y)|4dy, Wx ≡ �� |φ(x, y = 0)|2dx �2 / � |φ(x, y = 0)|4dx, and ˆLz = −i(x∂y − y∂x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Figure 2 displays these quanti- ties (along with IP = |φ|2 max and µ) versus N and g in the stability area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In panel 2(a1), the peak value saturates at (IP)sat ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='938 if N is sufficiently large, as is expected for the incompressible quantum fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (8), the analytical prediction of the equilibrium density is ne ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='942, in close agreement with (IP)sat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Panel 2(a2) shows that the chemical potential satisfies the Vakhitov- Kolokolov (VK) criterion, dµ/dN < 0, which is the well-known necessary stability condition for self-trapped modes [1–3, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' For large N the chemical potential sat- urates at µ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='889.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The analytical prediction given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (9) is µe ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='965, which is also close to the numer- ical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In panels 2(b1,b2), both IP and µ decay to zero at g → 2, which agrees well with the analytical pre- dictions provided by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (8, 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In panels 2(a3,b3), the effective area closely matches the analytical result (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In panels 2(a4,b4) the ellipticity remains smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='25, which indicates that the AVQDs manifest strong anisotropy with the elongation along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Fi- nally, when the AVQDs features a sufficiently anisotropic profile (namely, at El ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='2), the total momentum and ellipticity roughly satisfy ¯Lz ≈ 2El.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' We also find stationary AVQD solutions with higher vorticities, S ≥ 2, in the numerical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' However, sim- ulations demonstrate that they are fully unstable [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' It is known that zero-vorticity solitons in the dipolar BECs can rotate, adiabatically following slow in-plane rotation of the magnetic field which polarizes the mag- netic moments of the condensates [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The rotation can be introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (3) by replacing Θ → Θ+ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' When the rotation is sufficiently slow, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=', ω < ωcr, the AVQD is able to follow it, in the state of spinning motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Fig- ures 3(a-d) show an example of the steady rotation of an AVQD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Numerical simulations demonstrate that ωcr decreases with the increase of the size of the soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In- deed, the large size of the QD makes it difficult to syn- chronize the rotational motion of its core area and remote edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The intrinsic vorticity of the AVQD makes it more tolerant to the rotation in the direction of the inner vor- ticity than in the opposite one, therefore the simulations demonstrate two different critical values, ω(±) cr , as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 3(e,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 3: (a-d) Steady spinning of an AVQD with (N, g) = (1000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='25), which follows the rotation of the polarizing mag- netic field with angular velocity ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='25π ×10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The shape of the AVQD is displayed at t = 0 (a), 1000 (b) 2000 (c), 3000 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Panels (e) and (f): the largest angular velocity ���ω(±) cr ���, which admits the stable spinning of the AVQD in either of the two opposite directions, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' g (at N = 1000) and N (at g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='25), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 4: The collision of two AVQDs with identical vorticities, initiated, at t = 0, by input φ(x−x0, y)e−iηx+φ(x+x0, y)eiηx with x0 = 64, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='025, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='25, and norm of each AVQD N = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (a-d) Density patterns at t = 550 (a), 635 (b), 760 (c), and 900 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Equation (1) being invariant with respect to the Galilean boost, stable AVQDs can be set in motion by opposite kicks ±η applied along the x or y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Ac- cordingly, it is possible to simulate collisions between AVQDs moving in opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Results demon- strate a drastic difference from the usual scenario of colli- sions in local non-integrable systems, where the increase of η leads to a transition from inelastic collisions be- tween slow solitons to quasi-elastic outcomes for fast ones [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In the present setting, elastic collisions are only ob- served between AVQDs moving in the y-direction if kick η is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' If η, applied in the y-direction, is larger, or the head-on collision happens in other direc- 20 V0 20 50 0 50 50 0 50 50 0 50 50 0 50 x x x x (a) (b) (c) (d)40 t =0 t=1000 5 3 ) (-) c1 G cr (+) (+) 0 Wcr 4 L 40 (a) (b) 2 3 40 t=2000 t =3000 2 1 f(e) (f) 40 (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5 500 10001500 2000 40 0 40-40 0 40 g N x x2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='89 P 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='62 1 (b1) (a1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='35 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='8 (a2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='4 (b2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='8 1200 20000 eff 600 10000 A (a3) (b3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='25 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='20 (a4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='15 (b4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='33 (a5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='30 (b5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='3 500 1000 1500 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='0 N g4 tions in the (x, y) plane, the outcome is inelastic, leading to merger of AVQDs with identical (S1 = S2 = 1) or op- posite (S1 = −S2 = 1) vorticities into localized breathing modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' A noteworthy result is produced by the collision between the AVQDs with S1 = S2 = 1 traveling in the x- direction (in which the dipoles are polarized): formation of a transient state in the form of a breathing vortex- antivortex-vortex structure with three respective pivots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Eventually, this long-lived state transforms into a zero- vorticity breathing one (the present anisotropic system does not conserve the angular momentum, therefore the total vorticity is not conserved either).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' A typical exam- ple of such a collision is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 4, where the central pivot, which represents the antivortex, emerges in the beginning of the merger of the two colliding vortices [see panels (b) in the figure].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 5: (a) Density and phase patterns of the stable vortex- antivortex-vortex bound state with (N, g) = (1000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='25), x0 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='6 and µ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='8482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (b,c) The chemical potential of the states of this type versus g (at N = 1000) and N (at g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='25), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Solid and dashed parts of the curves represent, severally, stable and unstable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (d1- d4) Steady spinning of a vortex-antivortex-vortex bound state with (N, g) = (1000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='25), and the angular velocity of the ro- tational polarizing magnetic field is ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='2π × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The shape of the bound state is displayed at t = 0 (d1), 1250 (d2) 2500 (d3), 3750 (d4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The production of the above-mentioned long-lived vortex-antivortex-vortex breather by collisions along the x direction suggests that the system may support truly stationary bound states with a similar structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' They can indeed be produced by means of ITM, starting from input φ(0) = � ± A±˜r± · e−α±˜r2 ±+i˜θ± + A˜re−α˜r2−i˜θ, (11) where A± > 0 and α± > 0 are real constants, ˜r± ≡ � (x ± x0)2 + β2y2, ˜θ± ≡ arctan[βy/(x ± x0)], and x0 is an appropriately chosen separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' A typical example of such a stable bound state is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The family of stable bound states is characterized by dependences of the chemical potential on g and N, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 5(b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In the latter panels, stable bound state of the vortex-antivortex-vortex type populate ar- eas g < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='35 and 900 < N < 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Note also that the µ(N) relation satisfies the aforementioned VK criterion, dµ/dN < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Similar to what is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 3, it is possible to apply a rotating magnetic field to the bound states of the present type, and test a possibility of their steady spinning motion, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 5(d1-d4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In the underlying equation (1), which applies to the single-component dipolar BEC, nonlinearity constants g and γ are related by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Following works [23, 38], it is possible to extend the model for a two-component system, (ψ1, ψ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Limiting it to the symmetric set, with ψ1 = ψ2 = ψ/ √ 2, one arrives at the equation similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (1), but with independent coefficients in front of the cubic and quartic terms [39], i∂tψ = � −1 2∇2 + Φdd(r) + δg|ψ|2 + γ|ψ|3 � ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (12) Here δg ≡ g12 + √g11g22, with coefficients g11 = g22 ≡ g > 0 and g12 < 0 representing the strength of the self-repulsion and cross-attraction of the two compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Typical examples of a stable AVQD and vortex- antivortex-vortex bound state produced by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (12) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 6 for δg = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=', in the case when the MF interaction is exactly cancelled [40, 41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' These results indicate that robust AVQD and vortex-antivortex-vortex bound state exist in the binary condensate as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 6: Examples of a stable AVQD (left) and vortex- antivortex-vortex bound state (right) produced by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (12) with N = 1000, γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5, and δg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The upper and lower panels display density and phase patterns of the droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Conclusion We have constructed solutions for stable AVQDs in the effectively two-dimensional dipolar BEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The anisotropy and stability are stipulated by the choice of the polarization of atomic dipoles parallel to the sys- tem’s plane and perpendicular to the vortex’ axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The stability area of the AVQDs is identified in the system’s parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Characteristic features of the AVQDs, such as the peak density, chemical potential, effective area, ellipticity, and total angular momentum, are pre- sented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Spinning AVQDs can stably follow rotation of the polarizing magnetic field, provided that the rotation is not too fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Collisions between slow or fast mov- ing AVQDs are elastic or inelastic, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In the latter case, the colliding AVQDs merge into breathers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' In particular, these may be bound states of the vortex- antivortex-vortex type, which are also found as stable stationary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' (al) (b1) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='2 (a2) 10 2 (b2) 2 0 0 0 10 2 2 50 0 50 50 0 50 x x-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='82 10 (al) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5 (b) (c) >0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='6 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='86 10 (a2) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='7 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='8 10 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='40 1000 1500 2000 40 20 0 20 40 N g x 40 20 0 20 (dl (d4) (d2 (d3) 40 40 -20 0 20 40-40-20 0 20 40-40 -20 0 20 40-40 -20 20 40 x x x x5 The present analysis can be extended further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' First, it will be interesting to apply initial torque to an elongated AVQD mode, and simulate ensuing dynamics, which is expected to feature oscillations of the droplet’s orienta- tion around the original elongated direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Further, it may also be relevant to simulate motion of a spinning AVQD, driven by the rotating magnetic field, under the action of a kick applied to the AVQD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Another relevant possibility is to construct QD modes with hidden vortic- ity in the two-component system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=', bound states with vorticities ±1 in the two components with identical den- sity profiles, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' [34] The distance should be larger than 10 × 2R2, where R is the radial size of the droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' For example, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' 1(a) and 5(a), the radial sizes of the AVQD and the vortex- antivortex-vortex self-bound state are R ∼ 30 and 40, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The corresponding simulation times are ∼ 18000 and 32000, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' [35] N.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' [36] AVQDs with S ≥ 2 can be generated by initiating the ITM with ansatz (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' The result is splitting into an ar- ray of S unitary vortices set along the stretched axis of the AVQD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' However, these self-bound composite vortex states are completely unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WtE3T4oBgHgl3EQfFgks/content/2301.04305v1.pdf'} +page_content=' [37] Yu.' metadata={'source': 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https://git-lfs.github.com/spec/v1 +oid sha256:9f442370fb59471bbade21059f7a1f050e9344b3abcdcdd0013c10f0f5cf7a4f +size 5570605 diff --git a/XtE2T4oBgHgl3EQfuwhB/content/tmp_files/2301.04083v1.pdf.txt b/XtE2T4oBgHgl3EQfuwhB/content/tmp_files/2301.04083v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a762ee2a0f8cd57bbbd0051d5a17ac8b5a76ad7 --- /dev/null +++ b/XtE2T4oBgHgl3EQfuwhB/content/tmp_files/2301.04083v1.pdf.txt @@ -0,0 +1,4081 @@ +arXiv:2301.04083v1 [math.DS] 10 Jan 2023 +GEOMETRY OF THE SPACE OF MONODROMY DATA +Jean-Pierre Ramis *, Jacques Sauloy † +January 11, 2023 +Abstract +In [32], with Yousuke Ohyama, we defined and studied a “space of monodromy data” +underlying the well known derivation of q-Painlev´e VI equation from “q-isomonodromy” +conditions by Jimbo and Sakai [13]. In [16], Nalini Joshi and Pieter Roffelsen pursued our +work. However, both [32] and [16] are ambiguous on some foundational algebro-geometric +matters. We proceed here to provide sound bases. +R´esum´e +Dans [32], avec Yousuke Ohyama, nous avons d´efini et ´etudi´e un “espace des donn´ees +de monodromie” sous-jacent `a la c´el`ebre d´eduction de l’´equation de q-Painlev´e VI `a partir +de conditions de “q-isomonodromie” par Jimbo et Sakai [13]. Dans [16], Nalini Joshi and +Pieter Roffelsen ont prolong´e notre travail. Cependant, aussi bien [32] que [16] pr´esentent des +ambigu¨ıt´es sur certains aspects de leurs fondements alg´ebro-g´eom´etriques.Nous en proposons +ici des bases rigoureuses. +Contents +0 +Introduction +3 +1 +Preparatory material +5 +1.1 +General preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +1.1.1 +General notations and conventions . . . . . . . . . . . . . . . . . . . . . +5 +1.1.2 +The context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +1.2 +Some facts about quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +1.2.1 +Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +1.2.2 +Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +1.2.3 +Non separated quotients +. . . . . . . . . . . . . . . . . . . . . . . . . . +11 +1.2.4 +A useful criterion of isomorphy +. . . . . . . . . . . . . . . . . . . . . . +12 +*Institut de France (Acad´emie des Sciences) and Institut de Math´ematiques de Toulouse, CNRS UMR 5219, Uni- +versit´e Paul Sabatier (Toulouse 3), 118 route de Narbonne, 31062 Toulouse CEDEX 9, France; E-mail: ramis.jean- +pierre@wanadoo.fr +†Toulouse; E-mail:jacquessauloy@gmail.com +1 + +1.2.5 +An example of non separated geometric quotient +. . . . . . . . . . . . . +13 +1.3 +Some facts about toric varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +1.3.1 +Reminders on tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +1.3.2 +Toroidal embeddings and toric varieties . . . . . . . . . . . . . . . . . . +15 +1.3.3 +Important examples of toroidal embeddings . . . . . . . . . . . . . . . . +16 +1.4 +Invariants and quotients related to the Riemann-Hilbert-Birkhoff correspondence: +general case (n ≥ 2 arbitrary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +1.4.1 +Stabilizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +1.4.2 +Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +1.4.3 +Affine covering of V (∗) +S +. . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +1.4.4 +Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +1.4.5 +Algebras of invariants +. . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +2 +Invariants and quotients in the (possibly degenerate) JS case +23 +2.1 +Some general notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +2.2 +Non degenerate JS case: no zeroes allowed +. . . . . . . . . . . . . . . . . . . . +24 +2.3 +Degenerate JS case: one zero required . . . . . . . . . . . . . . . . . . . . . . . +25 +2.4 +A candidate craddle for partial patching: one zero allowed +. . . . . . . . . . . . +26 +2.5 +The patching +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +2.5.1 +Non degenerate part +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +2.5.2 +Degenerate part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +3 +Quadrics in the non degenerate JS case and functions on them +27 +3.1 +General notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . +27 +3.1.1 +Evaluation and linear forms +. . . . . . . . . . . . . . . . . . . . . . . . +28 +3.1.2 +Determinant and bilinear forms +. . . . . . . . . . . . . . . . . . . . . . +28 +3.1.3 +Application to FR,S,x and FR,S,x . . . . . . . . . . . . . . . . . . . . . . . +29 +3.2 +Rank 1 matrices in Mat2(C) and the Segre embedding . . . . . . . . . . . . . . . +29 +3.2.1 +Projective “coordinates” for rank 1 matrices . . . . . . . . . . . . . . . . +29 +3.2.2 +Special points and mixed projective coordinates . . . . . . . . . . . . . . +30 +3.3 +The maps ρk, k = 1,...,4, and [ρ] on F +. . . . . . . . . . . . . . . . . . . . . . +31 +3.3.1 +Definition of the ρk and [ρ] . . . . . . . . . . . . . . . . . . . . . . . . . +31 +3.3.2 +The image of [ρ] +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +3.4 +Some explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +33 +3.4.1 +Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +3.4.2 +The factor γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +35 +3.4.3 +The two quadrics are the same . . . . . . . . . . . . . . . . . . . . . . . +36 +3.4.4 +The discriminant and the singular locus . . . . . . . . . . . . . . . . . . +37 +3.4.5 +The locus detM = 0 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +39 +4 +Algebraic threefold and algebraic surface associated to a quadratic form +40 +4.1 +Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +4.2 +Smoothness properties +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +42 +4.3 +Pencils of quadrics +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +45 +2 + +5 +Projective embeddings of F +46 +5.1 +Embedding of F into K4 = +�� +P1(C) +�4 \Θ4 +� +/C∗ . . . . . . . . . . . . . . . . . +46 +5.2 +Embeddings of F into C6 and C4 +. . . . . . . . . . . . . . . . . . . . . . . . . +49 +5.2.1 +Embedding into C6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +49 +5.2.2 +Embedding into C4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +50 +5.3 +The Πk,l and Π′ +k,l, k,l = 1,...,4, of [32] . . . . . . . . . . . . . . . . . . . . . . +51 +5.3.1 +The Πk,l and Π′ +k,l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +5.3.2 +Parametrizations of F +. . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +5.4 +The 16 lines on the Segre surface . . . . . . . . . . . . . . . . . . . . . . . . . . +52 +5.4.1 +The 16 lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +52 +5.4.2 +Segre surfaces and del Pezzo surfaces of degree 4 . . . . . . . . . . . . . +53 +5.5 +Morphisms from F to +� +P1(C) +�6 and to +� +P1(C) +�3 . . . . . . . . . . . . . . . . . +55 +6 +The G model +56 +6.1 +A bijective morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +56 +6.1.1 +The regular map +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +56 +6.1.2 +The injectivity +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +57 +6.1.3 +The image Σ of V (∗)/H . . . . . . . . . . . . . . . . . . . . . . . . . . . +57 +6.2 +The image G of F +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +58 +7 +Conclusion +58 +7.1 +What we achieved in this article. . . . . . . . . . . . . . . . . . . . . . . . . . . +58 +7.2 +Open problems +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +59 +0 +Introduction +In [32] we attempted a description of the right-hand side of the Riemann-Hilbert correspondence +for a particular family of q-difference equations related to q-Painlev´e VI. We did that using a vari- +ant of the usual Birkhoff connection matrix, in which the local contributions at 0 and ∞ are ripped +off; the interest of such a procedure had been demonstrated (with somewhat different motivations) +in [37], where the idea was first introduced. In the first part of [32], we studied the space of such +matrices (up to relevant equivalence relations) from the point of view of algebraic geometry, using +mainly tools of (bi)linear algebra1. However, we did not reach a complete description of the space +of interest; in particular, we neither properly defined the quotient structures involved, nor proved +or disproved the smoothness properties that we were led to conjecture. We were not either able to +recognize among the standard list of classical algebraic surfaces (Segre, del Pezzo, Kummer, . . . ) +a candidate model2. +In [16], Nalini Joshi and Pieter Roffelsen take up our angle of attack but develop further the +necessary calculations, to the point of solving some of the questions we had left unanswered. +1In the second part of loc. cit. we introduced another approach through so-called Mano decomposition. This aspect +will appear here only in a particular case. +2We were not even able to decide if a good model is a rational surface or not. +3 + +However, we feel that some points stay unclear at the level of rigorous definitions: quotients are +introduced without any comment, let alone justification, about their nature (set theoretic ? analytic +? algebro- geometric ?); the authors call (improperly) embedding an injective map from the set of +monodromy data to a projective space whose image is a locally closed algebraic sub-variety and +do not compare their different “embeddings” from the point of view of algebraic geometry; etc. +In this work, we try to exploit the clever and efficient calculations of [16] while giving them +a sound basis in the domain of complex algebraic geometry. In particular, we properly define the +quotient spaces at stake and justify their existence and properties, and this allows us to give rigor- +ously a definition of the morphisms and a proof of their relevant properties. We define on the set +of monodromy data a structure of geometric quotient F (this is precisely stated in 3.1.3) and prove +that it is an affine, smooth, rational algebraic variety. We compare with F the different structures +on the set of monodromy data introduced in [32] and [16]. We get in particular an embedding, in +the sense of algebraic geometry3, of F in C4, whose image is the affine Segre surface introduced +in [16]. Therefore this affine Segre surface is a smooth algebraic surface. +In [32], using the Mano decomposition and in relation with some properties of partial re- +ducibility, we described 16 “lines” on the space of monodromy data. We prove that they corre- +spond to the 16 lines on the Segre surface. +Unfortunately, we cannot prove that F is isomorphic4 to the surface G defined in [32] as an +algebraic structure on the space of monodromy data. Therefore our conjecture about the smooth- +ness of G, under some generic hypothesis (cf. [32, Conjecture 7.10]), remains open5. We return to +the “model G” in section 6. +Although the heart of the article revolves around rank 2 linear systems considered for generic +values of the parameters, we propose (as we did in [32]) some more general results that could +allow for a future study of other cases. +Acknowledgements. +The first author thanks Nalini Joshi and Pieter Roffelsen for interesting discussions about the 16 +lines on a Segre surface. Both authors thank Yousuke Ohyama for sharing his knowledge of +Painlev´e equations. +3That is a morphism whose image is a locally closed subvariety isomorphic to the source. +4We have only a bijective morphism. +5The assertion in [16, Remark 2.19], is, in that sense, optimistic. What is proved in [16] is the existence of a structure +of a (possibly non separated) analytic manifold on the set of monodromy data: cf. Theorem 2.18. This does not imply +our Conjecture 7.10. +4 + +1 +Preparatory material +1.1 +General preliminaries +1.1.1 +General notations and conventions +In all the text, if E is a complex linear space, E∨ := LinC(E,C) denotes the dual of E. We +write E∗ := E \ {0}. If E is given as a product ∏Eα, then we define E(∗) := ∏E∗ +α (hopefully, +no ambiguity will arise). In a slight abuse, letting P(E) the projective space E∗/C∗, we shall set +P(E(∗)) := ∏P(Eα). +When a group G operates (on the left) on E, we write Gx ⊂ E the orbit of x ∈ E and Gx ⊂ G +its stabilizer (or isotropy subgroup). +We write P ∼ Q for the conjugacy relation in GLn(C) (actually, the symbol ∼ will be used for +some more equivalence relations) and Sp P the spectrum of P ∈ Matn(C). We denote Dn(C) the +subgroup of diagonal matrices in the linear group GLn(C). Diagonal matrices are abreviated as: +Diag(λ1,...,λn) := + + + + + +λ1 +0 +... +0 +0 +λ2 +... +0 +... +... +... +... +0 +0 +... +λn + + + + +. +For even shorter abreviations, we sometimes identify Dn(C) with C∗n and Diag(λ1,...,λn) ∈ +Dn(C) with λ := (λ1,...,λn) ∈ C∗n. +For any two c,d ∈ C∗, we write c ≡ d the congruence relation modulo the subgroup qZ: +∀c,d ∈ C∗ , c ≡ d ⇐⇒ +def c/d ∈ qZ := {qk | k ∈ Z} ⊂ C∗, +and c ̸≡ d its negation. The words “congruent”, “congruence”, etc, applied to elements of C∗, will +refer to that relation. +Whenever A(x), F(x) . . . , are invertible matrices of functions, we write σqF(x) := F(qx) and +F[A] := (σqF)AF−1 (“gauge transformation”) +1.1.2 +The context +In [32] we studied the following situation; let R := Diag(ρ1,...,ρn) ∈ Dn(C) and S := Diag(σ1,...,σn) ∈ +Dn(C) be fixed with the following strong non resonancy assumption: +∀i, j ∈ {1,...,n} , i ̸= j =⇒ (ρi ̸≡ ρ j and σi ̸≡ σ j). +5 + +Let also µ ∈ N∗, N := µn and x1,...,xN ∈ C∗ be pairwise noncongruent; we write x := {x1,...,xN}. +Then we introduced the following sets: +ER,S,x := + + + + + +A0 +···+Aµxµ �� + + + + + +all Ai ∈ Matn(C),A0,Aµ ∈ GLn(C), +A0 ∼ R,Aµ ∼ S, +detA(x1) = ··· = detA(xN) = 0, + + + + + +and the quotient set ER,S,x of ER,S,x by rational gauge equivalence relation ∼: +ER,S,x := ER,S,x +∼ +, where ∀A,B ∈ ER,S,x , A ∼ B ⇐⇒ +def ∃F ∈ GLn(C(x)) : B = F[A]. +Let V := {M ∈ Matn(O(C∗)) | σqM = RM(Sxµ)−1}, a complex linear space of dimension µn2 +(as we shall soon see); we also introduced its subset: +FR,S,x := {M ∈ V | detM ̸= 0,detM vanishes on x}. +The group Dn(C)×Dn(C) acts linearly on V through the formula: +(Γ,∆).M := ΓM∆−1. +The subset FR,S,x ⊂ V is stable under that action; letting ∼ the induced equivalence relation on it, +we define the quotient set (later abreviated as F , since the local data R,S,x will be fixed): +FR,S,x := FR,S,x +∼ · +We defined (and proved) a bijection (“Riemann-Hilbert-Birkhoff correspondence”) from ER,S,x +to FR,S,x. Our main goal here is to complete the geometric description of FR,S,x, mainly in the case +n = µ = 2 (the so-called “Jimbo-Sakai case” or “JS case” for short). +It is an essential feature of the target (“right hand side”) of our correspondence that the con- +dition σqM = RM(Sxµ)−1 on M = (mi,j)1≤i,j≤n ∈ V ⊂ Matn(O(C∗)) splits into n2 independent +conditions σqmi,j = (ρi/σ j)x−µmi,j, i.e. mi,j ∈ Vi,j, where Vi,j := Vµ,ρi/σj, such spaces being de- +fined as: +∀c ∈ C∗ , ∀k ∈ N∗ , Vk,c := {m ∈ O(C∗) | σqm = cx−km}, +each such linear space having dimension dimCVk,c = k. +So we have a natural identification +V := +∏ +1≤i,j≤n +Vi,j (whence our contention that dimCV = µn2). +Another essential feature is that the action of Dn(C) × Dn(C) on FR,S,x, which can naturally +extended to V ⊃ FR,S,x as a linear action, splits correspondingly: +(Γ,∆).(mi,j)1≤i,j≤n = ((γi/δ j)mi,j)1≤i,j≤n , +where Γ =: Diag(γ1,...,γn) and ∆ =: Diag(δ1,...,δn). These two features will somehow be “ax- +iomatized” in 1.4. +6 + +1.2 +Some facts about quotients +Since we intend to clarify the algebro-geometric structure of our spaces of interest, it seems fit +to make explicit our framework. We deal with the elementary theory of algebraic varieties over +C (not necessarily affine, nor irreducible, nor separated but always reduced) and linear algebraic +groups (automatically smooth but possibly reducible). +A word about terminology: since we do not require our algebraic varieties to be separated, +whenever they are, we call them separated varieties. Of course all affine, quasi-affine, projective +and quasi-projective varieties are separated, as will be all varieties X for which we seek to construct +a quotient. However, we shall in the end obtain some non separated quotients X → Y (i.e. X is +separated but Y is not). At any rate, in the whole of 1.2.1 and 1.2.2, all varieties are assumed to +be separated. This convention will be relaxed from 1.2.3 on (except in explicitly stated special +cases). +1.2.1 +Group actions +Basic formalism. +Unless otherwise stated, the words “closed”, “open”, “dense”, etc, refer to +Zariski topology. An affine algebraic variety X is totally determined (and conversely) by its affine +algebra C[X] (a finite type reduced C-algebra) and morphisms of affine varieties X →Y correspond +functorially to morphisms of C-algebras C[Y] → C[X]. A general morphism of varieties φ : X →Y +is locally given in the above form, i.e. by morphisms of C-algebras C[V] → C[φ−1(V)] where the +affine sets V cover Y. Important examples of non affine varieties, to keep in mind, are Cn \ {0} +and Pn−1(C) (whenever n ≥ 2). +All our algebraic groups are affine (as varieties), hence linear (i.e. realized as closed subgroups +of some GLn(C)); main references: [4, 12, 40], also see [39]. For any morphism of algebraic +groups f : G → H (i.e. f is a morphism of varieties and a group morphism), the image f(G) ⊂ H +is a closed subgroup (the kernel Ker f ⊂ G obviously is !); and, if f is injective, it is a closed +immersion. The neutral component G◦ (i.e. the connected component of the identity 1) is a closed +normal subgroup, the irreducible components of G are also its connected components gG◦ = G◦g +and the quotient group G/G◦ is finite. +From the structure theory of linear algebraic groups, we retain the following definitions and facts: +• The group G is semi-simple if its only connected normal solvable subgroup is trivial. The +special linear groups SLn(C) and the tori C∗n are semi-simple. +• The group G is reductive if its only connected normal unipotent subgroup is trivial. The +group GLn(C), and all semi-simple groups are reductive. +Rational actions. +Our main general references on algebraic group actions and quotients are +[8, 27] and, for some special facts, [33]; also note the survey [6] and the short summary [28] of +[27]. +Let G an algebraic group and X an algebraic variety. A rational action of G on X is a morphism +of varieties G×X → X, (g,x) �→ gx which is a group action (i.e. g′(gx) = (g′g)x and 1x = x). We +7 + +shall just speak of an action and say that X is a G-variety. Stabilizers Gx of elements x ∈ X are +automatically closed subgroups of G. Typical examples, to keep in mind, are: +1. C∗ acting on Cn, resp. on Cn \{0}, by homotheties; +2. C∗ acting on C2, resp. on C2 \{0}, by t.(x,y) := (tx,t−1y). +An equivariant morphism φ : X →Y of G-varieties, also called a G-morphism, is one such φ(gx) = +gφ(x) for all g ∈ G, x ∈ X. If Y is endowed with the trivial action (such that gy = y for all +g ∈ G, y ∈ Y) we say that φ is invariant (so φ(gx) = φ(x) for all g ∈ G, x ∈ X). If φ is invariant, +the induced morphism of algebras C[U] → C[φ−1(U)] (generally defined for every morphism of +varieties φ : X → Y and every open subset U ⊂ Y) has image in the subalgebra C[φ−1(U)]G of +G-invariant functions. +A (rational) G-module is a finite dimensional C-linear space V with a linear G-action (i.e. x �→ gx +is linear for all g ∈ G); equivalently, it is a linear representation and the corresponding group +morphism G → GL(V) is a morphism of algebraic groups. Then every algebraic subset of V which +is G-invariant is canonically a G-variety. Moreover, every affine G-variety can be obtained in this +way. (The definition of G-module is usually extended to infinite dimensional C-linear spaces V, +provided V = �Vi where all Vi are G-invariant and finite dimensional.) +Orbits. +Let X a G-variety. Then the orbit Gx of any x ∈ X is locally closed, i.e. it is open in its +Zariski closure Gx. Hence the complementary subset Gx \ Gx is a disjoint union of orbits, each +having dimension < dimGx. +We have dimGx = dimG−dimGx and the function x �→ dimGx is lower semicontinuous, i.e.: +{x ∈ X | dimGx ≤ n} is closed for every n ≥ 0. +Thus orbits of minimal dimension are closed and every orbit contains (at least) a closed orbit. +Note that if X is a G-variety and φ : X →Y an invariant morphism, every fiber φ−1(y) is closed and +G-invariant, whence a union of the orbits Gx such that φ(x) = y. So, the quotient map from the set +of orbits6 X +G to Y is well defined. Looking for a separated quotient (which a variety has to be), we +would hope that map to be bijective; this would at least require that all orbits be closed, which is +seldom the case. +1.2.2 +Quotients +Categorical quotients and orbit spaces. +There are two natural ways to define the quotient of a +G-variety X by the action of G: +• Consider the quotient set Y, endow it with the quotient topology and the sheaf of G-invariant +functions; this does not generally yield an algebraic variety. +• Categorically: this will be detailed next; by definition, if it exists, such a quotient is an +algebraic variety and unique up to isomorphism. But it may have bad geometrical properties. +6The notation is provisional and can be forgotten. For quotients, we shall rather use X/G and the like. +8 + +The morphism of varieties φ : X → Y is a categorical quotient if it is G-invariant (i.e. constant on +G-orbits) and initial for that property (i.e. any G-invariant morphism ψ : X → Z factors uniquely +as ψ = f ◦φ, where f : Y → Z is a morphism). +In the case of C∗ acting on Cn by homotheties, 0 belongs to the closure of all orbits, so all G- +invariant ψ : X → Z are constant, so the categorical quotient is trivial (a constant map to a point). +If the categorical quotient φ : X → Y separates orbits, i.e. the fiber φ−1(y) is an orbit for every +y ∈ Y, it is called an orbit space. +In the case of C∗ acting on Cn \{0} by homotheties, the natural projection Cn → Pn−1(C) is an +orbit space. +Good quotients and geometric quotients. +The following definition comes from [27]: +Definition 1.1 A good quotient is an affine7 morphism φ : X → Y which satisfies the following +properties: +(i) φ is surjective. +(ii) φ is G-invariant. +(iii) For every open subset U ⊂Y, the induced morphism C[U] → C[φ−1(U)]G is an isomorphism; +in particular, C[Y] can be identified with C[X]G, which completely defines Y. +(iv) For every closed G-invariant subset W ⊂ X, the set φ(W) ⊂ Y is closed. +(v) For every pair of closed G-invariant subsets W1,W2 ⊂ X, if W1∩W2 = /0 then φ(W1)∩φ(W2) = /0. +We write such a good quotient as X//G (thus omitting the structural morphism φ). +The following properties are consequences: +Corollary 1.2 For every open subset U ⊂Y, the morphism φ−1(U) →U is a categorical quotient. +(In particular, φ : X →Y is a categorical quotient.) If moreover the action of G on φ−1(U) is closed +(i.e. the orbits are closed), then it is an orbit space. +Moreover,“φ separates orbits as much as topologically feasible”: +Corollary 1.3 Let x1,x2 ∈ X. Then φ(x1) = φ(x2) ⇔ Gx1 ∩Gx2 ̸= /0. (The implication ⇐ is easy +and holds for all invariant morphisms.) +Corollary 1.4 The fibers of φ : X → X//G are closed if and only if they all have the same dimen- +sion, which in turn is equivalent to: the φ−1(y) are the orbits, i.e. X//G is an orbit space. +The following definition also comes from [27]: +Definition 1.5 A geometric quotient is a good quotient which is moreover an orbit space. +We shall write a geometric quotients as X/G. +The following characterization of a geometric quotient is taken as definition in [6, 33]: +7Recall that φ is said to be affine if for each affine open subset U ⊂ Y, its preimage φ−1(U) ⊂ X is affine. +9 + +Proposition 1.6 The morphism φ : X → Y is a geometric quotient if and only if: +(i) it is surjective and its fibers φ−1(y), y ∈ Y, are exactly the G-orbits; +(ii) Y has the quotient topology, i.e. U ⊂ Y is open if and only if φ−1(U) ⊂ X is open; +(iii) for every open subset U ⊂ Y, the induced morphism C[U] → C[φ−1(U)]G is an isomorphism. +Condition (ii) may be replaced by the equivalent one: (ii’) φ is an open map. +For example, Cn \{0} → Pn−1(C) (for n ≥ 2) is a geometric quotient, and so is G → G/H for any +closed subgroup of an affine algebraic group G. +At any rate, we see that the geometric quotient X/G is actually the topological quotient π : X → X +G +endowed with the sheaf (π∗OX)G, where OX denotes the structural sheaf on the variety X and +(π∗OX)G the invariant subsheaf under the obvious action of G on the direct image π∗OX. Last, we +quote from [33, theorem 4.2]: +Theorem 1.7 Let X a G-variety and φ : X → Y a surjective morphism such that its fibers φ−1(y) +are the orbits. Assume X irreducible and Y normal. Then φ is a geometric quotient. +Action of reductive groups on affine varieties. +Let G a reductive group and X an affine G- +variety. The following are proved in [27]. +Theorem 1.8 There exist an affine variety Y and a morphism φ : X → Y which is a good quotient. +It is the unique affine variety with affine algebra C[Y] := C[X]G. +The following precisions are proved in [6]: +Corollary 1.9 The algebra of invariants functions C[X]G is an affine algebra; let f1,..., fn a set of +generators. Then X//G can be described as the image of the map G → Cn, x �→ ( f1(x),..., fn(x)), +which is a closed subset of Cn. +Now let X′ ⊂ X a closed G-invariant subset and let φ′ : X′ → X′//G as in the previous theorem. +Since φ′ is a categorical quotient and since the restriction φ|X′ : X′ → X//G is G-invariant, it factors +through a morphism X′//G → X//G. +Corollary 1.10 X′//G → X//G is a closed immersion, so X′//G is (canonically identified to) a +closed subset of X//G. +Corollary 1.11 Let X′,X′′ ⊂ X a pair of closed G-invariant subsets; then φ(X′ ∩ X′′) = φ(X′) ∩ +φ(X′′). +Corollary 1.12 Every fiber φ−1(y) contains a unique closed orbit. +So X//G can be seen as “the space of closed orbits”. +Corollary 1.13 If X is irreducible, resp. normal, so is X//G. +10 + +Stable points and geometric quotients. +Let again G a reductive group and X an affine G- +variety. +Definition 1.14 The point x ∈ X is stable if its orbit Gx is closed and its stabilizer Gx is finite. +Theorem 1.15 (i) The set Xs of stable points is a (possibly empty) G-invariant open subset of X +and φ(Xs) ⊂ X//G is open too, so (provided Xs is non empty) φ(Xs) = Xs//G. +(ii) The restriction Xs → φ(Xs) = Xs//G is actually a geometric quotient Xs/G (again provided +Xs is non empty). +1.2.3 +Non separated quotients +We shall be confronted to the following situation: a separated G-variety X can be covered by +G-invariant open subsets Xi for which we are able to define geometric (resp. good, resp. cate- +gorical) quotients Xi → Yi. Can we patch together the Yi to obtain a geometric (resp. good, resp. +categorical) quotient X → Y ? Under some circumstances, the answer is yes, but Y might be non +separated. Since our basic way to construct geometric quotients involves affine varieties, we shall +only consider affine quotients Xi → Yi. +Glueing of affine varieties. +So we consider a family (Yi)i∈I of affine varieties. Following [8, +§8.2], we define glueing data for that family as: +1. for each (i, j) ∈ I ×I, an affine open subset Ui,j ⊂ Yi; +2. for each (i, j) ∈ I ×I, an isomorphism f j,i : Ui,j → Uj,i. +Those are subject to the following compatibility conditions: +1. for each i ∈ I, Ui,i = Yi and fi,i = IdYi; +2. for each (i, j,k) ∈ I ×I ×I, f j,i(Ui,j ∩Ui,k) ⊂ Uj,k; +3. for each (i, j,k) ∈ I ×I ×I, fk,j ◦ f j,i = fk,i on Ui,j ∩Ui,k. +Note that restriction to the domain Ui,j ∩Ui,k is a logical necessity in the third condition, and that +the second condition is then required for the third to have a meaning. +We can then define on the disjoint union �Yi an equivalence relation ∼ which is the smallest such +that yi ∼ f j,i(yi) for each (i, j) ∈ I × I and yi ∈ Ui,j. Then the quotient set +�Yi +∼ +admits a natural +structure of algebraic variety (possibly non separated) such that: +1. each Yi embeds into Y, so we shall identify Yi as an affine open subset of Y; +2. a regular function g on Y is the same thing as a family of compatible regular functions gi on +Yi (i.e. such that gj ◦ f j,i = gi on Ui,j). +Corollary 1.16 A morphism of algebraic varieties ψ : Y → Z is the same thing as a family of +compatible morphisms ψi : Yi → Z (i.e. such that ψj ◦ f j,i = ψi on Ui,j). +11 + +Corollary 1.17 Let X an algebraic variety, (Xi)i∈I an open covering of X and let φi : Xi → Yi a +family of morphisms such that φi(Xi ∩ Xj) ⊂ Ui,j for all i, j, and moreover supposed compatible +(i.e. such that φj ◦ f j,i = φi on Xi ∩ Xj). Then they can be uniquely patched (in an obvious sense) +into a morphism φ : X → Y. +Glueing of affine good and geometric quotients. +Let G a reductive group, X a G-variety and +(Xi)i∈I an affine open covering of X. We assume that the Xi are G-invariant and that8 Xi ∩ Xj is +affine for all i, j. Then, by 1.2.2, there are affine good quotients φi : Xi → Yi and their universal +properties provide glueing data on the Ui,j := φi(Xi ∩ Xj). From the above, we obtain a unique +morphism φ : X → Y such that each restriction φ|Xi is identified with the affine good quotient +φi : Xi → Yi ⊂ Y; and the Yi are an affine open covering of Y. +To obtain geometric quotients, we must extend the definition of stable points to non affine varieties: +Definition 1.18 The point x ∈ X is stable if belongs to a G-invariant affine subset of X, its orbit +Gx is closed and its stabilizer Gx is finite. +Now the following fact is stated in [27] for separated varieties, but it obviously extends to the case +the target Y is not separated, whether we use as definition 1.5 or 1.6; here we assume X to be +separated (but Y is arbitrary): +Proposition 1.19 Being a good or geometric quotient is a local property in the following sense: +(i) If φ : X →Y is a good, resp. a geometric quotient, then so is φ−1(U) →U for every open U ⊂Y. +(ii) If there is an open covering (Ui) of Y such that all the φ−1(Ui) → Ui are good, resp. geometric +quotients, then so is φ : X → Y. It follows that every map φ−1(U) → U is a categorical quotient +(and an orbit space if all orbits in φ−1(U) are closed). +Combining with theorem 1.15, we get: +Theorem 1.20 (i) The set Xs of stable points is a (possibly empty) G-invariant open subset of X +and φ(Xs) ⊂ X//G is open too, so (provided Xs is non empty) φ(Xs) = Xs//G. +(ii) The restriction Xs → φ(Xs) = Xs//G is actually a geometric quotient Xs/G (again provided +Xs is non empty). +1.2.4 +A useful criterion of isomorphy +Since we intend to consider non separated quotients, we shall have to use the following classical +criterion in that extended case, so we state and prove it. +Proposition 1.21 Let ϕ : X → Y be a bijective morphism of algebraic varieties. We suppose that +X is separated and that Y is normal, non necessarily separated. Then φ is an isomorphism. +Proof. - The problem is local on Y, therefore we can suppose that Y is separated. Then, we can +apply [24, Proposition (3.17),page 46]. It implies that ϕ is birational. It follows from Zariski’s +8Note that if we assume X to be separated (which will be the case in our applications), it is automatically true that +the intersection of two affine open subsets is affine. +12 + +Main Theorem [24, (3.20),page 46] that a finite birational morphism from a variety to a normal +variety is an isomorphism, therefore ϕ is an isomorphism. +□ +1.2.5 +An example of non separated geometric quotient +The following example will be needed later: it involves the diagonal action of C∗ on +� +P1(C) +�n. +First, some basic notations and a baby version (n = 1) of the example. We identify P1(C) with +ˆC := C∪{∞} by [a : b] �→ a/b. We write ρ, resp. ˜ρ for the “coordinate” a/b, resp. for its inverse +1/ρ = b/a. There is a natural action of C∗ on P1(C) given (for λ ∈ C∗) by [a : b] �→ [λa : b], i.e. by +ρ �→ λρ and ˜ρ �→ λ−1˜ρ. That action has two fixed points 0 and ∞. Let Θ1 := {0,∞} the set of fixed +points. All points of P1(C) \Θ1 = C∗ have trivial stabilizer and a non closed orbit in P1(C) but +a closed orbit in the invariant open subset P1(C) \Θ1. The action on that subset has a geometric +quotient: +� +P1(C)\Θ1 +� +/C∗ = C∗/C∗ = P0(C) = {•}. +Now we look at +� +P1(C) +�n for an arbitrary n ≥ 2. The ρ and ˜ρ coordinates of the components give +rise to maps ρ1,...,ρn and ˜ρ1,..., ˜ρn from +� +P1(C) +�n to ˆC. The diagonal action of C∗ on +� +P1(C) +�n +is characterized (for λ ∈ C∗) by ρi �→ λρi and ˜ρi �→ λ−1˜ρi for i = 1,...,n. The set of fixed points +is Θn := {0,∞}n. The only closed orbits are those of fixed points p ∈ Θn: they are singletons +C∗p = {p}. Every other point p ∈ +� +P1(C) +�n \ Θn has a trivial stabilizer. Its orbit has boundary +C∗p\C∗p ⊂ Θn, so it is closed in the invariant open subset +� +P1(C) +�n \Θn. +In the affine open subset Cn ⊂ +� +P1(C) +�n, the only fixed point is 0 := (0,...,0). All other +points have trivial stabilizer and an orbit which is closed in the invariant open subset Cn \{0}. The +latter has a geometric quotient under the diagonal C∗-action: +Cn \{0} → (Cn \{0})/C∗ = Pn−1(C). +More generally, let p ∈ Θn. We shall use the following notations for the complementary subsets +of indices: +I := {i ∈ {1,...,n} | ρi(p) = 0} and J := {j ∈ {1,...,n} | ρ j(p) = ∞} = {j ∈ {1,...,n} | ˜ρ j(p) = 0}. +Then we introduce an invariant open subset of +� +P1(C) +�n: +Up := +� +p ∈ +� +P1(C) +�n | +� +i ∈ I ⇒ ρi(p) ̸= ∞, +j ∈ J ⇒ ˜ρ j(p) ̸= ∞, +� +:= +� +p ∈ +� +P1(C) +�n | +� +i ∈ I ⇒ ρi(p) ̸= ∞, +j ∈ J ⇒ ρ j(p) ̸= 0, +� +so that in particular U0 = Cn. Note that the map sI which transforms ρ j into ˜ρ j for j ∈ J (and +leaves ρi invariant for i ∈ I is an automorphism of +� +P1(C) +�n which sends Up to Cn and p to 0 (and +conversely since it is an involution). Therefore Up ∩ Θn = {p} and all points of Up \ {p} have +13 + +trivial stabilizer and an orbit open in Up \{p}. So here again we have a geometric quotient Up of +Up \{p} and a commutative diagram: +Up \{p} +sI +� +� +Cn \{0} +� +Up +≃ +� Pn−1(C) +Now each Up ∩Up′, p ̸= p′ ∈ Θn, is a Zariski-dense affine open subset (actually a product of copies +of C and of C∗) in both Up \ {p} and Up′ \ {p′} (indeed p′ ̸∈ Up and conversely). It therefore +projects to a Zariski-dense open subset Up,p′ of Up and also to a Zariski-dense open subset Up′,p +of Up′, with a canonical isomorphism Up,p′ → Up′,p. We use those isomorphisms to glue the Up +along the Up,p′, yielding an irreducible algebraic variety Kn which is a geometric quotient: +� +p∈Θn +(Up \{p}) = +� +P1(C) +�n \Θn → Kn := +�� +P1(C) +�n \Θn +� +/C∗. +Note that the Up are compact (they are projective spaces) but they are Zariski-dense open subsets +of Kn, which is therefore not separated. +1.3 +Some facts about toric varieties +Some of our most interesting applications will fall under this heading. Exceptionally, in the whole +of 1.3, we denote A[X] the affine algebra of an affine variety X so as to avoid confusion with the +group algebra C[Λ] of a group Λ (or the monoid algebra C[M] of a monoid M). Conversely, an +affine algebra being given, we denote Spec A the corresponding affine variety; it can be canonically +realized with underlying set the set of all algebra morphisms A → C. +1.3.1 +Reminders on tori +General references for the structure of tori are [4, 12, 40, 39]. +Tori and their character groups. +Let Tn the linear algebraic group C∗n. Its affine algebra is +A[Tn] = C[X1,X−1 +1 ,...,Xn,X−1 +n ]. In more intrinsic terms, a (n-dimensional) torus T is a linear +algebraic group group isomorphic to Tn. Let: +X(T) := {λ ∈ A[T] | λ : T → C∗ is a group morphism} +the group of characters of T. Then Λ := X(T) is a finitely generated free abelian group and the +affine algebra of T is the group algebra of Λ: +A[T] = C[Λ] = +� +λ∈Λ +Cλ. +The group Λ∨ := {group morphisms Λ → C∗} has a natural structure of torus (if Λ = Zn, it is +canonically identified with C∗n) and the map T → Λ∨, g �→ (λ �→ λ(g)) is an isomorphism of +algebraic groups. Therefore T is completely determined (as a group and as a variety) by X(T). +14 + +Quotients of tori. +Let S ⊂ T a closed subgroup of the torus T. We keep the notation Λ := X(T). +Let S a closed subgroup of T and let Λ′ is the subgroup of Λ made of those characters which vanish +on S: thus Λ′ is a finitely generated free abelian group. Then T ′ := T/S has a natural structure of +torus with affine algebra A[T ′] = C[Λ′]. +Moreover, any generating set of the subgroup Λ′ is a family of equations defining S (i.e. a gener- +ating set of the ideal of S in T); and S is a torus if, and only if, the subgroup Λ′ is saturated, i.e. +Λ/Λ′ has no torsion (whence is free abelian). In that case, A[S] = C[Λ/Λ′]. +Note that in that case we have two dual exact sequences: +1 → S → T → T ′ → 1 and 0 → Λ′ → Λ → Λ/Λ′ → 0. +Example 1.22 The subgroup S := {(a,b) ∈ C∗2 | a2 = b2} of T := T2 is not a subtorus (it is no +even connected), but the subgroup S := {(a,b,c,d) ∈ C∗4 | ab = cd} of T := T4 is a subtorus +isomorphic to T3. +Linear representations of tori. +Let V is a finite dimensional space endowed with a rational9 +linear representation of T. Then: +V = +� +λ∈Λ +Vλ, where ∀λ ∈ Λ , Vλ := {v ∈ V | ∀g ∈ T , g.v = λ(g)v}. +If V is infinite dimensional but a union of finite dimensional T-invariant subspaces such that the +corresponding linear representations of T are rational, the same conclusion holds. +Actions of tori. +If a torus T = Λ∨ = Spec C[Λ] acts on an affine algebraic variety X, it acts du- +ally on A[X] (for g ∈ T and f ∈ A[X], we set gf : x �→ f(g−1x)) and this is an infinite dimensional +linear representation, whence a decomposition A[X] = � +λ∈Λ +A[X]λ. +In the particular case of the left multiplication action of T on itself, the corresponding decomposi- +tion is just A[T] = C[Λ] = � +λ∈Λ +Cλ. +We quote Sumihiro’s theorem from [30, p. 10]: if the torus T acts (rationally) on an irreducible +normal variety, then the later is the union of T-invariant affine open subsets. +1.3.2 +Toroidal embeddings and toric varieties +General references here are [6, 8] (for toric varieties) and [18, 29, 30] (for toroidal embeddings). +We mostly follow those sources, except that we distinguish toroidal embeddings from toric vari- +eties; and we do not systematically assume normality. +9This means that the corresponding map T → GL(V) is a morphism of algebraic groups. We shall omit the word +“rational” in the sequel. +15 + +Toroidal embeddings. +A toroidal embedding, of the torus T is a variety X admitting T as an +open dense subset (thus X is irreducible) and an action of T extending the left multiplication action +of T on itself, whence a commutative diagram (vertical arrows are inclusions, horizontal arrows +are actions): +T ×T +� +� +T +� +T ×X +� X +If X is affine, the dominant inclusion T ⊂ X induces an injection A[X] ⊂ A[T]. The splitting of +A[T] in eigenspaces A[T]λ induces a splitting of A[X] (under the action of T): +A[X] = +� +λ∈M +A[X]λ = +� +λ∈M +Cλ = C[M], +where M is a semigroup, the submonoid Λ∩A[X] of those characters of T which can be extended +as regular maps on the whole of X. The semigroup M is finitely generated (as a semigroup) and it +generates the group Λ. The variety X is normal if, and only if, M is saturated in Λ, i.e. for r ∈ N∗, +λ ∈ Λ, one has the implication rλ ∈ M ⇒ λ ∈ M. +The way M embeds into Λ admits a combinatorial description (“fans”) which encodes the way X′ +is completed into X by the addition of orbits of smaller dimension. +A morphism of toroidal embeddings of T is a T-equivariant morphism. Then there is a bijective +correspondence M �→ Spec C[M] from finitely generated sub semigroups of Λ to isomorphism +classes of affine toroidal embeddings of T; saturated semigroups correspond to normal affine +toroidal embeddings. More generally, morphisms of affine toroidal embeddings correspond (in a +contravariant way) to inclusions of semigroups (see [18, pp 4,6]). +Toric varieties. +Here we follow10 [6, §2.2]. +A toric variety is a normal irreducible T-variety X (for some torus T) containing an open orbit +(which is then Zariski dense since X is irreducible). +Let X′ = Tx, x ∈ X, such an orbit. Then the morphism T → X, g �→ gx is dominant, so A[X] is +(identified to) a subalgebra of A[T] = C[Λ]. More precisely, there is a finitely generated monoid +M ⊂ Λ such that A[X] = C[M]. +The surjective morphism T → X′, g �→ gx, realizes X′ as a homogeneous space under T. By [6, +§1.1], it is isomorphic (as a variety) to the torus T ′ := T/Tx (remember Tx is the stabilizer of x). +Let Λ′ the group of characters of T ′, so that A[T ′] = C[Λ′]. Interpreting characters λ : T → C∗ +which vanish on Tx as elements of Λ′, thus as regular functions on the orbit X′ = Tx, the elements +of M are those characters that can be extended to a regular function on the whole of X ⊃ X′. +Since Tx acts trivially on the dense subset X′ of X, it acts trivially on the whole of X and the action +of T on X factors into an action of T ′ on X. Identifying T ′ with its image X′, we see that the +general situation of toric varieties boils down to the case of toroidal embeddings. +1.3.3 +Important examples of toroidal embeddings +Two basic examples. +10The groups G, B, U of loc.cit. are here T, T, 1. +16 + +1. Recall from 1.2.5 the identification of P1(C) with ˆC := C ∪ {∞} by [a : b] �→ a/b, whence +an embedding of C∗ into P1(C), and then an embedding of Tn = C∗n into P1(C)n. The left +action of Tn on itself extends to an action on P1(C)n by: +(λ1,...,λn).([a1 : b1],...,[an : bn]) := ([λ1a1 : b1],...,[λnan : bn]), +making it a toroidal embedding. +2. We shall meet later the quadric hypersurface XY = ZT in P3(C). There is a natural action +of the torus T3 ≃ {(a,b,c,d) ∈ C∗4 | ab = cd} on that hypersurface, defined by: +(a,b,c,d)[X : Y : Z : T] := [aX : bY : cZ : dT], +again a toroidal embedding. +A significant example. +The following lemma is more or less obvious: +Lemma 1.23 Let T ֒→ X a toroidal embedding and let S ⊂ T a substorus such that T/S is itself a +torus. Assume that the quotient X/S is geometric. Then T/S ֒→ X/S is a toroidal embedding. +It follows that the variety Kn described in 1.2.5 is a non separated toric variety. Another example +will be provided by corollary 1.29. +1.4 +Invariants and quotients related to the Riemann-Hilbert-Birkhoff correspon- +dence: general case (n ≥ 2 arbitrary) +All the subsection 1.4, with general n ≥ 2 and µ ≥ 1, is intended to provide vocabulary and basic +tools for a future detailed study of the spaces FR,S,x. Only from section 2 on where we assume +that n = µ = 2, do we present substantial results; so this subsection may be skipped at first reading. +Let Vi,j, 1 ≤ i, j ≤ n, be non trivial finite dimensional complex vector spaces and, according to +our general conventions: +V := ∏ +1≤i,j≤n +Vi,j and V (∗) := ∏ +1≤i,j≤n +V ∗ +i,j. +We write their elements as M = (mi,j) (since they are here as models of the context described in +1.1.2). Let the 2n dimensional torus Dn(C)×Dn(C) act on V by: +(Γ, ∆).M := ΓM∆−1 = +� γi +δ j +mi,j +� +i,j=1,...,n +, where Γ =: Diag(γ1,...,γn) and ∆ =: Diag(δ1,...,δn). +The kernel of the action is clearly C∗(In,In) = {(γIn,γIn) | γ ∈ C∗} so we set: +H := Dn(C)×Dn(C) +C∗(In,In) +, +which acts faithfully on V. The group H is itself a (2n − 1)-dimensional torus: identifying the +obvious way Dn(C) × Dn(C) to C∗2n, it is the direct product of its algebraic subgroup C∗(In,In) +by anyone of the (2n − 1)-dimensional subtori C∗i × {1} × C∗ j, i + j = 2n − 1, so any of the +corresponding mappings C∗i ×{1}×C∗ j → H is an isomorphism. +17 + +1.4.1 +Stabilizers +(The class of) an element (λ,µ) ∈ Dn(C)×Dn(C) is in the stabilizer HM of M := (mi,j) ∈V if and +only if mi,j ̸= 0 ⇒ λi = µj. Those equations define a subtorus of Dn(C)×Dn(C), with dimension +the number of degrees of freedom left by those equations among the 2n coefficients λi, µj, and HM +is a torus of dimension that number minus 1. +Examples 1.24 (i) Let n = 2. If M has at most one zero component, HM is trivial. Otherwise it is +not. +(ii) Let n = 3. If M has zeroes at most at positions (1,1), (1,2), (2,1), (2,2), then HM is trivial. If +M has a null column or a null line, HM is not trivial. +So we define the support of M as S(M) := {(i, j) ∈ {1,...,n}2 | mi,j ̸= 0}. Then, for any subset +S ⊂ {1,...,n}2, consider a bipartite graph with set of vertices {1,...,n} ⊔ {1,...,n} (disjoint +union), where i on the left is connected to j on the right if and only if (i, j) ∈ S; and let χ(S) the +number of connected components of that graph. +Proposition 1.25 The dimension of the torus HM is χ(S(M))−1. +Proof. - The class of (Γ,∆) ∈ Dn(C)×Dn(C) is in HM if and only if γi = δ j for all (i, j) ∈ S(M). +This leaves one degree of freedom (in choosing all γi,δ j) for each connected component, so the +set of all such pairs (i.e. the preimage of HM in Dn(C)×Dn(C)) has dimension χ(S(M)). +□ +Examples 1.26 (i) Let n = 2. Then χ(S) = max(1,4−|S|). +(ii) Let n = 3. Then χ(S) > 1 if S ⊂ {i1,i2}×{1,2,3} for some i1,i2 or if S ⊂ {1,2,3}×{j1, j2} +for some j1, j2, while χ(S) = 1 if the complement of S is contained in {i1,i2}×{j1, j2} for some +i1,i2, j1, j2. +1.4.2 +Orbits +Let M ∈ V and S := S(M). We identify VS := +∏ +(i,j)∈S +Vi,j with the subspace of V defined by a zero +component at all (i, j) ̸∈ S. We denote (as in our general conventions) V (∗) +S +:= +∏ +(i,j)∈S +V ∗ +i,j, which we +identify to the subset of VS (thus V (∗) +S +is open in VS which is closed in V). Clearly the orbit H.M of +M is contained in V (∗) +S . More precisely, let us set: +L∗ +i,j(M) := C∗mi,j ⊂ V ∗ +i,j and L(∗) +S (M) := ∏ +(i,j)∈S +L∗ +i,j ⊂ V (∗) +S +. +Then: +H.M ⊂ ∏ +(i,j)∈S +L∗ +i,j(M) = L(∗) +S (M) ⊂ ∏ +(i,j)∈S +V ∗ +i,j = V (∗) +S . +The factors λi/µj effectively involved in the action on H.M are those such that (i, j) ∈ S. Under +the obvious isomorphism L(∗) +S (M) ≃ C∗S, the orbit H.M goes to: +TS := {(λi/µj)(i,j)∈S | all λi,µj ∈ C∗} ⊂ C∗S. +18 + +Proposition 1.27 TS is a closed subgroup of C∗S and a torus of dimension 2n−χ(S). +Proof. - It is the image of the map: +Dn(C)×Dn(C) → C∗S, (λ,µ) �→ (λi/µj)(i,j)∈S. +That map being a morphism of algebraic groups, the image is closed (Borel, chap 1, §1, cor. 1.4). +Since the kernel has dimension χ(S), the image has dimension 2n−χ(S). Also, being a connected +closed subgroup of the torus C∗S, it is a torus. +□ +We make this more precise as follows. Let πi,j : V ∗ +i,j → P(Vi,j) the natural projection and (with +a slight abuse of the notation P): +π := ∏ +(i,j)∈S +πi,j : V (∗) +S +→ P(V (∗) +S ) := ∏ +(i,j)∈S +P(Vi,j), +so that L(∗) +S (M) = π−1(π(M)) is closed inV (∗) +S +and H.M is closed in L(∗) +S (M). We have commutative +diagrams (in which horizontal arrows are canonical inclusions, vertical arrows are non canonical +isomorphisms): +H.M +� +� +L(∗) +S (M) +� +TS +� C∗S +Corollary 1.28 The orbit H.M is closed in V (∗) +S . +1.4.3 +Affine covering of V (∗) +S +The problem is that V (∗) +S +is, in general, not an affine variety: if W is a vector space, W ∗ is an affine +variety if and only if dimW = 1. However, for every linear form e ∈W ∨ (the dual), e ̸= 0, the open +subset W(e) := W \ Ker e is an affine variety, with affine algebra C[W(e)] = C[W][1/e]. More- +over, each W(e) is stable under the natural C∗-action and the W(e), e ∈ W ∨ \{0}, cover W ∗: the +corresponding quotients W(e)/C∗ are the natural affine charts P(W)e ⊂ P(W). (Actually, taking +the elements e in a basis of W ∨ is enough.) +So, writing for short V ∨∗ +S +:= +∏ +(i,j)∈S +(V ∨ +i,j \{0}), we define: +∀e := (ei,j)(i,j)∈S ∈ V ∨∗ +S +, VS(e) := ∏ +(i,j)∈S +Vi,j(ei,j) ⊂ V (∗) +S . +Those open subsets cover V (∗) +S , they are stable under the action of H and they are affine. Now, the +really useful result is: +19 + +Corollary 1.29 (i) If M ∈ VS(e), the orbit H.M is closed in VS(e). +(ii) If moreover χ(S) = 1, then all M ∈ VS(e) are stable (in the sense of Brion, definition 1.25). As +a consequence, there is a geometric quotient of V (∗) +S +by H. +(iii) V (∗) +S /H is a toric variety. +The last statement flows from lemma 1.23. +Example 1.30 This applies in particular to n = 2 and S = {1,2}× {1,2} or the same minus one +ordered pair (i, j), i.e. to the formats (each ⋆ stands for a non zero coefficient): +�⋆ +⋆ +⋆ +⋆ +� +, +�0 +⋆ +⋆ +⋆ +� +, +�⋆ +0 +⋆ +⋆ +� +, +�⋆ +⋆ +0 +⋆ +� +, +�⋆ +⋆ +⋆ +0 +� +. +We write down explicitly some affine algebras; for that, we endow each dual V ∨ +i,j with a partic- +ular basis (u(k) +i,j )1≤k≤di,j, where di,j := dimCVi,j. We let P(Vi,j)ei,j the image of Vi,j(ei,j) in P(Vi,j) +and P(V (∗) +S )e their product for (i, j) ∈ S, i.e. the image of VS(e) in P(V (∗) +S +). Then: +C[Vi,j] = C[(u(k) +i,j )1≤k≤di,j], +C[Vi,j(ei,j)] = C[(u(k) +i,j )1≤k≤di,j][1/ei,j], +C[VS] = +� +(i,j)∈S +C[(u(k) +i,j )1≤k≤di,j] = C[all u(k) +i,j ], +C[VS(e)] = +� +(i,j)∈S +C[(u(k) +i,j )1≤k≤di,j][1/ei,j] = C[all u(k) +i,j ][all 1/ei,j], +C[P(Vi,j)ei,j] = C[(u(k) +i,j /ei,j)1≤k≤di,j], +C[P(V (∗) +S )e] = C[all u(k) +i,j /ei,j]. +If e is fixed, we shall write x(k) +i,j := u(k) +i,j /ei,j, so that: +C[P(Vi,j)ei,j] = C[(x(k) +i,j )1≤k≤di,j] and C[P(V (∗) +S )e] = C[all x(k) +i,j ]. +Since each family (u(k) +i,j )1≤k≤di,j is a basis of V ∨ +i,j, there are linear relations: +ei,j = ∑ +1≤k≤di,j +α(k) +i,j u(k) +i,j =⇒ ∑ +1≤k≤di,j +α(k) +i,j x(k) +i,j = 1. +Note that, although each u(k) +i,j is a function on Vi,j and each x(k) +i,j is a function on Vi,j(ei,j), we +respectively consider them (in an obvious way) as functions on V (∗) +S , resp. on VS(e). +Example 1.31 In the JS case (for “Jimbo-Sakai”), i.e. for n = 2, S = {1,2} × {1,2} and all +dimVi,j = 2, we shall rather denote (ui,j,vi,j) the chosen basis of V ∨ +i,j; and xi,j := ui,j/ei,j, yi,j := +vi,j/ei,j. Then, a fixed e will be expressed as ei,j = αi,jui,j +βi,jvi,j, so αi,jxi,j +βi,jyi,j = 1. +Example 1.32 In degenerate JS case, which is the same except that S ⊂ {1,2}×{1,2} misses one +particular pair of indices (so that one coefficient is 0, the three other ones are ̸= 0), we keep those +notations for all (i, j) ∈ S. +20 + +1.4.4 +Equations +Since TS is a closed subgroup of the torus C∗S, it is defined by a set of monomial equations, actu- +ally a subgroup of the character group X(C∗S) = ZS. This subgroup is free, so it admits a basis and +there is a basis of monomial equations for TS of the form M(Xi,j)(i,j)∈S = 1. We shall determine in +a moment such a basis. +At any rate, letting: +X(C∗S) = +� +∏ +(i,j)∈S +Xri,j +i,j | all ri,j ∈ Z +� +≃ ZS, +the group of characters of C∗S, let MS ⊂ X(C∗S) such a set of monomials, i.e. a basis for X(C∗S/TS). +Then, for each M(Xi,j) ∈ MS and for each e := (ei,j)(i,j)∈S ∈V ∨∗ +S ), the substitutions e∗M := M(ei,j) +define regular functions on VS(e) constant on each orbit. +Remark 1.33 One can prove, e.g. by using the precise form of the basic cycles, that C∗S/TS is +itself a torus. +Lemma 1.34 Relations e∗M = constant, M ∈ MS, make up a complete set of equations for each +orbit H.M in L∗(M). +Proof. - Indeed, M = (mi,j)(i,j)∈S being fixed, we may choose the isomorphism L∗(M) ≃ C∗S send- +ing each N = (ni,j)(i,j)∈S to (ei,j(ni,j)/ei,j(mi,j))(i,j)∈S. Then the pullbacks of equations Mc = 1 are +equations e∗Mc(N) = e∗Mc(M). +□ +Since the dimension of TS is 2n−χ(S), the basis MS has c(S) := |S|−2n+χ(S) ´el´ements. This +number is the circuit rank or the cyclomatic number of our bipartite graph (see either [2, chap. 4], +or Wikipedia, heading “circuit rank”). Here is a way to generate equations from cycles. We write +C(S) a fixed basis of elementary cycles in the above bipartite graph. Every c ∈ C(S) has the form +of a loop i1 → j1 → i2 → j2 → ··· → ik → jk → i1, where i1,...,ik, resp. j1,..., jk are pairwise +distinct, and where the cycle c does not depend on the particular selected origin i1 of the loop. +Then the coordinates xi,j of an element of TS satisfy: +xi1,j1xi2,j2 ···xik,jk = λi1 ···λik +µj1 ···µik += xi1,jkxi2,j1 ···xik,jk−1 +for any antecedent (λ,µ). We then set: +Mc := Xi1,j1Xi2,j2 ···Xik,jk +Xi1,jkXi2,j1 ···Xik,jk−1 +∈ MS. +This is almost well defined from c, independent of the point of departure (chosen among the i- +indices, i.e. in the left factor of {1,...,n}×{1,...,n} ⊃ S). The only freedom left comes from the +change of orientation on the cycle, which can transform Mc to M−1 +c . We can freeze it by deciding +21 + +for instance that i1 is the smallest possible in the cycle, then j1 is the smallest possible for this i1. +That being set, almost all statements until the end of 1.4.4, as well as in 1.4.5, flow from +elementary combinatorial arguments and we leave their proof to the reader. +Proposition 1.35 TS is exactly the closed subset of C∗S defined by these equations, which are +independent. +Corollary 1.36 H.M is exactly the closed subset of L∗(M) defined by equations e∗Mc = constant, +c ∈ C(S), which are independent. +Example 1.37 In the JS case, there is only one equation for TS, namely x1,1x2,2 = x1,2x2,1, so the +equations of orbits are x1,1x2,2/x1,2x2,1 = constant. +Example 1.38 In the degenerate JS case, there are no cycles and no equations: H.M = L∗(M). +We have the equations of H.M in L∗(M), now we look for the equations of L∗(M) in VS(e). +Since L∗(M) = π−1(π(M)), using that: +π(N) = π(M) ⇐⇒ ∀(i, j) ∈ S , ∀u ∈ V ∨ +i,j , u(M) = u(N), +so, resorting to the selected bases: +π(N) = π(M) ⇐⇒ ∀(i, j) ∈ S , ∀k = 1,...,di,j , u(k) +i,j (M) = u(k) +i,j (N). +Lemma 1.39 A complete set of equations of L∗(M) in VS(e) is given by the u(k) +i,j = constant. +Corollary 1.40 H.M is exactly the closed subset of VS(e) defined by equations e∗Mc = constant, +c ∈ C(S), and u(k) +i,j = constant, (i, j) ∈ S, k = 1,...,di,j, which are independent. +The above statements will be made more precise by computing algebras of invariants. +1.4.5 +Algebras of invariants +We know that C +� +C∗S� += C +� +(Xi,j,X−1 +i,j )(i,j)∈S +� +. From now on, we take the set MS := {Mc | c ∈ +C(S)} as a basis of equations of TS in C∗S. +Lemma 1.41 The affine algebra of TS is: +C[TS] = C +� +{M,M−1 | M ∈ MS} +� += C +� +{Mc,M−1 +c +| c ∈ C(S)} +� +. +Proposition 1.42 The algebra of invariants on VS(e) is: +C[VS(e)/H] = C[VS(e)]H = C[all x(k) +i,j , all e∗Mc,e∗M−1 +c ], +with |S| relations ∑ +k +α(k) +i,j x(k) +i,j = 1. +22 + +Corollary 1.43 The quotient VS(e)/H is isomorphic to P(V (∗) +S )e ×TS. +Theorem 1.44 The quotient VS/H is a fibered space over P(V (∗) +S ) with fibers isomorphic to TS. +Proof. - The affine spaces P(V (∗) +S +)e, e ∈ V ∨∗ +S , patch up into P(V (∗) +S ). For e,e′ ∈ V ∨∗ +S , the functions +φi,j := e′ +i,j/ei,j from P(V (∗) +S )e ∩ P(V (∗) +S )e′ to C∗ define the transition functions on the affine atlas. +They yield the transition functions on the fibers defined by the Mc(φi,j). +□ +Example 1.45 In the JS case (recall that this means n = µ = 2), we get a C∗-fibration on P1(C)4. +The transition functions are the +(e′ +1,1/e1,1)(e′ +2,2/e2,2) +(e′ +1,2/e1,2)(e′ +2,1/e2,1)· +In the degenerate JS case, we get P1(C)3. +2 +Invariants and quotients in the (possibly degenerate) JS case +2.1 +Some general notations +Here n = 2 and each Vi,j has dimension 2. The group H is D2(C)×D2(C) +C∗(I2,I2) +· +For i, j ∈ {1,2}, we fix a basis (ui,j,vi,j) of V ∨ +i,j. +For ei,j = αi,jui,j +βi,jvi,j ∈ V ∨ +i,j \{0}, we set Vi,j(ei,j) := Vi,j \Ker ei,j. Those form an affine +covering of Vi,j. The corresponding affine algebras are: +C(Vi,j(ei,j)) = C[ui,j,vi,j][e±1 +i,j ]. +The associated projective space P(Vi,j) is covered by the affine charts P(Vi,j)ei,j images of the +C(Vi,j(ei,j)); the corresponding affine algebras are: +C +� +P(Vi,j)ei,j +� += C[ui,j,vi,j][e±1 +i,j ] = C[xi,j,yi,j][e±1 +i,j ], +where we have set xi,j := ui,j/ei,j and yi,j := vi,j/ei,j (the letters x,y are fixed although the variables +xi,j,yi,j actually depend on the linear form ei,j and on the particular affine chart). They are not +independent variables: +αi,jxi,j +βi,jyi,j = αi,jui,j/ei,j +βi,jvi,j/ei,j = 1. +Remarks 2.1 (i) Let V := V1,1 ×V1,2 ×V2,1 ×V2,2. Then the only invariants in C(V) under the +H-action are the constants (this will easily follow from the calculations to come). Thus the cate- +gorical quotient V/H of the affine variety V by the reductive group H is trivial. +(ii) On the other hand, e1,1,e1,2,e2,1,e2,2 being chosen as above, all the corresponding xi,j,yi,j (con- +sidered as rational functions onV) are invariant under the H-action and so is φ := (e1,1e1,2)/(e2,1,e2,2). +It will also follow from the calculations to come that they together generate the C(V)H (where +C(V) is the fraction field of C(V)); so C(V)H has transcendence degree 5. +23 + +2.2 +Non degenerate JS case: no zeroes allowed +Here S = {1,2}×{1,2}. We have: +VS = V (∗) = V ∗ +1,1 ×V ∗ +1,2 ×V ∗ +2,1 ×V ∗ +2,2. +Let e := (e1,1,e1,2,e2,1,e2,2), each ei,j ∈ V ∨ +i,j \{0}. Then: +VS(e) = V (∗)(e) = V1,1(e1,1)×V1,2(e1,2)×V2,1(e2,1)×V2,2(e2,2). +Those open subsets of V (∗) form an affine covering. The corresponding affine algebras are: +C +� +V (∗)(e) +� += +� +i,j +C[xi,j,yi,j][e±1 +i,j ] = C[all xi,j,yi,j][all e±1 +i,j ]. +To study the invariants under H, we set the torus: +TS = C∗4 with affine algebra C(TS) = C[all X±1 +i,j ,i, j = 1,2] = +� +M∈M ∗ +CM, +where M ∗ = MS is the free abelian group over the Xi,j, i.e. the set of all monomials M := +Xr1,1 +1,1 Xr1,2 +1,2 Xr2,1 +2,1 Xr2,2 +2,2 , all ri,j ∈ Z. In particular we distinguish Φ := X1,1X2,2 +X1,2X2,1 +· For M ∈ M ∗ and for e +as above, we set: +e∗M := M(ei,j) = er1,1 +1,1 er1,2 +1,2 er2,1 +2,1 er2,2 +2,2 . +In particular e∗Φ = φ := e1,1e2,2 +e1,2e2,1 +, which is H-invariant. (Of course ambiguous notation φ depends +on the particular e, we shall try to avoid confusions.) +Lemma 2.2 (i) C +� +V (∗)(e) +�H = C[all xi,j,yi,j][φ±1]. +(ii) There is a geometric quotient: +V (∗)(e)/H = P(V (∗))e ×C∗ where P(V (∗))e := ∏P(Vi,j)ei,j. +Proof. - (i) Every P ∈ C +� +V (∗)(e) +� +admits a unique expansion as: +P = ∑ +M∈M ∗ +PM(all xi,j,yi,j)e∗M. +The factors PM are H-invariants, while the factors e∗M are semi-invariants, the only invariant ones +being the powers of e∗Φ. (This is actually the decomposition of a linear representation flowing +from the semi-simplicity of H.) +(ii) We know that the categorical quotient has algebra C +� +V (∗)(e) +�H (affine by reductive). It is a +geometric quotient by [6, p 9], because orbits are closed and stabilizers are trivial (this was proved +before). +□ +24 + +Proposition 2.3 The quotient V (∗)/H is (the total space of) a fibration with fiber C∗ over the base +P(V (∗)) := ∏P(Vi,j) ≃ P1(C)4. It is a geometric quotient. +Proof. - The affine charts P(V (∗))e, P(V (∗))f on P(V (∗)) are patched along their intersection +P(V (∗))e ∩ P(V (∗))f by the transition functions (ei,j/fi,j)i,j. The corresponding transition func- +tions on the C∗ factors are the e∗Φ +f ∗Φ· The possibility of patching geometric quotients is guaranteed +by [27, prop. 3.10 (b), p 71]. One just has to check first that there is a well defined morphism +V (∗) → Y, Y the candidate quotient (the fiber space described above); and then to check that it +localizes to geometric quotients on a covering of Y. The first step is easy, the second step comes +from the lemma. +□ +Corollary 2.4 The space V (∗)/H is separated. +2.3 +Degenerate JS case: one zero required +We fix (for instance) the position of the zero coefficient at (1,1), so that here: +S = ({1,2}×{1,2}) \{(1,1)} = {(1,2),(2,1),(2,2)}. +We thus look at the H-action on: +VS = V 0 = {0}×V ∗ +1,2 ×V ∗ +2,1 ×V ∗ +2,2. +We shall write e′ := (e1,2,e2,1,e2,2), each relevant ei,j ∈V ∨ +i,j \{0} (here and later, “relevant” means +that (i, j) = (1,1) is omitted). Then: +VS(e′) = V 0(e′) = {0}×V1,2(e1,2)×V2,1(e2,1)×V2,2(e2,2). +Those open subsets of V 0 form an affine covering. The corresponding affine algebras are (same +notations as in the non degenerate JS case): +C +� +V 0(e′) +� += C[x1,2,y1,2,x2,1,y2,1,x2,2,y2,2][e±1 +1,2,e±1 +2,1,e±1 +2,2]. +Lemma 2.5 (i) C +� +V 0(e′) +�H = C[x1,2,y1,2,x2,1,y2,1,x2,2,y2,2]. +(ii) There is a geometric quotient: +V 0(e′)/H = P(V 0)e′ := P(V1,2)e1,2 ×P(V2,1)e2,1 ×P(V2,2)e2,2. +Proof. - (i) Every P ∈ C +� +V 0(e′) +� +admits a unique expansion as a sum ∑PM(relevant xi,j,yi,j)M(relevant ei,j), +but here the only invariant monomial M is the trivial one (because the relevant λi/µj are three in- +dependent complex numbers). +(ii) Follows as in the non degenerate case. +□ +Proposition 2.6 There is a geometric quotient: +V 0/H = P(V 0) := P(V1,2)×P(V2,1)×P(V2,2) ≃ P1(C)3. +25 + +2.4 +A candidate craddle for partial patching: one zero allowed +We fix again the position of the allowed zero coefficient at (1,1), so that we are looking at the +H-action on: +V ′ := V1,1 ×V ∗ +1,2 ×V ∗ +2,1 ×V ∗ +2,2 = V (∗) ⊔V 0. +For e′ := (e1,2,e2,1,e2,2), each relevant ei,j ∈ V ∨ +i,j \{0}, we set: +V ′(e′) = V1,1 ×V1,2(e1,2)×V2,1(e2,1)×V2,2(e2,2). +Those open subsets of V ′ form an affine covering. To describe the corresponding affine algebras, +we use the same notations as before, plus a new one: +x′ +1,1 = u1,1 +e2,2 +e1,2e2,1 +, +y′ +1,1 = v1,1 +e2,2 +e1,2e2,1 +· +They are H-invariant and, whenever e is involved, related by equalities: +� +x′ +1,1 = x1,1 e∗Φ, +y′ +1,1 = y1,1 e∗Φ, +=⇒ α1,1x′ +1,1 +β1,1y′ +1,1 = e∗Φ. +The affine algebras are then given by: +C +� +V ′(e′) +� += C[x′ +1,1,y′ +1,1,x1,2,y1,2,x2,1,y2,1,x2,2,y2,2][e±1 +1,2,e±1 +2,1,e±1 +2,2]. +Then by the same arguments as before: +Lemma 2.7 (i) C(V ′(e′))H = C[x′ +1,1,y′ +1,1,x1,2,y1,2,x2,1,y2,1,x2,2,y2,2]. +(ii) There is a categorical quotient: +V ′(e′)/H = P(V ′)e′ := C2 ×P(V1,2)e1,2 ×P(V2,1)e2,1 ×P(V2,2)e2,2. +Proof. - (i) Is easy as in the previous calculations. +(ii) Follows by the general statement about a reductive group acting on an affine variety. +□ +Proposition 2.8 There is a categorical quotient V ′/H; it is a vector bundle of rank 2 over P(V 0). +It is actually the cartesian square of a line bundle with transition functions the f2,2/( f1,2 f2,1) +e2,2/(e1,2e2,1)· +Proof. - Indeed, those are the laws of transformation independently of variables x′ +1,1,y′ +1,1. We +again use the patching argument from [27] already used in the proof of proposition 2.3. +□ +Remark 2.9 The above bundle is actually an “external tensor product” of line bundles O(±1). +2.5 +The patching +To see how the quotients V (∗) → V (∗)/H and V 0 → V 0/H embed into the quotient V ′ → V ′/H, we +use two different principles. +26 + +2.5.1 +Non degenerate part +Let e′ be extracted from e omitting e1,1. We have an open immersion of affine varieties V (∗)(e) → +V ′(e′) induced by the dual morphism of their affine algebras: +C[x′ +1,1,y′ +1,1,all other xi,j,yi,j][e±1 +1,2,e±1 +2,1,e±1 +2,2] → C[all xi,j,yi,j][all e±1 +i,j ] +From the equality u1,1 +e2,2 +e1,2e2,1 += u1,1 +e1,1 +e1,1e2,2 +e1,2e2,1 +and the similar one with v1,1, we see that it sends x′ +1,1 +to x1,1φ and y′ +1,1 to y1,1φ. It restricts to the H-invariant subalgebras as the morphism sending x′ +1,1 +to x1,1φ and y′ +1,1 to y1,1φ: +C[x′ +1,1,y′ +1,1,all other xi,j,yi,j] → C[all xi,j,yi,j][φ±1]. +The open immersion being H-equivariant, it induces an immersion of the quotients from P(V (∗))e× +C∗ to C2 ×P(V1,2)e1,2 ×P(V2,1)e2,1 ×P(V2,2)e2,2 dual to the above morphism of algebras. Patching +up, we get an open immersion: +V (∗)/H = P(V (∗))×C∗ → V ′/H = plane bundle over P(V 0), +described in coordinates by the above morphism. So V (∗) → V (∗)/H is the restriction of the cate- +gorical quotient V ′ → V ′/H to an invariant open subset. +2.5.2 +Degenerate part +Let e′ be as usual. We have a closed immersion of affine varieties V 0(e′) → V ′(e′) induced by the +dual morphism of their affine algebras sending x′ +1,1 and y′ +1,1 to 0: +C[x′ +1,1,y′ +1,1,all other ][e±1 +1,2,e±1 +2,1,e±1 +2,2] → C[x1,2,y1,2,x2,1,y2,1,x2,2,y2,2][e±1 +1,2,e±1 +2,1,e±1 +2,2]. +It restricts to a similar morphism of the invariant subalgebras: +C[x′ +1,1,y′ +1,1,x1,2,y1,2,x2,1,y2,1,x2,2,y2,2] → C[x1,2,y1,2,x2,1,y2,1,x2,2,y2,2]. +The dual morphism is a closed immersion to the {0} ⊂ C2 section: +P(V1,2)e1,2 ×P(V2,1)e2,1 ×P(V2,2)e2,2 → C2 ×P(V1,2)e1,2 ×P(V2,1)e2,1 ×P(V2,2)e2,2 +Patching up, we get a closed immersion to the null section: +V 0/H = P(V 0) → V ′/H = plane bundle over P(V 0). +3 +Quadrics in the non degenerate JS case and functions on them +3.1 +General notations and conventions +This section aims at translating in terms of (bi)linear algebra the properties of the monodromy +matrices in FR,S,x (cf. the paper [32]). This yields some objects subject to some axioms, for which +we briefly recall each time the justification from [32]. From now on in this article, n = µ = 2 so +that N = 4. The following conditions will be systematically assumed: +27 + +• (NR) Strong non resonancy: +∀i, j ∈ {1,...,2} , i ̸= j =⇒ (ρi ̸≡ ρ j and σi ̸≡ σ j), +∀k,l ∈ {1,...,4} , k ̸= l =⇒ xk ̸≡ xl. +• (FR) Fuchs relation: +(−1)Nx1 ···xN = ρ1 ...ρn +σ1 ...σn +· +(Recall that we denote ≡ the congruence modulo qZ in C∗.) Also the following conditions will be +frequently invoked: +• (NS) Non splitting: +∀i, j ∈ {1,2} , ρi/σ j ̸≡ x1x2. +• (SC) Special condition: +ρ1 +σ1 +̸≡ ρ2 +σ2 +and +ρ2 +σ1 +̸≡ ρ1 +σ2 +. +For comments around (NR), (FR) and (NS), see [32, 5.1]; for (SC), see [32, 4.4]. +3.1.1 +Evaluation and linear forms +We are given on each Vi,j four particular non trivial linear forms respectively corresponding to the +evaluation maps m �→ m(xk), k = 1,2,3,4. They define four families in the V ∨ +i,j. +Axiom. - For each i, j ∈ {1,2}, the four given forms are pairwise linearly independent. +Explanation. - If mi,j(xk) = mi,j(xl) = 0 with mi,j ̸= 0, then xkxl ≡ ρi/σ j, contradicting (NS). +With the previous notations for bases (ui,j,vi,j) of the Vi,j, we can decide for instance that ui,j +and vi,j respectively correspond to the evaluations at x1 and x2. We denote wi,j, resp. w′ +i,j, the +evaluation at x3, resp. at x4 and express them as: +� +wi,j = λi,jui,j +µi,jvi,j, +λi,j,µi,j ̸= 0, +w′ +i,j = λ′ +i,jui,j +µ′ +i,jvi,j, +λ′ +i,j,µ′ +i,j ̸= 0, +with moreover the conditions λi,jµ′ +i,j −λ′ +i,jµi,j ̸= 0. +Through the natural projections from V := V1,1 ×V1,2 ×V2,1 ×V2,2 to each Vi,j, and the dual +injections V ∨ +i,j → V ∨, we may (and will) intepret the above as linear forms on V. In this way, we +obtain four linear maps u, v, w, and w′, from V to Mat2(C). +3.1.2 +Determinant and bilinear forms +Corresponding to the evaluations (m1,1m2,2 − m1,2m2,1)(xk) = m1,1m2,2(xk) − m1,2m2,1(xk) of the +determinant, we define on V twelve bilinear forms as follows: +A+ := u1,1u2,2, +A− := u1,2u2,1, +A := A+ −A− = detu, +B+ := v1,1v2,2, +B− := v1,2v2,1, +B := B+ −B− = detv, +C+ := w1,1w2,2, +C− := w1,2w2,1, +C := C+ −C− = detw, +C′ ++ := w′ +1,1w′ +2,2, +C′ +− := w′ +1,2w′ +2,1, +C′ := C′ ++ −C′ +− = detw′. +28 + +Axiom. - There exists α,β,γ ∈ C∗ such that C+ = αA++βB++γC+ and C− = αA− +βB−+γC−. +It follows that C = αA+βB+γC. +Explanation. - Fuchs relations x1x2x3x4 ≡ ρ1ρ2 +σ1σ2 +imply that, if m ∈ V4, ρ1ρ2 +σ1σ2 vanishes at three of the +xk then it vanishes at the fourth. So this is really a relation among linear forms on V4, ρ1ρ2 +σ1σ2 . +3.1.3 +Application to FR,S,x and FR,S,x +We want to translate the condition (from [32]) that det M vanishes at the xk, k = 1,...,4, but +does not vanish identically; equivalenly, because of the Fuchs relations and the fact that det M ∈ +V4, ρ1ρ2 +σ1σ2 : the matrices M(xk) all have rank 1 (otherwise det M would have a multiple zero). So this +means that det M(xk) = 0 but no M(xk) = 0. Also it implies that M has neither a null line nor +a null column. At any rate, the subset F ⊂ V is locally closed (det ̸= 0, open condition, within +det M(xk) = 0, closed conditions) and also stable under the action of H. We write the space of +interest (in which we recognize FR,S,x): +F := A−1(0)∩B−1(0)∩C−1(0)\W, +W := det−1(0). +Lemma 3.1 F ⊂ V (∗) = V ∗ +1,1 ×V ∗ +1,2 ×V ∗ +2,1 ×V ∗ +2,2. +Proof. - If for instance m1,1 = 0, then m1,2m2,1 ̸= 0 (lest det M be 0) but it vanishes at all xk +(because det M does), so at least one of m1,2, m2,1, vanishes at at least two of the xk, contradicting +(NS). A similar argument applies to all mi,j. +□ +The image F of F under the geometric quotient map V (∗) → V (∗)/H (in which we recognize +FR,S,x) is a subvariety of V (∗)/H. +Theorem 3.2 (i) The map FR,S,x → FR,S,x is a geometric quotient. +(ii) The monodromy data space FR,S,x is a separated variety. +Proof. - (i) Using the lemma, it is an application of corollary 1.29. +(ii) This follows from corollary 2.4. +□ +Another useful consequence of the lemma is: +Corollary 3.3 For M ∈ F, and k ̸= l, matrices M(xk) and M(xl) cannot have a zero at the same +position. +Proof. - If for instance m1,1 vanishes at xk and xl, k ̸= l, then, by (NS), m1,1 = 0, which was just +excluded. +□ +3.2 +Rank 1 matrices in Mat2(C) and the Segre embedding +3.2.1 +Projective “coordinates” for rank 1 matrices +The image of the map (C,L) �→ CL from Mat2,1(C) × Mat1,2(C) to Mat2(C) is the subset of sin- +gular matrices. The restriction to Mat2,1(C)∗ × Mat1,2(C)∗ arrives in Mat2(C)∗, its image is the +29 + +set Mat2(C)1 of matrices of rank 1. Note that the latter is locally closed in Mat2(C), so it is a +quasi-affine variety, actually a C∗-cone, so the notation P(Mat2(C)1) is licit (it denotes a quasi- +projective variety). Going to the projective quotients, we obtain: +P(Mat2,1(C))×P(Mat1,2(C)) → P(Mat2(C)), +which can be identified to the Segre embedding P1(C)×P1(C) → P3(C), an isomorphism of the +source with a smooth projective quadric surface, the Segre surface. +Mat2,1(C)∗ ×Mat1,2(C)∗ +� +� +Mat2(C)1 +� +� +� +Mat2(C)∗ +� +P(Mat2,1(C))×P(Mat1,2(C)) +� +� +P(Mat2(C)1) +� +≃ +� +P(Mat2(C)) +� +P1(C)×P1(C) +≃ +� Segre surface +� P3(C) +The Segre surface corresponds isomorphically to the image P(Mat2(C)1) of Mat2(C)1 in +P(Mat2(C)). We write (ρ,ρ′) the composite map (the dotted arrow in the previous diagram): +Mat2(C)1 → P(Mat2(C)1) → Segre surface → P1(C)×P1(C), +CL �→ (class of C,class of L). +So ρ and ρ′ are both regular maps from the variety Mat2(C)1 to P1(C). +Concretely, let A := +�a1,1 +a1,2 +a2,1 +a2,2 +� +∈ Mat2(C)1. Then ρ(A),ρ′(A) ∈ P1(C) are given by the +formulas: +(3.3.1) +ρ(A) = [a1,1 : a1,2] = [a2,1 : a2,2] and ρ′(A) = [a1,1 : a2,1] = [a1,2 : a2,2]. +In the first formula, one of the expressions [a1,1 : a1,2], [a2,1 : a2,2] may be undefined (if one line of +A vanishes): then just dismiss the corresponding equality; same thing for the second formula. +3.2.2 +Special points and mixed projective coordinates +It is immediate by inspection of CL that: +• The closed subset ρ−1(0) of Mat2(C)1 consists of matrices with trivial first line; +• The closed subset ρ−1(∞) of Mat2(C)1 consists of matrices with trivial second line; +• The closed subset ρ′−1(0) of Mat2(C)1 consists of matrices with trivial first column; +• The closed subset ρ′−1(∞) of Mat2(C)1 consists of matrices with trivial second column. +Now, let Mi := CiLi ∈ Mat2(C)1, i = 1,2, be such that neither ρ(M1) = ρ(M2) = 0 nor ρ(M1) = +ρ(M2) = ∞. Denoting Ci := +� fi +gi +� +̸= 0, one easily checks that f1g2 = f2g1 = 0 would imply either +30 + +f1 = f2 = 0 or g1 = g2 = 0, which are both excluded by assumption. Therefore [ f1g2 : f2g1] ∈ +P1(C) is well defined. Using the fact that CL =C′L′ ⇒C′,C proportional, we see that [ f1g2 : f2g1] +depends on M only. We call it Π(M1,M2). Using lines instead of columns we define similarly +Π′(M1,M2). This way, we define two regular maps: +� +Π : Mat2(C)1 ×Mat2(C)1 \ +� +(ρ−1(0)×ρ−1(0))∪(ρ−1(∞)×ρ−1(∞)) +� +→ P1(C), +Π′ : Mat2(C)1 ×Mat2(C)1 \ +� +(ρ′−1(0)×ρ′−1(0))∪(ρ′−1(∞)×ρ′−1(∞)) +� +→ P1(C). +3.3 +The maps ρk, k = 1,...,4, and [ρ] on F +We try to adapt the first approach in [16]. Independently of the more or less “axiomatic” presen- +tation adopted here (unification will come later), we summarize some basic facts about F and its +quotient F : +1. F is a subset of the 8-dimensional linear space V, defined by three homogeneous quadratic +equations detM(xk) = 0, k = 1,2,3 (the fourth one is a consequence by Fuchs relation (FR)) +and one inequation detM ̸= 0. Therefore F is a quasi-affine variety of dimension ≥ 5. +2. Because of the non splitting condition (NS), we know that F ⊂ V (∗). It is also clearly stable +under the action of H. Since we know that there is a geometric quotient V (∗)/H, we see that +F is the image of F in V (∗)/H and that it is a quasi-affine variety of dimension ≥ 2. +3.3.1 +Definition of the ρk and [ρ] +Let Γ,∆ ∈ D2(C). Then, with the usual notations M = CL: +ρ(ΓM) = ρ(ΓCL) = ρ((ΓC)L) = class of L = ρ(CL) = ρ(M), +while: +ρ(M∆) = ρ(CL∆) = ρ(C(L∆)) = class of L∆ = χ(∆).ρ(CL) = χ(∆).ρ(M), +where11 χ(Diag(d,d′)) := d/d′ and we have C∗ act on P1(C) by c.[x : y] := [cx : y]. Symmetrically: +ρ′(ΓM) = χ(Γ).ρ′(M), +ρ′(M∆) = ρ′(M). +We have defined F so that it is sent into Mat2(C)1 by either one of u, v, w, w′. Therefore we +have a well defined regular map: +(ρ1,ρ2,ρ3,ρ4) := (ρ◦u,ρ◦v,ρ◦w,ρ◦w′) : F → +� +P1(C) +�4 . +It is equivariant under the action of H in the following sense: +M �→ (r1,r2,r3,r4) ∈ +� +P1(C) +�4 =⇒ ΓM∆−1 �→ (c.r1,c.r2,c.r3,c.r4), where c := χ(Γ) +χ(∆)· +Recall F , the image of F under V (∗) → V (∗)/H. The above induces a map: +[ρ] : F → +� +P1(C) +�4 /C∗. +11Of course this character χ is in no way related to the function χ on graphs introduced in 1.4.1. +31 + +Proposition 3.4 The map [ρ] is injective. +Proof. - We want to show that if M,N ∈ F are such that ρk(N) = cρk(M) for some c ∈ C∗ and +k = 1,...,4, then N = ΓM∆ for some Γ,∆ ∈ D2(C). First, changing M to M∆ for some ∆ ∈ D2(C) +(e.g. Diag(1,c)), we can assume that c = 1. Then, from the lemma that follows applied in turn to +the four equalities ρk(M) = ρk(N), i.e. ρ(M(xk)) = ρ(N(xk)), we draw that for all i, j = 1,2: +mi,1nj,2 −mi,2nj,1 ∈ V4,ρiρj/(σ1σ2) vanishes at all xk. +For i = j, using properties (NR) (non resonancy) and (FR), we have: +ρ2 +i /(σ1σ2) ̸≡ ρ1ρ2/(σ1σ2) =⇒ ρ2 +i /(σ1σ2) ̸≡ x1x2x3x4, +so an element of V4,ρ2 +i /(σ1σ2) which vanishes at all xk must be 0. Therefore we have mi,1ni,2 = +mi,2ni,1 for i = 1,2. Let γi := ni,1/mi,1 = ni,2/mi,2 (recall that all mi,j and ni,j are ̸= 0). Both γi +are non zero elliptic functions of order ≤ 2. But, because of (NR), mi,1 and mi,2 cannot have two +common zeroes. So actually γ1,γ2 ∈ C∗ and Γ := Diag(γ1,γ2) ∈ D2(C) is such that N = ΓM. +□ +Lemma 3.5 Let A := (ai,j),B := (bi,j) ∈ Mat2(C)1. Then: +ρ(A) = ρ(B) ⇐⇒ ∀i, j = 1,2 , ai,1bj,2 −ai,2bj,1 = 0. +Proof. - Note that, generally speaking, [a1 : a2] = [b1 : b2] ⇔ a1b2 = a2b1 and apply the “concrete +description” (3.3.1) of ρ(A),ρ(B) at the end of 3.2.1. +□ +We would like to consider [ρ] as a regular map, but it is not clear in what sense the quotient +� +P1(C) +�4 /C∗ is defined beyond its obvious set-theoretic sense. For instance, generic orbits under +the C∗ action on +� +P1(C) +�4 have dimension 1 while there are 16 invariant points Θ4 := {0,∞}4, +etc. However, we saw in 1.2.5 that, extracting the set Θ4, we obtain a (non separated) geometric +quotient K4 := +�� +P1(C) +�n \Θn +� +/C∗, which is moreover a toric variety (cf. 1.3.3). +3.3.2 +The image of [ρ] +Instead of ρ1,ρ2,ρ3,ρ4, we shall write ρ,σ,τ,τ′ in harmony with our notations u,v,w,w′ (and in +slight contradiction with the previous use of ρ, but it does not seem to matter much). We do the +case of F1, so (ρ,σ,τ,τ′) sends it to Y ′ +1 = C∗ ×C∗ ×C×C∞. We have defining equalities: +� +u1,2 = ρu1,1, +u2,2 = ρu2,1, +� +v1,2 = σv1,1, +v2,2 = σv2,1, +� +w1,2 = τw1,1, +w2,2 = τw2,1, +� +w′ +1,2 = τ′w′ +1,1, +w′ +2,2 = τ′w′ +2,1. +Only in the special case that τ′ = ∞ must we rather use τ′′ := τ′−1 (to be considered at τ′′ = 0) and +replace the last pair of equalities by +� +τ′′w′ +1,2 = w′ +1,1, +τ′′w′ +2,2 = w′ +2,1, +. We also recall that +� +wi,j = λi,jui,j +µi,jvi,j, +w′ +i,j = λ′ +i,jui,j +µ′ +i,jvi,j, +for i, j = 1,2, whence: +� +λ1,2ρu1,1 +µ1,2σv1,1 = τ(λ1,1u1,1 +µ1,1v1,1), +λ2,2ρu2,1 +µ2,2σv2,1 = τ(λ2,1u2,1 +µ2,1v2,1), +and +� +λ′ +1,2ρu1,1 +µ′ +1,2σv1,1 = τ′(λ′ +1,1u1,1 +µ′ +1,1v1,1), +λ′ +2,2ρu2,1 +µ′ +2,2σv2,1 = τ′(λ′ +2,1u2,1 +µ′ +2,1v2,1). +32 + +Of course, the last two equalities must be modified in an obvious way at τ′ = ∞ (i.e. put a τ′′ factor +at left instead of a τ′ factor at right). +We translate the above into linear systems; first we do that in the case τ′ ̸= ∞: +� +(λ1,2ρ−λ1,1τ)u1,1 +(µ1,2σ−µ1,1τ)v1,1 = 0, +(λ′ +1,2ρ−λ′ +1,1τ′)u1,1 +(µ′ +1,2σ−µ′ +1,1τ′)v1,1 = 0, +and +� +(λ2,2ρ−λ2,1τ)u2,1 +(µ2,2σ−µ2,1τ)v2,1 = 0, +(λ′ +2,2ρ−λ′ +2,1τ′)u2,1 +(µ′ +2,2σ−µ′ +2,1τ′)v2,1 = 0. +Noting that we can have neither (u1,1,v1,1) = (0,0) nor (u2,1,v2,1) = (0,0) (again because of (NR) +and (NS)), we deduce the vanishing of the determinants: +� +(λ1,2ρ−λ1,1τ)(µ′ +1,2σ−µ′ +1,1τ′)−(λ′ +1,2ρ−λ′ +1,1τ′)(µ1,2σ−µ1,1τ) = 0, +(λ2,2ρ−λ2,1τ)(µ′ +2,2σ−µ′ +2,1τ′)−(λ′ +2,2ρ−λ′ +2,1τ′)(µ2,2σ−µ2,1τ) = 0. +Those can be written in the form of quadratic equations; for i = 1,2: +Aiρσ+Biττ′ +Ciρτ−Diρτ′ −Eiστ+Fiστ′ = 0, +where the coefficients are the following (note that they are all ̸= 0): +Ai := λi,2µ′ +i,2 −λ′ +i,2µi,2,Bi := λi,1µ′ +i,1 −λ′ +i,1µi,1,Ci := λ′ +i,2µi,1,Di := λi,2µ′ +i,1,Ei := λ′ +i,1µi,2,Fi := λi,1µ′ +i,2. +We shall see soon (see 3.4.3) that the quadratic equations for i = 1 and for i = 2 are actually pro- +portional, so they define the same quadrics. +Near τ′ = ∞, i.e. near τ′′ = 0, the equations become: +Aiρστ′′ +Biτ+Ciρττ′′ −Diρ−Eiσττ′′ +Fiσ = 0, +which, near τ′′ = 0, is a graph: +ρ = Biτ+Fiσ−Eiσττ′′ +Di −(Aiσ+Ciτ)τ′′ , +whence the quadrics is nonsingular at points such that τ′′ = 0. +3.4 +Some explicit formulas +They are based on the following particular choices12 of bases for V2,ρi/σj. We set: +∀i, j = 1,2 , ∀k = 1,...,4 , ek +i,j := θq(−x/xk,−xxkσ j/ρi). +(We use the standard convention θq(a,b) := θq(a)θq(b).) Then the linear forms ui,j,vi,j,... on Vi,j +(evaluations at x1,x2,...) satisfy: +� +ui,j(e1 +i,j) = 0, +ui,j(e2 +i,j) = θq(−x1/x2,−x1x2σ j/ρi), +� +vi,j(e1 +i,j) = θq(−x2/x1,−x1x2σ j/ρi), +vi,j(e2 +i,j) = 0, +12Note that our theta functions differ from those in [16] by the change of variable x ↔ −x. +33 + +� +wi,j(e1 +i,j) = θq(−x3/x1,−x1x3σ j/ρi), +wi,j(e2 +i,j) = θq(−x3/x2,−x2x3σ j/ρi), +� +w′ +i,j(e1 +i,j) = θq(−x4/x1,−x1x4σ j/ρi), +w′ +i,j(e2 +i,j) = θq(−x4/x2,−x2x4σ j/ρi). +Now, applying equalities wi,j = λi,jui,j +µi,jvi,j and w′ +i,j = λ′ +i,jui,j +µ′ +i,jvi,j to all ek +i,j, we readily +get: +λi,j = θq−1(−x1/x2,x1x2σ j/ρi)θq(−x3/x2,x2x3σ j/ρi), +µi,j = θq−1(−x2/x1,x1x2σ j/ρi)θq(−x3/x1,x1x3σ j/ρi), +λ′ +i,j = θq−1(−x1/x2,x1x2σ j/ρi)θq(−x4/x2,x2x4σ j/ρi), +µ′ +i,j = θq−1(−x2/x1,x1x2σ j/ρi)θq(−x4/x1,x1x4σ j/ρi). +3.4.1 +Notational conventions +For convenience, we shall use the following abreviations: +T(x) := θq(−x) and T +�x,y,... +z,... +� +:= T(x)T(y)... +T(z)... +; +also: +[kl. ji] := xkxlσ j/ρi, k,l = 1,2,3,4, i, j = 1,2. +Note that, since xθq(1/x) = θq(x), we have xy = 1 ⇒ T(y) = −yT(x). In order to combine that +with Fuchs relation x1x2x3x4 = ρ1ρ2 +σ1σ2 +, we introduce the complementarity abreviations: +i, j ∈ {1,2} =⇒ +def {i,i′} = {j, j′} = {1,2} and k,l ∈ {1,2,3,4},k ̸= l =⇒ +def {k,l,k′,l′} = {1,2,3,4}. +Then [kl. ji][k′l′. j′i′] = 1, whence the complementarity relations: +T +� [kl. ji] +[k′l′. j′i′] +� += −[kl. ji] = −xkxlσ j/ρi. +From loc. cit. we immediately draw: +λ2,j +λ1,j += T +�[23. j2],[12. j1] +[12. j2],[23. j1] +� +, +µ2,j +µ1,j += T +�[13. j2],[12. j1] +[12. j2],[13. j1] +� +, +λ′ +2,j +λ′ +1,j += T +�[24. j2],[12. j1] +[12. j2],[24. j1] +� +, +µ′ +2,j +µ′ +1,j += T +�[14. j2],[12. j1] +[12. j2],[14. j1] +� +, +and: +λ′ +i,j +λi,j += T +�x4/x2 +x3/x2 +� +T +�[24. ji] +[23. ji] +� +, +µ′ +i,j +µi,j += T +�x4/x1 +x3/x1 +� +T +�[14. ji] +[13. ji] +� +. +34 + +3.4.2 +The factor γ +As an immediate application, let13 γ0 := T +�x4/x2,x4/x1 +x3/x2,x3/x1 +� +. Then: +λ′ +1,1µ′ +2,2 +λ1,1µ2,2 += γ0T +�[24.11],[14.22] +[23.11],[13.22] +� +, +λ′ +2,2µ′ +1,1 +λ2,2µ1,1 += γ0T +�[24.22],[14.11] +[23.22],[13.11] +� +, +λ′ +1,2µ′ +2,1 +λ1,2µ2,1 += γ0T +�[24.21],[14.12] +[23.21],[13.12] +� +, +λ′ +2,1µ′ +1,2 +λ2,1µ1,2 += γ0T +�[24.12],[14.21] +[23.12],[13.21] +� +. +All second factors of the right hand sides have the form T +� +u,v +v−1,u−1 +� += uv = x1x2x2 +4 +σ1σ2 +ρ1ρ2 += x4 +x3 +, +so all the left hand sides have the same value γ = γ0 +x4 +x3 +· +This γ is actually the one which appears in the relations of 3.1.2. Indeed, by the argument +sketched there, we have more generally a linear relation among the evaluations at the xk: +∃α,β,γ ∈ C∗ : ∀ f ∈ V4,(ρ1ρ2)/(σ1σ2) , f(x4) = αf(x1)+β f(x2)+γf(x3). +(This is in essence what was presented as an axiom in 3.1.2). In particular: +w′ +1,1w′ +2,2 = αu1,1u2,2 +βv1,1v2,2 +γw1,1w2,2 and w′ +1,2w′ +2,1 = αu1,2u2,1 +βv1,2v2,1 +γw1,2w2,1. +Substituting +λi,jui,j +µi,jvi,j for wi,j and λ′ +i,jui,j +µ′ +i,jvi,j for w′ +i,j, +and assuming moreover (SC), so that the families: +(u1,1u2,2,u1,1v2,2,v1,1u2,2,v1,1v2,2) and (u1,2u2,1,u1,2v2,1,v1,2u2,1,v1,2v2,1) +are free in W, we get by identification: +λ′ +1,1λ′ +2,2 = α+γλ1,1λ2,2, +λ′ +1,2λ′ +2,1 = α+γλ1,2λ2,1, +µ1,1µ2,2 = β+γµ1,1µ2,2, +µ′ +1,2µ′ +2,1 = β+γµ1,2µ2,1, +λ′ +1,1µ′ +2,2 = γλ1,1µ2,2, +λ′ +1,2µ′ +2,1 = γλ1,2µ2,1, +λ′ +2,2µ′ +1,1 = γλ2,2µ1,1, +λ′ +2,1µ′ +1,2 = γλ2,1µ1,2. +Remark 3.6 It is perhaps possible to relax here assumption (SC) using an argument of analytic +continuation. +13This looks like a cross ratio, and it actually degenerates into one when q → 1. +35 + +3.4.3 +The two quadrics are the same +We prove here that A2/A1 = B2/B1 = C2/C1 = D2/D1 = E2/E1 = F2/F1, so that the two quadrics +obtained in 3.3.2 are actually identical. We begin with the four latter quotients, which are simpler. +Let us denote φ := T +�[12.11],[12.21] +[12.12],[12.22] +� += φ1φ2, where φj := T +�[12. j1] +[12. j2] +� +, j = 1,2; and φ′ := +x1x2x3x4 +σ2σ1 +ρ2 +2 += ρ1 +ρ2 +· The following equalities are readily checked (using formulas from 3.4.1): +C2 +C1 += +λ′ +2,2µ2,1 +λ′ +1,2µ1,1 += T +�[24.22],[12.21],[13.12],[12.11] +[12.22],[24.21],[12.12],[13.11] +� += φT +�[24.22],[13.12] +[24.21],[13.11] +� += φφ′ +D2 +D1 += +λ2,2µ′ +2,1 +λ1,2µ′ +1,1 += T +�[23.22],[12.21],[14.12],[12.11] +[12.22],[23.21],[12.12],[14.11] +� += φT +�[23.22],[14.12] +[23.21],[14.11] +� += φφ′ +E2 +E1 += +λ′ +2,1µ2,2 +λ′ +1,1µ1,2 += T +�[24.12],[12.11],[13.22],[12.21] +[12.12],[24.11],[12.22],[13.21] +� += φT +�[24.12],[13.22] +[24.11],[13.21] +� += φφ′ +F2 +F1 += +λ2,1µ′ +2,2 +λ1,1µ′ +1,2 += T +�[23.12],[12.11],[14.22],[12.21] +[12.12],[23.11],[12.22],[14.21] +� += φT +�[23.12],[14.22] +[23.11],[14.21] +� += φφ′. +Now we go for A2/A1 and B2/B1. Let (temporarily) u := σ j/ρi; then: +λi,jµ′ +i,j −λ′ +i,jµi,j = T +�x3/x2,x2x3u +x1/x2,x1x2u +� +T +�x4/x1,x1x4u +x2/x1,x1x2u +� +−T +�x4/x2,x2x4u +x1/x2,x1x2u +� +T +�x3/x1,x1x3u +x2/x1,x1x2u +� += +Φ(x4) +T (x1/x2,x2/x1,x1x2u,x1x2u), +where Φ(x) := T (x3/x2,x2x3u)T (x/x1,x1ux) −T (x3/x1,x1x3u)T (x/x2,x2ux). +Lemma 3.7 +Φ(x) = x3 +x1 +T (x1/x2,x1x2u)T (x/x3,x3ux). +Proof. - Although this follows from a general three terms relation for Theta products (see remark +below), we give a direct argument. The function Φ is holomorphic over C∗ and vanishes at x = +x3. It moreover satisfies the functional equation σqΦ = u−1x−2Φ, so it takes the form Φ(x) = +CT(x/x3)T(x3ux) for some C ∈ C, which can be determined by evaluation at (say) x = x2: +C = +Φ(x2) +T(x2/x3)T(x2x3u) = T (x3/x2,x2x3u)T (x2/x1,x1x2u) +T(x2/x3)T(x2x3u) +, +because the second term in Φ(x2) includes the vanishing factor T (x/x2,x2ux). Thus: +C = T (x3/x2) +T(x2/x3) T (x2/x1,x1x2u) = −x3 +x2 +T (x2/x1,x1x2u) = x3 +x1 +T (x1/x2,x1x2u), +by the (now) customary transformation rules. The stated formula follows. +□ +36 + +Proposition 3.8 +λi,jµ′ +i,j −λ′ +i,jµi,j = x3 +x1 +T +�x4/x3 +x2/x1 +� +T +�x3x4u +x1x2u +� += x3 +x1 +T +�x4/x3 +x2/x1 +� +T +�[34. ji] +[12. ji] +� +. +Proof. - Combining the lemma with the formula that precedes it, we get: +λi,jµ′ +i,j −λ′ +i,jµi,j = x3 +x1 +T +�x1/x2,x1x2u,x4/x3,x3x4u +x1/x2,x2/x1,x1x2u,x1x2u +� += x3 +x1 +T +�x4/x3,x3x4u +x2/x1,x1x2u +� +· +□ +Corollary 3.9 +Ai = x3 +x1 +T +�x4/x3 +x2/x1 +� +T +�[34.2i] +[12.2i] +� +, +Bi = x3 +x1 +T +�x4/x3 +x2/x1 +� +T +�[34.1i] +[12.1i] +� +, +A2/A1 = T +�[34.22],[12.21] +[34.21],[12.22] +� +, +B2/B1 = T +�[34.12],[12.11] +[34.11],[12.12] +� +. +Using the complementarity relations, we eventually get the identity of our two quadrics: +Corollary 3.10 +A2/A1 = B2/B1 = ρ1 +ρ2 +T +�[12.11],[12.21] +[12.12],[12.22] +� += C2/C1 = D2/D1 = E2/E1 = F2/F1. +Remark 3.11 The formula of the lemma can also be thought as reflecting the fact that V2,u−1 has +dimension 2. Along the same lines one can prove a general three terms relation; we exhibit here +three different forms of the said relation. +cT(a)T(b/c)T(x/a)T(x/bc)+aT(b)T(c/a)T(x/b)T(x/ac)+bT(c)T(a/b)T(x/c)T(x/ab) = 0. +T(b)T(a/c)T(x/b)T(x/ac)−T(a)T(b/c)T(x/a)T (x/bc) = b +cT(c)T(a/b)T(x/c)T(x/ab). +cT(a/b)T(d/c)T(x/ab)T (x/cd)+aT(a/c)T(d/b)T(x/ac)T(x/bd)+dT(c/b)T(a/d)T(x/ad)T (x/bc) = 0. +We are not sure whether or not those relations are related to “Fay’s trisecant formula” (cf. [25, +chapter IIIb]). +3.4.4 +The discriminant and the singular locus +The discriminant of the quadratic form is (for i = 1,2): +det + + + + +0 +Ai +Ci +−Di +Ai +0 +−Ei +Fi +Ci +−Ei +0 +Bi +−Di +Fi +Bi +0 + + + + = A2 +i B2 +i +C2 +i F2 +i +D2 +i E2 +i −2AiBiCiFi −2AiBiDiEi −2CiDiEiFi += (AiBi −CiFi −DiEi)2 −4CiDiEiFi +37 + +Using the previously obtained expressions, we find: +the above = +� +(λi,2µ′ +i,2 −λ′ +i,2µi,2)(λi,1µ′ +i,1 −λ′ +i,1µi,1)−λ′ +i,2µi,1λi,1µ′ +i,2 −λi,2µ′ +i,1λ′ +i,1µi,2 +�2 −4λ′ +i,2µi,1λi,2µ′ +i,1λ′ +i,1µi,2λi,1µ′ +i,2 += (−λ′ +i,2µi,2λi,1µ′ +i,1 −λi,2µ′ +i,2λ′ +i,1µi,1)2 −4λ′ +i,2µi,1λi,2µ′ +i,1λ′ +i,1µi,2λi,1µ′ +i,2 += (λ′ +i,1λi,2µi,1µ′ +i,2 −λi,1λ′ +i,2µ′ +i,1µi,2)2 = ∆2 +i , +where we have set ∆i := λ′ +i,1λi,2µi,1µ′ +i,2 −λi,1λ′ +i,2µ′ +i,1µi,2. +Remark 3.12 The quadratic equation can be written: +(Aiρ−Eiτ+Fiτ′)(Aiσ+Ciτ−Diτ′)+(EiCiτ2 +DiFiτ′2 +(AiBi −CiFi −DiEi)ττ′) = 0, +i.e. XY + ZZ′ = 0, where X := Aiρ − Eiτ + Fiτ′, Y := Aiσ +Ciτ − Diτ′, Z := λi,1λ′ +i,2τ − λ′ +i,1λi,2τ′ +and Z′ := µi,1µ′ +i,2τ−µ′ +i,1µi,2τ′. We shall see next (see corollary 3.14 further below) that ∆i ̸= 0, so +that X,Y,Z,Z′ can be taken as coordinates. +Now we use again the explicit formulas. We write ∆i as t1 −t2 (“first and second term”) where: +t1 = λ′ +i,1λi,2µi,1µ′ +i,2 += T +�x3/x2,x4/x2,x3/x1,x4/x1 +x1/x2,x1/x2,x2/x1,x2/x1 +� +T +�[23.1i],[24.2i],[13.2i],[14.1i] +[12.1i],[12.1i],[12.2i],[12.2i] +� +, +t2 = λi,1λ′ +i,2µ′ +i,1µi,2 += T +�x3/x2,x4/x2,x3/x1,x4/x1 +x1/x2,x1/x2,x2/x1,x2/x1 +� +T +�[23.2i],[24.1i],[13.1i],[14.2i] +[12.1i],[12.1i],[12.2i],[12.2i] +� +. +Setting u := σ1/ρi and v := σ2/ρi, we see that t1 and t2 have a common factor: +common factor = T +�x3/x2,x4/x2,x3/x1,x4/x1 +x1/x2,x1/x2,x2/x1,x2/x1 +� +T +� +1 +x1x2u,x1x2u,x1x2v,x1x2v +� +, +so we compute: +∆i = (common factor) ×(T(x2x3u,x1x4u,x2x4v,x1x3v)−T(x2x4u,x1x3u,x2x3v,x1x4v)). +Again, instead of the general three terms relation, we favor a direct argument. The second factor +on the right hand side of ∆i is Ψ(x4), where: +Ψ(x) := T(x2x3u,x1x3v)T(x1ux,x2vx)−T(x1x3u,x2x3v)T(x2ux,x1vx). +This function is holomorphic on C∗, vanishes at x = x3 and satisfies σqΨ = cΨ, where c := +1 +x1x2uv· +From this, Ψ(x) = CT(x/x3)T(x1x2x3uvx) for some C ∈ C which can be determined through eval- +uation at x = 1/(x2u) (so that the second term in Ψ vanishes), thus yielding: +Ψ(x) = +Ψ(1/(x2u)) +T(1/(x2x3u))T(x1x3v)T(x/x3)T(x1x2x3uvx) += T(x2x3u,x1x3v)T(x1/x2,v/u) +T(1/(x2x3u))T(x1x3v) +T(x/x3)T(x1x2x3uvx) +=⇒ Ψ(x4) = T(x2x3u,x1x3v)T(x1/x2,v/u) +T(1/(x2x3u))T(x1x3v) +T(x4/x3)T(x1x2x3uvx4) += x2x3vT (x1/x2,σ1/σ2,x4/x3,ρi′/ρi). +38 + +For the last equality, we applied the usual transformation rules for T along with the obvious equal- +ities u/v = σ1/σ2 and x1x2x3uvx4 = x1x2x3x4σ1σ2/ρ2 +i = ρi′/ρi (by Fuchs relation). In the end, we +get one of the possible totally factored forms for (the square root of) the discriminant: +Proposition 3.13 +∆i = [23.2i]T +�x3/x2,x4/x2,x3/x1,x4/x1,x4/x3 +x1/x2,x2/x1,x2/x1 +� +T +� +σ1/σ2,ρi′/ρi +[12.1i],[12.1i],[12.2i],[12.2i] +� +. +Corollary 3.14 ∆i ̸= 0. +3.4.5 +The locus detM = 0 +We describe the image by [ρ] of the locus detM = 0 inside V (∗). So let M := +�m1,1 +m1,2 +m2,1 +m2,2 +� +such +that mi,j ̸= 0 and σqmi,j/mi,j = (ρi/σ j)x−2, i, j = 1,2; and m1,1m2,2 = m1,2m2,1. +Lemma 3.15 (i) There areCi,j ∈ C∗ and ai,bj ∈ C∗, i, j = 1,2, such that mi,j =Ci,jθq(x/ai)θq(x/bj). +(ii) One has aibj = ρi/σ j, i, j = 1,2, and C1,1C2,2 = C1,2C2,1. +Proof. - (ii) flows directly from (i) by the usual arguments, plus the fact that here: +detM = (C1,1C2,2 −C1,2C2,1)θq(x/a1)θq(x/a2)θq(x/b1)θq(x/b2). +(i) We first see that each mi,j has a common zero with his line neighbour mi,j′ and his column +neighbour mi′,j (recall the “complementarity conventions” 1′ := 2, 2′ := 1). Indeed, the relation +m1,1m2,2 = m1,2m2,1 entails (for instance) m1,2/m1,1 = m2,2/m2,1. The left hand side has poles +among the zeroes of m1,1, the product of which must be ≡ ρ1/σ1, so they cannot be the same as +the poles of the right hand side, so there must be simplifications, i.e. m1,2,m1,1 have a common +zero (no more for exactly the same reason) and likewise m2,2,m2,1 have a common zero. +Then we see that there cannot be a common zero to all mi,j; indeed, if a was such a zero, the +same argument applied to +1 +θq(−x/a)M would lead (among the same lines) to a contradiction. The +conclusion easily follows. +□ +Proposition 3.16 The image by [ρ] of the locus detM = 0 inside V (∗) can be parameterized (up +to the C∗ action on +� +P1(C) +�4, by C∗ ∋ t �→ ( f1(t), f2(t), f3(t), f4(t)), where fk(t) := θq(σ2xkt) +θq(σ1xkt), +k = 1,2,3,4. +Proof. - In the direct sense, it is easy to check that any such ( f1(t), f2(t), f3(t), f4(t)) can indeed +be realized as some [ρ](M). Conversely, first note that M as described in the lemma is equivalent, +modulo the D2(C) × D2(C)-action, to the same with all Ci,j = 1. So we take it in that form. +Then m1,2/m1,1 = m2,2/m2,1 = θq(x/b2)/θq(x/b1) has to be a solution of σq f/f = σ1/σ2, so +b2/b1 = σ1/σ2, so we can define: +t := +1 +σ1b1 += +1 +σ2b2 +=⇒ m1,2/m1,1 = m2,2/m2,1 = θq(σ2xt) +θq(σ1xt), +39 + +hence the corresponding ( f1(t), f2(t), f3(t), f4(t)) is the image of M. +□ +Theorem 3.17 The image by [ρ] of the locus detM = 0 inside V (∗) lays inside all the quadrics of +a two-parameters pencil. +Proof. - First note that each fk(x) := θq(σ2xkx)/θq(σ1xkx) satisfies σq fk = (σ1/σ2) fk, so each +gk,l := fk flθq(σ1x1x)θq(σ1x2x)θq(σ1x3x)θq(σ1x4x) satisfies σqgk,l = cx−4gk,l with: +c := (σ1/σ2)2 +1 +σ4 +1x1x2x3x4 += +1 +ρ1ρ2σ1σ2 +, +since, by (FR), x1x2x3x4 = (ρ1ρ2)/(σ1σ2). For 1 ≤ k < l ≤ 4, the denominator of fk fl is chased +by the theta product and gk,l ∈ O(C∗). Therefore we get six such functions gk,l in V4,c, which has +dimension 4. Thus, chasing denominators, there is (at least) a two-dimensional space of linear +relations of the form: +Aρσ+Bττ′ +Cρτ−Dρτ′ −Eστ+Fστ′ = 0. +Among them, of course, are those found in 3.3.2 (which amount to the same by 3.4.3). +□ +4 +Algebraic threefold and algebraic surface associated to a quadratic +form +In all this part A = (A,B,C,D,E) ∈ (C∗)4, We return to the coordinates ρi and set: +ΦA := Aρ1ρ2 +Bρ3ρ4 +Cρ1ρ3 −Dρ1ρ4 −Eρ2ρ3 +Fρ2ρ4. +on C4 = U0. We will say that A is regular if the discriminant of the quadratic form ΦA is not +identically 0 and singular on the contrary. +The notations Up and Up come from 1.2.5. +4.1 +Generalities +Lemma 4.1 The rank of the quadratic form ΦA is 4 or 3. More precisely: +(i) A is regular if and only if the rank of ΦA is 4, then, up to a linear change of coordinates, the +equation of CA is XY = ZT and the only singular point of CA is (0,0,0,0); the quadric QA is +smooth. +(ii) A is singular if and only if the rank of ΦA is 3 and, up to a linear change of coordinates, +the equation of CA is XY = Z2. The singular points are the λ(a,b,c,d)|λ ∈ C for some +(a,b,c,d) ∈ (C∗)4; the quadric QA has a unique singular point. +Proof. - +40 + +(i) See remark 3.12 and corollary 3.14. +(ii) We suppose that A is singular, then, up to a linear change of coordinates, the equation of S[A] +is XY = Z2. More precisely Z = αρ3 +α′ρ4, with +Z2 = ECρ2 +3 +DFρ2 +4 +(AB−CF −DE)ρ3ρ4. +As EC ̸= 0 and DF ̸= 0, we have αα′ ̸= 0. We verify easily that X,Y,Z are independent. +□ +We consider the homogeneous form of ΦA: +ΦAhom = Aρx +1ρx +2ρy +3ρy +4 +Bρy +1ρy +2ρx +3ρx +4 +Cρx +1ρy +2ρx +3ρy +4 −Dρx +1ρy +2ρy +3ρx +4 −Eρy +1ρx +2ρx +3ρy +4 +Fρy +1ρx +2ρy +3ρx +4 +on +� +(C2)∗�4. The quadratic form ΦA defines an affine quadratic cone CA in C4 = U0, invariant +under the action of C∗ defined by ρ �→ λρ. The geometric quotient C∗ +A/C∗ is a quadric QA in +U0 ≈ P3(C). Similarly, the quadratic form: +�ΦA := A˜ρ1 ˜ρ2 +B˜ρ3˜ρ4 +C˜ρ1˜ρ3 −D˜ρ1˜ρ4 −E ˜ρ2˜ρ3 +F ˜ρ2 ˜ρ4 +defines an affine quadratic cone ˜CA in C4 = U∞. The geometric quotient ˜C∗ +A/C∗ is a quadric ˜QA in +U∞ ≈ P3(C). +The homogeneous form ΦAhom defines a closed hypersurface SA in +� +P1(C)4� +invariant by C∗. +This hypersurface contains the quadratic cones CA and ˜CA, more precisely it is the Zariski closure +of each cone. +The image of SA \Θ4 in the quotient K4 is denoted SA. It is an algebraic surface. It contains +the quadrics QA and ˜QA . More precisely, it is the Zariski closure of each quadric. The quadrics +are compact and not equal, therefore SA is not separated. +For i = 1,2,3,4, we denote S0 +A,i = SA ∩{σi = 0}, S∞ +A,i = SA ∩{σi = ∞}, and, for i ̸= j, S0,∞ +A,i,j = +S0 +A,i ∩S∞ +A,j. +We have: +SA = CA ∪ +� +i=1,2,3,4 +S∞ +A,i = ˜CA ∪ +� +i=1,2,3,4 +S0 +A,i = CA ∪ ˜CA ∪ +� +i,j=1,2,3,4;i̸= j +S0,∞ +A,i,j, +SA = CA ∪ +� +i=1,2,3,4 +S ∞ +A,i = ˜CA ∪ +� +i=1,2,3,4 +S 0 +A,i = QA ∪ ˜QA ∪ +� +i,j=1,2,3,4;i̸= j +S 0,∞ +A,i,j. +We will see later that the S 0,∞ +A,i,j are points which do not belong to QA ∪ ˜QA. +We will now describe the S0 +A,i, S∞ +A,i, S 0 +A,i, S∞ +A,i as union of “simple” pieces. We will see in +particular that the S 0 +A,i (resp. S ∞ +A,i) are the Zariski closures in SA of some projective lines. (They are +non separated unions of smooth rational curves.) +We will describe the set of the points of SA admitting at least a coordinate equal to ∞. We +consider all the possible cases. +41 + +1. Exactly one coordinate equal to ∞. We can suppose that it is ρ4. We consider: +Ψ1 := Aρ1ρ2 ˜ρ4 +Bρ3 +Cρ1ρ3 ˜ρ4 −Dρ1 −Eρ2ρ3 ˜ρ4 +Fρ2. +As ˜ρ4 = 0, we have Ψ1 = Bρ3−Dρ1+Fρ2 = 0. This equation defines a plane of +� +P1(C) +�3× +{∞}. The set of the points whose the only ∞ coordinate is ρ4 is a plane of C3 ×{∞} +2. Exactly two coordinates equal to ∞. We will prove that this cannot happen. We can suppose +that it is ρ3 and ρ4. We consider: +Ψ2 := Aρ1ρ2 ˜ρ3 ˜ρ4 +B+Cρ1˜ρ4 −Dρ1˜ρ3 −Eρ2ρ3 ˜ρ4 +Fρ2 ˜ρ3. +As ˜ρ4 = ˜ρ3 = 0, we have Ψ2 = B = 0. This is impossible. +3. Exactly three coordinates equal to ∞. We can suppose that it is ρ2, ρ3 and ρ4. We consider: +Ψ3 = Aρ1˜ρ3 ˜ρ4 +B˜ρ2 +Cρ1 ˜ρ2 ˜ρ4 −Dρ1˜ρ2 ˜ρ3 −E ˜ρ4 +F ˜ρ3. +As ˜ρ4 = ˜ρ3 = ˜ρ3 = 0, Ψ3 = 0 for all ρ1 ∈ C. +4. Four coordinates equal to ∞: this is the point (∞,∞,∞,∞) ∈ Θ4 +We remark that the 4 lines L0 +i,j,k = {σi = σ j = σk = 0}, i < j < k, and the 4 lines L∞ +i,j,k = {σi = +σ j = σk = ∞}, i < j < k, are contained in SA for all the values of A. We denote p0 +i,j,k (resp. p∞ +i,j,k the +images of the L0 +i,j,k \Θ4 (resp. L∞ +i,j,k \Θ4) in the quotient SA. They are points. We have p0 +i,j,k ∈ QA +and p∞ +i,j,k ∈ ˜QA. +4.2 +Smoothness properties +We will describe the singular points of SA and SA respectively in the cases A non-singular and +singular. We recall that we supposed A ∈ (C∗)4. +Theorem 4.2 We suppose that A is non-singular. Then: +(i) the only singularities of SA are the points (0,0,0,0) and (∞,∞,∞,∞). They are normal of +type A1; +(ii) SA is a (non separated) smooth algebraic surface. +Proof. - +(i) The proof is a variant, in an abstract setting, of a proof in [16, 5.2]. We have SA ∩U0 = CA +and we know that the only singularity of CA is {(0,0,0,0)}. (The situation is similar for the +singularities in U∞, the only singularity is {(∞,∞,∞,∞)}.) +We will search singular points with one (at least) coordinate equal to ∞. We return to the +coordinates ρi and we check each case (as in [16, 5.2, p. 37]). +42 + +1. Exactly one coordinate equal to ∞. We can suppose that it is ρ4. We consider: +Ψ1 := Aρ1ρ2 ˜ρ4 +Bρ3 +Cρ1ρ3 ˜ρ4 −Dρ1 −Eρ2ρ3 ˜ρ4 +Fρ2. +We compute the gradient ∇Ψ1 on {˜ρ4 = 0}: +∇Ψ1{˜ρ4=0} = (−D,F,B,Aρ1ρ2 +Cρ1ρ3 −Eρ2ρ3). +As D,F,B ̸= 0, ∇Ψ1 ̸= 0 on {˜ρ4 = 0}, there are no singularity on {˜ρ4 = 0} (i.e. ρ4 = +∞). +2. Exactly two coordinates equal to ∞. We know that this does not happen (cf. page 42, +at the end of 4.1, item 2 of the enumeration). +3. Exactly three coordinates equal to ∞. We can suppose that it is ρ2, ρ3 and ρ4. We +consider: +Ψ3 = Aρ1˜ρ3 ˜ρ4 +B˜ρ2 +Cρ1 ˜ρ2 ˜ρ4 −Dρ1˜ρ2 ˜ρ3 −E ˜ρ4 +F ˜ρ3. +We compute the gradient ∇Ψ3 on {˜ρ2 = ˜ρ3 = ˜ρ4 = 0}: +∇Ψ3{˜ρ2=˜ρ3=˜ρ4=0} = (0,B,F,−E). +As B,F,E ̸= 0, ∇Ψ3 ̸= 0 on {˜ρ2 = ˜ρ3 = ˜ρ4 = 0}. There are no singularity on {˜ρ2 = +˜ρ3 = ˜ρ4 = 0}. In particular (0,∞,∞,∞) is not singular. +4. Four coordinate equal to ∞. The point (∞,∞,∞,∞) is singular (it is clear using the +coordinates ˜ρi). +(ii) We can consider the smooth algebraic threefold SA \Θ4 as an analytic manifold. The group +C∗ operates on it without fixed point, therefore the corresponding action of its its Lie group +C defines a foliation and we can put on the quotient (SA \Θ4)/C∗ (interpreted as a set) the +analytic structure of the space of leaves of the foliation. We get an analytic manifold but a +priori it could be non Haussdorf14, and in fact it is the case. This is very briefly sketched in +[16] (cf. 3.2, page 38), without reference to the problem of separation15. On the other side, +there is a structure of analytic space on SA deduced from the algebraic quotient structure by +GAGA [38]. +We denote this space by S an +A . If we could prove that S an +A +and the analytic manifold of +leaves coincide, then S an +A would be smooth and it would follow that SA is smooth, using +the proposition 4.4 recalled below. Unfortunately we cannot prove a priori the equality of +the two analytic structures. Therefore we will use another approach and prove that the (non +separated) algebraic surface SA is smooth directly ”by hand”. We will also give an abstract +proof which will allow us to prove a posteriori the equality of the two analytic structures. +The surface SA is the union of the two quadrics QA and ˜QA and of a finite set of points. +The two quadrics are smooth, therefore it remains only to look at the points. These points +14In general spaces of leaves are non Haussdorf and therefore quite pathological. +15In the usual results on the quotient of a manifold by a free G-action, the group G is supposed compact, but C∗ is +not compact. +43 + +correspond to the images of the points of SA admitting one (and only one) 0 coordinate and +one (and only one) ∞ coordinate. Up to a coordinate permutation, all the cases are the same, +therefore it is sufficient to consider only one case. Such cases already appeared above (at +the end of 3.3.2). +We reformulate in ρi coordinates: +Near ρ4 = ∞, i.e. near ˜ρ4 = 0, the equation +Aρ1ρ2 +Bρ3ρ4 +Cρ1ρ3 −Dρ1ρ4 −Eρ2ρ3 +Fρ2ρ4 = 0, +becomes: +Aρ1ρ2 ˜ρ4 +Bρ3 +Cρ1ρ3 ˜ρ4 −Dρ1 −Eρ2ρ3 ˜ρ4 +Fρ2 = 0, +which, near ˜ρ4 = 0, is a graph: +ρ1 = Bρ3 +Fρ2 −Eρ2ρ3 ˜ρ4 +D−(Aρ2 +Ciρ3)˜ρ4 +, +whence the surface SA is non singular at points such that ρ4 = ∞, and in particular at the +points defined by ρ4 = ∞ and ρ2 = 0. If (ρ1,0,ρ3,∞) ∈ SA \Θ4, then Bρ3 −Dρ1 = 0, ρ3 ̸= 0 +and we can write: +ρ1/ρ3 = B+Fρ2/ρ3 −Eρ2˜ρ4 +D−(Aρ2/ρ3 +C)ρ3 ˜ρ4 +, +Let p ∈ K4 be the image of S0,∞ +A,2,4 \Θ4, then, in a neighborhood of p in K4, we can choose as +coordinates (ξ1 = ρ1/ρ3,ξ2 = ρ2/ρ3, ˜ξ4 = ρ3 ˜ρ4) and, near the point p = (ξ1 = B/D,ξ2 = +0, ˜ξ4 = 0), the surface SA is a graph: +ξ1 = B+Fξ2 −Eξ2 +D−(Aξ2 +C)˜ξ4 +, +therefore it is non singular at p. +□ +We will give now an “abstract” proof of the smoothness of SA, using the Luna’s slice theorem +(cf. [22]) It will give more information and allow us to compare the structure of analytic manifold +associated by GAGA to the algebraic structure of SA and the structure of analytic manifold used +in [16] (cf. proof of theorem 2.18). +Proposition 4.3 Let G be a reductive group acting on an affine variety X. Let X′ ⊂ X be the subset +of stable points. We suppose that G operates freely on X′ (i. e. without fixed point). Then we +have a geometric quotient π : X′ → Y ′ = X′/G and X′ is a G-principal bundle over Y ′. Moreover, +if x ∈ X′ is a smooth point, then π(x) is a smooth point of Y ′. +Proof. - Cf. [10, Proposition 5.7]. +□ +44 + +We can apply the above proposition to the Xp = SA ∩Up (cf. 1.2.5). We have X′ +p = SA ∩(Up \ +{p}). The Y ′ +p are smooth and cover SA. Therefore SA is smooth. +We can consider the analytic manifolds (SA \Θ4)an and S an +A associated by GAGA to SA \Θ4 +and SA. Then SA \ Θ4)an → S an +A is a locally trivial analytic principal G-bundle and the analytic +manifold structure on SA defined by the space of leaves (as in [16]) is the same than S an +A . +We recall the following result. +Proposition 4.4 Let X be an algebraic variety. Let Xan be the associated analytic space. Then X +is smooth if and only if Xan is smooth +Proof. - It follows from [38, no 6, prop. 3, p 9], [38, no 24, prop. 27, p 39] and [5, chapter +VIII, §5, no 2, cor. of prop. 1]. +□ +Proposition 4.5 We suppose that SA is singular. Then: +(i) The set of singular points of SA \Θ4 is the orbit: C∗(a1,a2,a3,a4), for some (a1,a2,a3,a4) ∈ +(C∗)4. +(ii) The surface SA is a non-separated normal surface with a unique singular point. This singular +point is of type A1. +Proof. - +(i) We already know the singularities in CA = SA ∩U0: {(0,0,0,0)} ∪ C∗(a,b,c,d)16. The +points of SA which do not belong to CA ∪ ˜CA ∪Θ4 are smooth: the proof is the same than in +the regular case. +(ii) The quadric QA is a normal surface with a unique singular point. This singular point is of +type A1. It is also the unique singular point of ˜QA. The points of SA which do not belong to +QA ∪ ˜QA are smooth: the proof is the same than in the regular case. +□ +4.3 +Pencils of quadrics +We recall basics on pencils of quadrics in P3(C). For details and complements, see [11]. +We consider two (non colinear) quadratic forms Φ1 and Φ2 on C4 and the pencil of quadrics of +P3(C) defined by: {Φλ := λ1Φ1 +λ2Φ2|(λ1,λ2) ∈ (C2)∗}. We set Ci = {Φi = 0} and we denote +Qi its image in P3(C). +The base of the pencil is the curve B := Q1 ∩ Q2. If Q is a quadric vanishing on B, then Q +belongs to the pencil. +By definition the rank of a pencil is maxi=1,2 rankΦi (it is independant of the generators). If +the rank of a quadric of the pencil is the rank of the pencil, we will say that this quadric is generic. +There are three possibilities. +16The situation is similar for the singularities in U∞. +45 + +1. The rank of the pencil is 4. Then all the quadrics of the pencils, except a finite number +(corresponding to a vanishing discriminant: discrΦλ = 0), are smooth. +2. The rank of the pencil is 3. Then all the quadrics of the pencil, except a finite number, have +a unique singular point . +3. The rank of the pencil is ≤ 2. +In the following, all the pencils of quadrics are in the first case. +5 +Projective embeddings of F +In all this part, we will fix σ2, µ1, µ2, x1, x2, x3, x4 and put σ1 = ωσ2, with ω ∈ C∗ arbitrary17. We +will suppose (FR) and (NS). We will suppose (NR), except for (σ1,σ2): we will allow ω ∈ qZ. +For us embedding is allways in the sense of algebraic geometry. A morphism f : X → Y is +an embedding if it is injective and if induces an isomorphism between X and its image f(X), the +algebraic structure on f(X) being the structure induced by Y. Be careful: it is different from the +notion of embedding used in [16]18. +5.1 +Embedding of F into K4 = +�� +P1(C) +�4 \Θ4 +� +/C∗ +We recall the two quadratic forms (i = 1,2, cf. 3.3.2): +Aiρστ′′ +Biτ+Ciρττ′′ −Diρ−Eiσττ′′ +Fiσ = 0, +In the following, we will consider only the case i = 1 and set A = (A,B,C,D,E,F) = (A1,B1,C1,D1,E1,F1). +Then A depends on ω = σ1/σ2 and, if necessary, we will denote A(ω). We fix a value of +(σ1,σ2) such that σ1 = σ2 and we denote A′ the corresponding value of A. We denote A′ = A(1). +We return to the coordinates ρi and set: +ΦA := Aρ1ρ2+Bρ3ρ4+Cρ1ρ3−Dρ1ρ4−Eρ2ρ3+Fρ2ρ4 = Aρσ+Bττ′+Cρτ−Dρτ′−Eστ+Fστ′ +It is easy to compare with the definitions in [16]. To T = (Ti,j) of [16], we associate AT := +(T12,T34,T13,−T14,−T23,T24). +Lemma 5.1 We have QAT = QA. +Proof. - Let R ⊂ SA be the image of F in K4. We have R∩U0 ⊂ QA and, using [16], R∩U0 ⊂ +QAT . Moreover, R∩U0 is a Zariski open subset of QA and, therefore dim(R∩U0) = 2. +We consider R′ := QA ∩QAT . The quadric QA is irreducible, therefore we have two possibili- +ties: +1. R′ = QA, +17In [16], the parameter is κ0: ω = σ1/σ2 = κ2 +0. +18In [16], X is a set, Y an algebraic variety, and f an injective map whose image is a locally closed surface of Y. +46 + +2. dimR′ ≤ 1. +We have R∩U0 ⊂ R′ and therefore dimR′ ≥ 2. Therefore the only possibility is 1 and QAT = QA. □ +Remark 5.2 Using a similar argument, we get another proof of QA1 = QA2. +We denote X the Zariski closure in +� +P1(C) +�4 of [ρ](W) (the image by [ρ] of {detM = 0}. We +have: +X \Θ4 = X′ ∪ +� +i,j,k;1≤i< j 5 × 108 g cm−3, significant +amounts of stable iron group elements (IGEs) such as 58Ni +are produced (Seitenzahl & Townsley 2017). These central +densities correspond to WD masses of ≳ 1.2 𝑀⊙, where the +thermonuclear runaway must start via compressional heating +in the center of the WD (Seitenzahl & Townsley 2017). +Traditionally there have been fewer studies of SNe Ia in +the longer near-infrared (NIR) and mid-infrared (MIR) wave- +lengths compared to the optical. +However, recent efforts +have shown that these longer wavelengths offer additional, +and sometimes better, information about the physics of SN +explosions (Meikle et al. 1993; Höflich et al. 2002; Marion +et al. 2009; Hsiao et al. 2013; Graham et al. 2017; Diamond +et al. 2018; Wilk et al. 2018; Hoeflich et al. 2021; Lu et al. +2022; Kumar et al. 2022; Hoeflich et al. 2023). This is due, +in part, to the fact that the location of the photosphere is +wavelength-dependent, and that different diagnostic spectral +lines are revealed at longer wavelengths (Hoeflich et al. 1991, +1995; Wheeler et al. 1998; Kasen 2006; Ashall et al. 2019a,b). +Prior to the launch of the James Webb Space Telescope +(JWST), there were only seven MIR (𝜆 > 5 𝜇m) spectral +observations of SNe Ia across four different objects. Three +spectrawereobtainedwiththeSpitzer SpaceTelescope(SST); +one of SN 2003hv at ∼ +375 days (relative to estimated explo- +sion), one of SN 2005df at ∼ +135 days (Gerardy et al. 2007), +and one of SN 2006ce at +120 days (GO-30292, PI: W.P. +Meikle; Kwok et al. 2022). Four MIR spectra of SN 2014J +were obtained with CanariCam on the 10.4-m Gran Telesco- + +SN 2021aefx: High-density Burning in SNe Ia +3 +pio Canarias (GTC) between 57 − 137 days after explosion +(Telesco et al. 2015). +Despite the small sample size it is +apparent that the MIR contains many diagnostics to differ- +entiate between leading explosion scenarios. For example, +nebular phase MIR spectral observations, which probe the +high-density central layers, can reveal the presence and dis- +tribution of stable Ni. These lines are direct indicators of +high-density burning. +With the successful launch of JWST, high-S/N MIR spec- +tral observations during the nebular phase of SNe Ia are now +possible. The first spectrum of a SN Ia obtained with JWST +was that of SN 2021aefx at +255 days after maximum light +(MJD=59801.4; Kwok et al. 2022). +Here we present and +analyze a spectrum of SN 2021aefx taken +323 days after +maximum light (MJD=59871.6). In contrast to the work of +Kwok et al. (2022), who focused primarily on line identifica- +tions and determination of observed velocities, we interpret +the explosion physics of SN 2021aefx through comparisons +to a self-consistent set of non-local thermodynamic equilib- +rium (NLTE) radiation hydrodynamic models of SNe Ia. This +allows us to provide a set of line IDs specific to SN 2021aefx in +addition to a consistent picture of the explosion based on our +newly observed spectrum and models. In § 2 we describe our +observations, and in § 3 the details of our spectral reduction. +Line identifications from full NLTE models are performed +in § 4, while an analysis of their velocities is presented in +§ 5. § 6 discusses the details of our chosen NLTE models +and a comparison to the observations. Alternative explosion +scenarios are discussed in § 6.4. Finally, we summarize our +findings in § 7. +2. OBSERVATIONS +SN +2021aefx +was +discovered +on +2021 +Nov +11.3 +(MJD=59529.5) by the Distance Less Than 40 Mpc Survey +(DLT40; Tartaglia et al. 2018) and classified as a young +SN Ia (Bostroem et al. 2021; Hosseinzadeh et al. 2022). +SN 2021aefx was subsequently followed by several groups, +including a multi-band optical and spectroscopic follow-up +campaign by the Precision Observations of Infant Supernova +Explosions Collaboration (POISE, Burns et al. 2021; Ashall +et al. 2022). POISE’s detailed photometric observations re- +vealed an early blue excess, which may be explained by a +rapid change in the velocities of spectral lines (Ashall et al. +2022). An analysis of the complete POISE data set reveals +the basic light curve properties of SN 2021aefx, including +a decline rate of Δm15(B) = 1.01 ± 0.06 mag, and a peak +absolute magnitude of 𝑀𝐵 = −19.28 ± 0.49 mag, which +places SN 2021aefx in the normal part of the luminosity- +width relation (Phillips 1993; Ashall et al. 2022; C. Stevens +et al., in prep.). SN 2021aefx is located 105′′.3 south, 37′′.0 +west from the center of its host NGC 1566 at a redshift of +0.005 (𝛼 = 04ℎ20𝑚00𝑠.42, 𝛿 = −54◦56′16′′.10; Allison et al. +Table 1. JWST/MIRI Observation Details +Parameter +Value +Acquisition Image +Filter +F1000W +Exp Time [s] +89 +Readout Pattern +FASTGRPAVG8 +SN 2021aefx Spectrum +Mode +LRS +Exp Time [s] +1493 +𝑇obs [MJD] +59871.6 +Epocha [days] +322.71 +Groups per Integration +134 +Integrations per Exp. +2 +Exposures per Dither +1 +Total Dithers +2 +Note— a Rest frame days relative to 𝐵-band +maximum of MJD = 59547.25 (C. Stevens et +al., in prep.). +2014). NGC 1566 is a face-on spiral galaxy with systemic +recessional velocity of 1500 km s−1, and a rotational velocity +of 65 ± 60 km s−1 at the location of the SN (Elagali et al. +2019). All figures showing observed spectra of SN 2021aefx +have been corrected for the combined recessional and ratio- +nal velocities of 1550 km s−1 at the location of the SN in the +host. This low rotational velocity implies that any observed +off-center lines (i.e. lines shifted relative to the line-of-sight +velocity) are intrinsic to the progenitor system itself, and not +attributable to a peculiar velocity within the host galaxy. +We present a MIR observation of SN 2021aefx ob- +tained through program GO-JWST-2114 (P.I. Ashall) from +∼ 4 − 14 𝜇m. The data were obtained using JWST’s Mid- +Infrared Instrument (MIRI) in its Low Resolution Spec- +troscopy (LRS) configuration. In this mode, MIRI/LRS ob- +tains slit spectroscopy of objects with a spectral resolving +power (𝑅 = 𝜆/Δ𝜆) of 𝑅 ∼ 100 at 7.5 𝜇m, varying from +𝑅 ∼ 40 at 5 𝜇m to 𝑅 ∼ 160 at 10 𝜇m (Kendrew et al. 2015, +2016; Rigby et al. 2022). The instrumental configuration is +identical to that of Kwok et al. (2022). The spectral observa- +tions were performed with a 2-point dither strategy. For each +grating setting there were 134 groups per integration, 2 inte- +grations per exposure and 1 exposure per dither. This results +in an exposure of 734.5 seconds at each dither position, which +are combined for a total exposure time of 1493 seconds. Full +details of our observational set-up are found in Table 1. +3. DATA REDUCTION + +4 +DerKacy, Ashall, Hoeflich, Baron et al. +4 +6 +8 +10 +12 +14 +Rest Wavelength [µm] +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +Flux [mJy] +[Co II] +[Ni II] +[Ar II] +[Ni I] +[Ni III] +[Ni IV] +[Ni IV] +[Ar III] +[Co II] +[Ni II] +[Co III] +Figure 1. JWST/MIRI LRS spectrum of SN 2021aefx at +323 days relative to 𝐵-band maximum. The ions responsible for the most prominent +features in the spectrum are labeled. A full set of line identifications is plotted in Figure 3 and shown in Table 2. +The data were obtained on 2022 Oct 19.6 (MJD=59871.6) +and reduced with the JWST calibration pipeline1, version +1.8.1 (Bushouse et al. 2022). +Both raw (Stage 1 cal- +ibrated) and fully reduced data were retrieved from the +Mikulski Archive for Space Telescopes (MAST)2. The raw +data was processed with a local installation of the ver- +sion 1.8.1 pipeline for comparison to the fully reduced +data from MAST, using the spec_mode_stage_2 and +spec_mode_stage_3 Jupyter notebooks as templates for the +reduction. Both reductions used the most up-to-date wave- +length (jwst_miri_specwcs_0005.fits) and flux calibration +(jwst_miri_photom_0085.fits) files. These calibration files +produce a wavelength solution accurate to ∼ 0.05 − 0.02 𝜇m, +varying from short to long wavelengths, and a flux calibra- +tion accurate to a ∼ 2 – 5% global offset between 5 – 12 𝜇m. +(Gordon et al. 2022, S. Kendrew, private communication). +Furthermore, Kwok et al. (2022) found that the flux calibra- +tion of their MIRI spectrum was accurate to 2%. +1 https://jwst-pipeline.readthedocs.io/en/stable/jwst/introduction.html +2 https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html +Using the LRS Optimal Spectral Extraction notebook3, the +spectra were re-extracted using multiple techniques. This re- +extraction was necessary to properly center the position of the +spectrum in the science aperture, as the pipeline-derived aper- +ture produced a poor extraction at long wavelengths. After +proper re-extraction with the Optimal Extraction notebook, +no significant differences were found between the locally re- +duced data and the fully calibrated (but un-extracted) data +available from MAST. Future updates to the JWST calibra- +tion files are expected to further improve the accuracy of the +automated extractions (S. Kendrew, private communication). +4. DATA COMPARISON & LINE IDENTIFICATIONS +Figure 1 presents the spectrum of SN 2021aefx acquired +on 2022 Oct 19.6 (corresponding to +323 days after 𝐵-band +maximum light) from 4 − 14 𝜇m. At these phases, the ejecta +are optically thin and dominated by emission lines. +The +strongest of these lines are labeled in Figure 1. +Figure 2 +shows our spectrum of SN 2021aefx compared to the MIR +spectra of SNe 2005df (Gerardy et al. 2007), 2006ce (Kwok +et al. 2022), 2014J (Telesco et al. 2015), and the earlier spec- +3 https://spacetelescope.github.io/jdat_notebooks/notebooks/MIRI_LRS_ +spectral_extraction/miri_lrs_spectral_extraction.html + +SN 2021aefx: High-density Burning in SNe Ia +5 +6 +8 +10 +12 +14 +Rest Wavelength [µm] +0 +1 +2 +3 +4 +5 +Normalized Flux + Constant +SN 2014J (+57) +SN 2014J (+81) +SN 2014J (+108) +SN 2014J (+137) +SN 2006ce (+120) +SN 2005df (+135) +SN 2021aefx (+255) +SN 2021aefx (+323) +Figure 2. Comparison of +323 day spectrum of SN 2021aefx to +other MIR spectral observations of SNe Ia, including SNe 2005df +(Gerardy et al. 2007), 2006ce (Kwok et al. 2022), 2014J (Telesco +et al. 2015), and the +255 days spectrum of 2021aefx (Kwok et al. +2022). +The primary difference between the +255 and +323 day +spectra of SN 2021aefx is the increased strength of other features +relative to the peak at ∼ 11.9 𝜇m. +trum of SN 2021aefx at +255 days (Kwok et al. 2022). From +Figure 2 it is clear that SN 2021aefx is similar to other previ- +ously observed SNe Ia, but the size and sensitivity of JWST +produces a high S/N spectrum with a quality that was previ- +ously impossible to obtain. Comparing the two JWST spectra +of SN 2021aefx, the most noticeable difference is the decrease +in the relative strength of the ∼ 11.9 𝜇m profile compared to +the other features caused by the radioactive decay of 56Co. +To assist in line identifications, we use a suite of full NLTE +radiation transport models. These models reproduce both the +early and late time properties of SN 2021aefx, and an in-depth +discussion of the models with respect to the MIR observables +can be found in § 6. +A detailed examination of the SN 2021aefx MIR spectrum +reveals four prominent wavelength regions of line formation, +which are described individually in the following subsections. +Detailed line identifications in each of these regions are plot- +ted in Figure 3, while Table 2 lists the lines that contribute +significantly to the spectrum. +Table 2. Mid-Infrared Line Identifications from Model 25 +S +𝜆 [𝜇m ] +Ion +S +𝜆 [𝜇m ] +Ion +∗ ∗ +6.214 +[Co II] +8.555 +[Fe III] +∗ +6.273 +[Co I] +∗ ∗ +8.611 +[Fe III] +∗ +6.274 +[Co II] +∗ +8.644 +[Co II] +∗ ∗ +6.383 +[Ar III] +∗ ∗ +8.733 +[Fe II] +∗ ∗ ∗ +6.636 +[Ni II] +∗ ∗ +8.945 +[Ni IV] +6.636 +[Ni II] +∗ ∗ ∗ +8.991 +[Ar III] +∗ ∗ +6.920 +[Ni II] +∗ +9.618 +[Ni II] +∗ ∗ ∗ +6.985 +[Ar II] +∗ +10.080 +[Ni II] +6.985 +[Ar II] +∗ ∗ ∗ +10.189 +[Fe II] +7.045 +[Co I] +∗ ∗ +10.203 +[Fe III] +6.985 +[Ar II] +∗ ∗ ∗ +10.523 +[Co II] +7.103 +[Co III] +∗ ∗ ∗ +10.682 +[Ni II] +7.147 +[Fe III] +∗ +11.002 +[Ni III] +7.272 +[Fe III] +∗ ∗ +11.130 +[Ni IV] +∗ ∗ +7.349 +[Ni III] +∗ ∗ +11.167 +[Co II] +∗ ∗ ∗ +7.507 +[Ni I] +∗ ∗ +11.307 +[Ni I] +7.773 +[Co I] +∗ ∗ ∗ ∗ +11.888 +[Co III] +∗ ∗ ∗ +7.791 +[Fe III] +∗ +11.978 +[Fe III] +∗ +8.044 +[Co II] +∗ ∗ +12.001 +[Ni I] +∗ +8.063 +[Ni II] +∗ ∗ +12.255 +[Co I] +8.114 +[Co I] +∗ +12.261 +[Mn II] +∗ ∗ +8.211 +[Fe III] +∗ ∗ ∗ +12.642 +[Fe II] +8.282 +[Ni I] +∗ ∗ +12.681 +[Co III] +∗ ∗ ∗ +8.405 +[Ni IV] +∗ ∗ ∗ +12.729 +[Ni II] +∗ ∗ +8.489 +[Co III] +13.058 +[Co I] +Note—For each transition, the markers correspond to domi- +nant (∗ ∗ ∗ ∗), strong (∗ ∗ ∗), moderate (∗ ∗), weak (∗), +and scarcely detectable ( ) on top of the quasi-continuum +formed by a large number of lines. The relative strength S +is estimated by the integral over the envelope, +∫ +𝐴𝑖 𝑗𝑛 𝑗 𝑑𝑉 +where 𝑛 𝑗 is the particle density of the upper level. The list +is based on the simulations described in § 6. +4.1. The 6.0 − 8.0 𝜇m Region +The 6.0 − 8.0 𝜇m region is dominated by emission lines +of stable Ni, the most prominent of which is a blend of +[Ni III] 7.349 𝜇m and [Ni I] 7.507 𝜇m that defines the +red edge of the feature. +The blue edge of this peak is +blended with several other weaker lines, creating a series of +shoulders, extending from ∼ 6.5 𝜇m to ∼ 7.2 𝜇m. Mov- +ing from red to blue, these shoulders are comprised of +[Ar II] 6.985 𝜇m, [Ni II] 6.920 𝜇m, and [Ni II] 6.636 𝜇m. +Finally, there is a small bump associated with a combination +of [Co II] 6.214 𝜇m, [Co I] 6.273 𝜇m, and [Co II] 6.274 𝜇m. +4.2. The 8.0 − 9.5 𝜇m Region + +6 +DerKacy, Ashall, Hoeflich, Baron et al. +6.0 +6.5 +7.0 +7.5 +8.0 +Rest Wavelength [µm] +0.0 +0.2 +0.4 +0.6 +Normalized Flux +[Co II] +[Co I] +[Co II] +[Ni II] +[Ni II] +[Ar II] +[Ni III] +[Ni I] +8.0 +8.5 +9.0 +9.5 +Rest Wavelength [µm] +0.0 +0.1 +0.2 +0.3 +0.4 +Normalized Flux +[Ni IV] +[Fe II] +[Ni IV] +[Ar III] +10.0 +10.5 +11.0 +11.5 +Rest Wavelength [µm] +0.0 +0.1 +0.2 +0.3 +0.4 +Normalized Flux +[Fe II]+[Fe III] +[Co II] +[Ni II] +[Ni III] +[Co II] +[Ni IV] +[Ni I] +11.5 +12.0 +12.5 +13.0 +Rest Wavelength [µm] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Flux +[Co III] +[Fe III] [Ni I] +[Co I] +[Fe II] +[Co III] [Ni II] +Figure 3. Detailed line identifications in the four prominent feature regions based on the lines from Model 25 (Hoeflich 2017) included in +Table 2. The color intensity of the vertical lines corresponds to the strength of the spectral lines, with 4-star lines the most intense and 1-star +lines being the faintest. Dashed lines correspond to ground state ions, solid lines singly-ionized species, dotted lines doubly-ionized species, +and dash-dotted lines triply-ionized species. + +SN 2021aefx: High-density Burning in SNe Ia +7 +The 8.0−9.5 𝜇m region is dominated by two features whose +edges are blended with other weaker lines at ∼ 8.7 𝜇m. The +bluer of the two is due to the emission of [Ni IV] 8.405 𝜇m, +while the redder feature is dominated by the [Ar III] 8.991 𝜇m +line. The [Ar III] line shows a distinct flat-topped profile, +which increases in flux moving from blue to red. Tilted flat- +topped profiles are connected to both an ion’s velocity distri- +bution in the ejecta and the viewing angle of the explosion +(see § 5.1 and § 6.3.1, also Hoeflich et al. 2021). Small contri- +butions from the weak [Fe II] 8.733 𝜇m and [Ni IV] 8.945 𝜇m +lines may also add to the observed flux at the 10% level. +4.3. The 9.5 − 11.5 𝜇m Region +The 9.5 − 11.5 𝜇m region shows a structure reminiscent +of the 6.0 − 8.0 𝜇m region; with one dominant blended fea- +ture and a series of smaller bumps and shoulders blended +into the wings. +The strongest peak arises from a blend +of [Co II] 10.523 𝜇m and [Ni II] 10.682 𝜇m. +A blend +of [Fe II] 10.189 𝜇m and [Fe III] 10.203 𝜇m forms a +shoulder that is partially blended into the blue wing of the +[Co II]+[Ni II] blend. Blended into the red wing is a series +of three other weaker features. +The first feature, centered +near ∼ 10.85 𝜇m is not associated with any strong lines in +our model. +The next feature in the series arises from the +comparatively weak [Ni III] 11.002 𝜇m line, while a blend of +[Ni IV] 11.130 𝜇m and [Co II] 11.167 𝜇m forms a shoulder +on the red wing of the [Ni III] line. Finally, there may be +a small contribution to the red wing of the [Ni IV]+[Co II] +shoulder from [Ni I] 11.307 𝜇m. +4.4. The 11.5 − 13.0 𝜇m Region +The 11.5 − 13.0 𝜇m region contains the only relatively +isolated, un-blended feature in the MIR spectrum, the +[Co III] 11.888 𝜇m resonance line which produces the +strongest line in the entire MIR spectrum. +Our model +shows weak contributions from [Fe III] 11.978 𝜇m and +[Ni I] 12.001 𝜇m, however they only produce ∼1% of the +flux and do not alter the line profile in a significant man- +ner (again see Table 2). +A small shoulder at the edge +of the red wing of the [Co +III] line is attributable to +[Co I] 12.255 𝜇m. A series of peaks between 12.5−13.0 𝜇m +suggests the presence of multiple weak lines, however the +low S/N in this region prevents us from unambiguously iden- +tifying any lines. We tentatively identify the first peak with +[Fe II] 12.642 𝜇m and [Co III] 12.681 𝜇m, and the second +peak with [Ni II] 12.729 𝜇m. Our model shows no strong +lines in the vicinity of the third and final peak in the series. +5. VELOCITY DISTRIBUTIONS AND LINE PROFILES +In this section, we discuss the velocity distributions and +line profiles of three important species in the ejecta: Ar, Co, +and Ni. In discussing these velocities and profiles we reiter- +ate that the current wavelength calibration of the MIRI/LRS +observations is accurate to 0.05 − 0.02 𝜇m, with lower errors +at longer wavelengths. This corresponds to an error on the +order of ∼ 500 km s−1 in the [Co III] 11.888 𝜇m line, and +∼ 1400 km s−1 in the [Ni III] 7.349 𝜇m line. Future updates to +the JWST pipeline calibration files may increase the precision +of these results. +5.1. [Ar III] 8.991 𝜇m +Ar traces the transition region between incomplete oxy- +gen burning and nuclear statistical equilibrium (NSE) in +the ejecta; thereby providing details about the chemical +distribution between the 56Ni and Si-group layers. +The +[Ar III] 8.991 𝜇m line profile is plotted in Figure 4 in ve- +locity space. The profile is flat-topped with an increasing tilt +from blue to red wavelengths, which we refer as a “flat-tilted” +profile hereafter. Flat-topped profiles are indicative of a cen- +tral hole or void in the emitting material — that is, a shell +of line emitting material (Beals 1929; Menzel 1929; Struve +1931). For [Ar III] the flat-top component of the feature starts +at ∼ −7000 km s−1 and extends to ∼ 8000 km s−1. The feature +increases in flux by 10% across the profile from the blue to red +side, and the flat-topped component of the profile indicates +that there is a central hole in the ejecta of ∼ ±8000 km s−1 +which does not contain Ar. This is because Ar is destroyed in +high temperature regimes of the NSE where 𝑇 ≥ 6 × 109 K, +and there is a lack of strong mixing during the explosion, +consistent with explosion models of near 𝑀Ch WDs (see § 6 +for details). +5.2. [Co III] 11.888 𝜇m +SNe Ia are powered by the nuclear decay chain of 56Ni +to 56Co to 56Fe. Since the [Co III] 11.888 𝜇m feature is a +resonance line, most of the de-excitation and recombination +of Co passes through this transition, making it a direct tracer +of the distribution and amount of 56Ni in the ejecta. This +feature covers a width of ∼ ±10000 km s−1. If the shape of +the line is assumed to be symmetric, and thus well described +by a Gaussian profile, it peaks at 740 ± 200 km s−1 with a +FWHM of 4840 ± 170 km s−1 (see Figure 5). Combining the +error in the line-of-sight velocity (recessional plus rotational; +∼ 60 km s−1) and the estimated error in the wavelength cali- +bration of ∼ 500 km s−1 with that of the fit error yields a total +estimated error of 544 km s−1. The fact that this resonance +line is not located at the kinematic center of the explosion +indicates that the bulk of the 56Ni is off-center, at the 1.4𝜎 +level. The [Co III] 11.888 𝜇m profile also shows hints of a +flat-tilted profile, peaking to the red at ∼ 2000 km s−1 (see +Figure 4), although the low resolution prevents a definitive +identification of this profile. Similarly, hints of this flat-tilted +peak are also seen in the spectrum at the earlier epoch of +SN 2021aefx (see Kwok et al. 2022). If real, this flat-tilted +profile may extend from ∼ −1000 km s−1 to ∼ 2000 km s−1. + +8 +DerKacy, Ashall, Hoeflich, Baron et al. +−10 +0 +10 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +Normalized Flux +[Ar III] +8.991 µm +−10 +0 +10 +Velocity [103 km s−1] +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Flux +[Co III] +11.888 µm +Figure 4. +Line profiles of the [Ar III] (top) and [Co III] lines +(bottom) in velocity space. +The blue boxed region around 𝑣 = +0 km s−1 in the rest frame denote the 1𝜎 error in the rest wavelength +for the given line. Red vertical lines mark the left and right edges of +the flat-tilted profiles in both panels. +Similar to the [Ar III] 8.991 𝜇m line, the flat-tilted profile +of the [Co III] feature may imply a central hole of 56Ni in +the ejecta. This hole is smaller than that of Ar, and would +only be a few thousand km s−1 across (Telesco et al. 2015; +Diamond et al. 2015). Note that unlike Ar, which is pro- +duced by nuclear burning that has a steep temperature depen- +dence leading to a sharp cutoff in velocity extent and thus +flat line profiles, electron capture is nearly temperature in- +dependent, so its effects follow the density profile leading to +somewhat rounder line profiles. The increase in flux across +the [Co III] 11.888 𝜇m profile is 10%, the same as that in +the [Ar III] 8.991 𝜇m feature, implying the distribution of the +two elements are linked. Since Ar is produced at the edge +−10 +−5 +0 +5 +10 +Velocity [103 km s−1] +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Flux +[Co III] 11.888 µm +Fit +Figure 5. The [Co III] 11.866 𝜇m line compared to a Gaussian fit. +The Gaussian peaks at 11.91±0.01 𝜇m, 𝜎 = 0.19±0.04 𝜇m, which +in velocity space corresponds to a peak at 740 ± 200 km s−1 and +𝜎 = 4840 ± 170 km s−1. +of the 56Ni region (see § 6), it is reasonable that Ar and Co +have similar changes in flux across their profiles. +In § 6, +we discuss the [Co III] 11.888 𝜇m line profile in the context +of off-center 56Ni distributions in the explosion. However, +in order to confirm that the [Co III] 11.888 𝜇m feature is +truly asymmetrical and off-center, higher resolution spectra +are required (such as those obtainable by the Medium Res- +olution Spectrograph (MRS) of JWST/MIRI) and improved +wavelength calibrations are also needed. +5.3. Stable Ni +Multiple ionization states of Ni have forbidden emission +lines which occur in the MIR, making nebular phase MIR +spectra an invaluable resource for probing the explosion +physics and corresponding nucleosynthesis of SNe Ia. Since +the 56Ni that powers the early light curves of SNe Ia has +a half life of 6.1 days, any emission from Ni at these late +phases comes from isotopes of stable Ni (e.g. 58Ni) and not +from radioactive isotopes like 56Ni. Figure 6 presents three +of these regions in velocity space. The left panel shows the +[Ni III] 7.349 𝜇m line, which appears to be red-shifted in ve- +locity space with an apparent maximum around 3000 km s−1. +However, this feature is blended with [Ni I] 7.506 𝜇m, such +that the velocity extent of [Ni III] 7.349 𝜇m appears to be +larger than its true distribution. The middle panel shows the +[Ni III] 8.405 𝜇m line profile in velocity space, while the +right panel depicts the [Ni III] 11.002 𝜇m feature within a +much larger series of blended lines. +At higher resolution + +SN 2021aefx: High-density Burning in SNe Ia +9 +−10 +−5 +0 +5 +10 +Velocity [103 km s−1] +0.3 +0.4 +0.5 +0.6 +0.7 +Normalized Flux +[Ni I] 7.506 µm +[Ni III] 7.349 µm +−10 +−5 +0 +5 +10 +Velocity [103 km s−1] +0.0 +0.1 +0.2 +0.3 +Normalized Flux +[Ni IV] 8.405 µm +−10 +0 +10 +Velocity [103 km s−1] +0.0 +0.1 +0.2 +0.3 +0.4 +Normalized Flux +[Ni II] 10.682 µm +[Ni IV] 11.130 µm +[Ni I] 11.307 µm +[Ni III] 11.002 µm +Figure 6. Velocity space profiles of the three spectral regions with prominent Ni lines. Vertical lines indicate the ionization and line strength +as in Figure 3. The left panel shows the [Ni III] 7.349 𝜇m region is contaminated by the [Ni I] 7.506 𝜇m feature. The right panel, centered on +the [Ni III] 11.002 𝜇m line, shows evidence for multiple stable Ni lines contributing to the series of weak features and shoulders. +these blends, including the 11.002 𝜇m line, are likely to be +resolved. +6. NUMERICAL MODELING AND IMPLICATIONS +FOR EXPLOSION SCENARIOS +To explore the explosion physics of SN 2021aefx we turn +to detailed comparisons with NLTE radiation hydrodynam- +ical models. +The goals of these comparisons are: (1) to +demonstrate that MIR spectral features and line profiles can +be used as a critical tool to determine the explosion physics +and progenitor scenario of SNe Ia, (2) to show that JWST has +opened up a new frontier in MIR SN science and that there +is a need to test and calculate atomic models and processes, +including cross sections to improve future models. Specifi- +cally, we address how the data allows us to measure the mass +of the exploding WD, the chemical asymmetries in the initia- +tion of the explosion, and small-scale mixing processes in the +ejecta. When taken in total, these measurements allow us to +determine the most likely explosion scenario of SN 2021aefx. +As discussed in § 5.3 and shown by Kwok et al. (2022), +SN 2021aefx presents many spectral lines of stable Ni (Fig- +ure 3). This Ni requires high-density burning in the ejecta, +above 5 × 108 g cm−3, which must originate from a massive +WD, making the explosion either a near-𝑀Ch WD where the +explosion is triggered by compressional heating in the center +of the explosion, or the detonation of a high-mass, sub-𝑀Ch +larger than 1.15–1.2 𝑀⊙ (Höflich et al. 1998; Hoeflich & +Khokhlov 1996; Seitenzahl & Townsley 2017, but see also +Blondin et al. 2022). Such massive WDs can only be pro- +duced via accretion (Kippenhahn et al. 2013). Therefore, we +limit our comparisons to models within this region of param- +eter space. +6.1. Numerics +The simulations employ modules of the HYDrodynami- +cal RAdiation (HYDRA) code. +HYDRA solves the time- +dependent radiation transport equation (RTE) and positron +transport (Penney & Hoeflich 2014), including the rate equa- +tions that calculate the nuclear reactions based on a network +with 211 isotopes and statistical equations for the atomic level +populations, the equation of state, the matter opacities, and the +hydrodynamic evolution as applied to SN 2020qxp (Hoeflich +et al. 2021; Hristov et al. 2021, and references therein). De- +tailed atomic models and line lists are based on the database +for bound-bound transitions of van Hoof (2018)4, supple- +mented by additional forbidden lines from Diamond et al. +(2015) and Telesco et al. (2015). For details on modeling +of nebular phase spectra with HYDRA, see Hoeflich et al. +(2021), and for more general discussions on modeling the +nebular phase and downward cascading of high-energy parti- +cles and photons by Monte Carlo, see also Spencer & Fano +(1954), Axelrod (1980), Kozma & Fransson (1992), Fransson +(1994), Fransson & Jerkstrand (2015), Botyánszki & Kasen +(2017), Wilk et al. (2018), Shingles et al. (2020), and Wilk +et al. (2020). The models include transitions for ionization +stages I-IV of C, O, Ne, Mg, Si, S, Cl, Ar, Ca, Sc, Ti, V, Cr, +Mn, Fe, Co, and Ni. Though most of the prominent features in +the MIR are caused by forbidden lines, the underlying quasi- +continuum is formed by allowed lines in the inner layers well +above the critical density. At these phases, the iron-rich layers +are still partially optically thick at UV wavelengths, meaning +the inclusion of permitted lines is important to fully charac- +terize the ionization balance via Rosseland cycles (Mihalas +1978). +6.2. A Delayed-Detonation Model for SN 2021aefx +4 Version v3.00b3, https://www.pa.uky.edu/~peter/newpage/ + +10 +DerKacy, Ashall, Hoeflich, Baron et al. +Table 3. Model 25 Parameters +Parameter +Value +Mej +∼1.38 𝑀⊙ +𝜌𝑐 +1.1 × 109 g cm−3 +Mtr +0.24 𝑀⊙ +MDDT +0.5 𝑀⊙ +B(WD) +106 G +We compare SN 2021aefx to new simulations of off-center +𝑀Ch mass explosion models, based upon the spherical model +of the Model 25-series from Hoeflich (2017), as it produces +early light-curve properties and a maximum light luminosity +very similar to those of SN 2021aefx. +These new simu- +lations are parameterized explosion models, using a spher- +ical delayed-detonation to constrain the global parameters +of the explosion. +Fine-tuning these models is not nec- +essary to achieve the goals of this study as we focus on +spectra rather than high-precision photometry. The model +produces Δ𝑚15(𝑉) = 0.68 mag (for reference, Δ𝑚15(𝑉) = +0.64 ± 0.01 mag for SN 2021aefx, which is within the error +of the model), and ∼0.6 𝑀⊙ of 56Ni. +The model originates from a C/O WD with a main-sequence +progenitor mass of 5 𝑀⊙, solar metallicity, and a central den- +sity 𝜌𝑐 = 1.1 × 109 g cm−3. We adopt this 𝜌𝑐 due to the line +width and shape of the [Co III] 11.888 𝜇m line and due to +the strength of the stable Ni lines in the MIR spectrum (see +§ 5.3). In this model, burning starts as a deflagration front +near the center and transitions to a detonation (Khokhlov +1991b). The deflagration–detonation transition is triggered +when the density at the burning front drops below 2.5×107 g +cm−3, when ∼ 0.24 𝑀⊙ of the material has been burned by +the deflagration front, and is induced by the mixing of un- +burned fuel and hot ashes (Khokhlov 1991b). The model has +a magnetic field of B(WD) = 106 G, which has been found +in magneto-hydrodynamical simulations, suggesting that tur- +bulent magnetic fields are produced during the deflagration +phase (Diamond et al. 2018; Hristov et al. 2021). The basic +model parameters are given in Table 3. +6.2.1. Off-center 56Ni and Abundance Distributions +To investigate the line profiles and asymmetries, we con- +sider the Model 25-series which includes off-center DDTs. +For the construction of the off-center DDT we follow the +description of Livne (1999) that has also been employed by +Höflich et al. (2006), Fesen et al. (2015), Hoeflich et al. +(2021), and Hoeflich et al. (2023). The DDT is triggered at +𝑀DDT = 0.5 𝑀⊙. Note that due to the buoyancy of flame +fronts in the explosion, the DDT can be triggered at a dif- +ferent mass coordinate relative to the total integrated mass of +the deflagration burning. This leads to asymmetric abundance +distributions of all elements produced during the detonation +phase (see Fig. 2 in Hoeflich et al. 2021). +In principle, the use of multiple resolved line profiles allows +us to determine the value of 𝑀DDT as well as the viewing +angle. In the case of SN 2021aefx we use the two strongest +features: [Co III] 11.888 𝜇m and [Ar III] 8.991 𝜇m. +As +shown in Figure 4 we see a consistent tilt in the [Ar III] and +[Co III] lines. We can determine the viewing angle from the +tilt of these features. The value of 𝑀DDT determined here was +also consistent with the spectrophotmetric observations of the +normal SN Ia 2019np (Hoeflich et al. 2023). Most normal +SNe Ia have very similar polarization properties (Cikota et al. +2019). +6.2.2. Overall Abundance Distribution +The angle-averaged abundance structure and the 56Ni dis- +tribution of Model 25 are shown in Figure 7. In the model, +the region of high electron capture is spherical because we as- +sume central ignition, no fragmentation during the 56Ni decay +over the first week after the explosion (e.g. Fesen et al. 2015), +and that Rayleigh-Taylor instabilities are largely suppressed +by high magnetic fields (Hristov et al. 2018). The most no- +table results in the abundance distribution are: (1) ∼ 6 × 10−2 +of 58Ni is produced in the center of the ejecta, (2) the velocity +extent of the central hole in 56Ni is ∼ 3200 km s−1, (3) the +velocity extent of the 56Ni region produced in NSE ranges +from ∼ 3200 – 10000 km s−1, and (4) the size of the shell of +the Ar region covering a range of ∼ 8000 – 15000 km s−1. We +note that simulating the point of the DDT in multi-dimensions +does not lead to a strong rarefaction wave (Gamezo et al. 2005; +Fesen et al. 2015) as seen in all spherical delayed-detonation +models (Khokhlov 1991b; Höflich et al. 2002; Hoeflich et al. +2021). +The off-center DDT at a point in an already-expanding +medium results in a run-time effect which yields an asym- +metric distribution of burning products. The material closer +to the DDT burns under higher density than the opposite side +because the front reaches the corresponding layer 0.5 − 1 s +later. The result is a bulge of all elements that undergo only +Si and O burning including Ca, Ar, and 56Ni (see Figure 7). +For a more complete depiction of this, see Fig. 7 of Fesen +et al. (2007). These asymmetries are aligned along the axis +defined by the center and the DDT ignition point. +6.3. Spectral Modeling +6.3.1. Determining the Inclination Angle +We begin our discussion of Model 25’s fit to the obser- +vations by illustrating its ability to determine the inclination +angle of the explosion relative to our line of sight. Remember +that to first order, the [Co III] profile can be fit with a Gaus- +sian of a half-width ≈ 4800 km s−1, emission wings ranging + +SN 2021aefx: High-density Burning in SNe Ia +11 +Figure 7. (Left:) The chemical composition of our best fit model, Model 25 from Hoeflich et al. (2017) and Hoeflich et al. (2023). The model +has a chemically stratified ejecta. EC elements (e.g. 58Ni with M(58Ni) ≈ 5.9 × 10−2 𝑀⊙) are located in the center of the ejecta, followed +by 56Ni further out in velocity space. The Ar distribution goes between 8000 – 15000 km s−1 and the lightest elements (e.g. O and C) are +located in the outermost layers. For illustration, the thin red line at expansion velocities larger than 3200 km s−1 shows the EC distribution after +microscopic mixing applied (see text). (Right:) The distribution of the IGEs of the off-center delayed detonation Model 25 at a point (black dot). +The bulk of the 56Ni is in a ring-like structure between 3000 – 9500 km s−1 as well as a bulge produced at the point of the delayed detonation +transition. Depending upon the viewing angle, differently shaped line profiles will be produced in the [Co III] feature. These profiles are shown +in Figure 8. +from −10000 – +10000 km s−1, and an offset from the rest +wavelength of +740 km s−1 (see § 5 and Figure 5). This is +consistent with the overall 56Ni distribution seen in the model +(see Figure 7), but we note that assuming an emission feature +is a Gaussian makes an implicit assumption about the under- +lying chemical distribution of an element within the ejecta, +and should be used with caution. As previously discussed, +the host galaxy is seen face on and has a very small projected +rotation (65 ± 60 km s−1), implying that host rotation plays a +minor role in this offset. This leaves the peculiar motion of +the progenitor system and the orbital velocity of the progen- +itor as remaining potential sources of this offset. However, +if these were the dominant factors, one would expect a con- +sistent velocity offset in all of the spectral lines, contrary to +observations. +On +the +other +hand, +the +observed +flux +in +the +[Co III] 11.888 𝜇m line center changes by ∼ 10% across the +peak of the feature (see § 5), consistent with expectations of +flux arising from the asymmetric ejecta of an off-center DDT +model when viewed from a specific angle (Hoeflich et al. +2021). As previously shown in § 5, the [Co III] 11.888 𝜇m +feature appears to show a flat-tilted profile, where the velocity +extent of the central tilted region corresponds to the region +in velocity space of partial burning in quasi-statistical equi- +librium (QSE) (≈ 1800 km s−1 across in the angle averaged +spectrum). This flat-tilted profile is seen in both the +255 and ++323 day JWST spectra. In Model 25, the inner size of the +electron capture region and the distribution of 56Ni produce +different line profiles when viewed at different angles. Three +specific viewing angles, −90◦, −30◦ and +30◦ are shown in +Figure 8. From the bottom left panel, we see that the ob- +servations are well matched by a viewing angle of ≈ −30◦, +including replicating the ∼ 10% change in flux across the peak +seen in the observations. While the high signal-to-noise (S/N +≈ 100) of both JWST spectra of SN 2021aefx suggests that +the flat-tilted profile is real and significant, future planned +observations with JWST/MIRI MRS (JWST-GO-2114, PI: +Ashall) will better resolve the line profiles. +6.3.2. Overall MIR Spectra +Having determined the inclination angle, we now compare +our full model spectrum to our observations, as seen in Fig- +ure 9. Model spectra are shown with and without mixing of +electron capture elements on the scale of the pressure scale +height of the WD (Höflich & Stein 2002). We examine mi- +croscopic mixing (i.e. smaller than the mean free path of +the positrons) in the center of the explosion to constrain the +position (e.g. central, off-center, or multi-spot) of the ther- +monuclear runaway ignition (Niemeyer et al. 1996; Höflich +& Stein 2002; Calder et al. 2004; Livne et al. 2005; Röpke +et al. 2007; Ma et al. 2013). +The most dominant lines produced by the models are shown +in Table 2, and have been successfully identified in Figure 3. +For NIR lines outside the observed range, see Appendix A. +Identification of weaker lines within the spectrum will be +possible after the acquisition of MIRI/MRS data. +The model reproduces the observations overall, including +all four regions of prominent spectral lines and does especially +well in reproducing the blends of the 6.0 − 8.0 𝜇m and 8.0 − +9.5 𝜇m regions in addition to the [Co III] 11.888 𝜇m line. + +36 +C +40Ca +0 +44Ti +0.8 +Si +56Ni +S +Ne +EC +Mg +0.6 +0.4 +X +0.2 +10 +15 +20 +25 +5 +[1000 km/sec]1 +X, ++30° +.5 +-30° +0. +.06-12 +DerKacy, Ashall, Hoeflich, Baron et al. +−10 +−5 +0 +5 +10 +Velocity [103 km s−1] +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Flux ++255 d ++323 d +−10 +−5 +0 +5 +10 +Velocity [103 km s−1] +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Flux ++323 d ++30◦ +−30◦ +−90◦ +−10 +−5 +0 +5 +10 +Velocity [103 km s−1] +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Flux ++323 d +−30◦ +−10 +−5 +0 +5 +10 +Velocity [103 km s−1] +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Flux ++255 d ++323 d ++30◦ +−30◦ +−90◦ +Figure 8. Top Left: Comparison of the [Co III] 11.888 𝜇m line profile at +255 (dashed grey) and +323 days (solid black). Top Right: +Dependence of the [Co III] 11.888 𝜇m line profile as a function of inclination in comparison with the +323 day spectrum (solid black). Note +that the profile and the red-shift of the observed peak are consistent with the off-center DDT model seen at −30◦ (bottom left) and the −30◦ +model is also the best fit to both observations (bottom right). + +SN 2021aefx: High-density Burning in SNe Ia +13 +6 +7 +8 +9 +10 +11 +12 +13 +14 +Rest Wavelength [µm] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized Flux +SN 2021aefx (+255 d) +SN 2021aefx (+323 d) +Model 25 w/ EC mixing +Model 25 w/ no EC mixing +Figure 9. Comparison of the synthetic MIR spectrum of the off-center Model 25 seen from −30◦ without (blue) and with (red) mixing of the +EC elements (see Figure 6) and the JWST/MIRI LRS spectrum of SN 2021aefx at +255 (dashed grey) and +323 (solid black) days relative to +𝐵-band maximum. The angle-averaged spectra would look similar, but they would show a flat-topped rather than a flat-tilted [Ar III] 8.991 𝜇m +feature. Though [Ni II] lines are present in both the synthetic spectra, the sensitivity to microscopic mixing should be noted. In particular, the +[Ni II] 6.6 𝜇m line shows a strong variation with mixing (see text). +While the exact contribution of each ion may vary with the +underlying explosion model, the synthetic spectra have been +obtained without further tuning and in general are in good +agreement with the observations. The similarity between the +mixed and unmixed model shows the stability of the synthetic +spectral features. +Most of the Ni features are in blends with other iron-group +elements of similar strengths. +In light of uncertainties in +the atomic models and cross sections, the photons at a given +wavelength may couple to elements other than Ni (through +fluorescence; Morrison & Sartori 1966). Thus, many of the +line IDs of weaker features in low resolution spectra are model +dependent. The easiest way to separate the elements is by +comparing the mixed and unmixed models. In the unmixed +models, the electron capture elements are effectively shielded +from non-thermal excitations from radioactive decay, thus the +electron capture features will be weaker. Features dominated +by Ni show variations in the region between 6.4 – 8.5 𝜇m and +weak variation at longer wavelengths, for example at 10.5 𝜇m. +One major effect of mixing can be seen in the 6.5 – 7.4 𝜇m +region. In the unmixed models, EC and radioactive elements +are mostly separate, leading to weak spectral features. This +due to the locally trapped positrons dominating the excita- +tion in the EC region, with 𝛾-rays responsible for ∼ 10% of +the excited ions (Penney & Hoeflich 2014). In turn, micro- +scopic mixing produces narrow features, for example, [Ni II] +at 6.636 𝜇m has a half-width of ≈ 4000 km s−1 determined +by the numerical resolution. In contrast, the observed broad +features suggest a central ignition of the WD. +The broad feature at ∼ 9.0 𝜇m dominated by Ar is a strong +diagnostic of the point of the transition from the deflagra- +tion to the detonation. Ar is located in a shell with a large +central hole because it is destroyed in moderately high den- +sity burning environments (𝜌 ≳ 1 − 2 × 107 g cm−3). Our +model (Figure 8) is consistent with the estimates for the +minimum Ar velocity, strongly suggesting a high mass WD. +The [Ar III] 8.991 𝜇m feature shows the same slope as the +[Co III] 11.888 𝜇m, providing evidence of an off-center DDT. +Line profiles are strongly affected by the ionization balance. +Typically for normal SNe Ia at this phase, iron group elements +are dominated by doubly ionized ions, and the ionization frac- +tion decreases with increasing density because the recombi- + +14 +DerKacy, Ashall, Hoeflich, Baron et al. +nation rate scales with the square of the density. Only in the +center do we see effects at the 10% level of singly ionized +iron group elements that produce a strong resonance [Ni I] +feature at ≈ 3 𝜇m (Kwok et al. 2022; Fisher 2022). In the +line forming regions, the ionization balance hardly changes. +Therefore, our results are insensitive to the differences in the +ionization structure. More detailed discussions are given in +Hoeflich et al. (2021), Wilk et al. (2020), and § 6.1. +In the synthetic spectrum, the feature at ∼ 11.9 𝜇m +agrees well with the observations, it is dominated by +[Co +III] 11.888 +𝜇m and has minor line blends of +[Ni I] 12.001 𝜇m and [Co I] 12.255 𝜇m in the red wing +at the 1% level relative to the [Co III] peak (see the line +strengths given in Table 2). +However, our models tend to show features from singly ion- +ized elements that are too strong by about 10 − 20% as can be +seen from the ratio of the [Co III] 11.888 𝜇m and the [Co II] +10.523 𝜇m plus [Ni II] 10.682 𝜇m blend. The discrepancies +in the ionization balance are not unexpected, due to uncer- +tainties in the atomic data and the ∼ 2 − 5% accuracy of the +flux calibration of the observed spectrum (see § 3). Uncertain +ionization and excitation by non-thermal leptons and uncer- +tainties in the recombination rates lead to ionization balance +uncertainties. Though we treat the cascades in energy within +our Monte-Carlo scheme, missing and uncertain atomic lev- +els are likely responsible for some of the discrepancies (Wilk +et al. 2018; Shingles et al. 2020, and see Appendix A of +Hoeflich et al. 2021). +Finally, we discuss other observables obtainable at a higher +spectral resolution that may support our interpretation. Off- +center delayed-detonation models predict an offset between +electron capture elements (e.g., 58Ni) produced during the +deflagration phase and elements (e.g., 56Ni, Fe, Co, Si, S, Ar) +synthesized during the detonation. The former are created +in a subsonic deflagration resulting in a slow pre-expansion +phase of the WD with an almost spherical density structure, +whereas the latter are formed in a weak detonation with a +burning speed close to the speed of sound. This offset may +be seen with higher-resolution spectra. +Note that the unresolved small flux variations near 11 𝜇m +and in the wavelength range 12 − 14 𝜇m seen in MIR spectra +are at a 1𝜎 level which, if confirmed by MRS spectra, have im- +portant implications. The computational results indicate that +these small variations signal the presence of a caustic struc- +ture in density and abundance of the inner electron-capture +core as has been observed by Fesen et al. (2007) and Fesen +et al. (2015). At this epoch, positrons remain local for the +high 𝐵-field required (Penney & Hoeflich 2014; Mera Evans +et al. 2022; Hristov et al. 2021). +6.4. Alternative Explosion Scenarios +A detailed discussion of explosion scenarios and progenitor +systems of SNe Ia is beyond the scope of this work. +For +reviews, we refer to Alsabti & Murdin (2017) and Hoeflich +et al. (2021). The total mass of the exploding WD is one of +the parameters separating different explosion scenarios such +as He-triggered detonations of sub-𝑀Ch, dynamical mergers, +violent mergers, collisions of two WDs in a triple system, and +𝑀Ch explosions. Dynamical mergers go through a loosely +bound hydrostatic WD state and are unlikely to synthesize +EC elements because the peak density during merging is too +low (Benz et al. 1990; García-Berro et al. 2017). Collisions of +two WDs may result in high density burning, high masses, and +may produce EC elements; but also produce large polarization +signatures and a 90◦ flip in the polarization angle (Höflich +1995; Bulla et al. 2016), both of which are not observed in any +SNe Ia (Cikota et al. 2019). Similarly, violent mergers would +be expected to yield high continuum polarization (∼ 1%) at +∼ 10 days before the 𝐵−band maximum light (Bulla et al. +2016), which is also not seen in observations (Yang et al. +2020; Patra et al. 2022). +Therefore, we focus on sub-𝑀Ch He-triggered detonations +of C/O WDs as an alternative viable candidate to produce +SN 2021aefx. For normal SNe Ia sub-𝑀Ch models have WD +masses of 1 – 1.05 𝑀⊙ and for bright SNe Ia they have WD +masses up to 1.1 𝑀⊙ (Shen et al. 2018; Blondin et al. 2022). +Stable Ni has been tentatively suggested to have been seen in +observations of several SN Ia based upon: (1) heavily blended +optical features (e.g. Mazzali et al. 2020), (2) the 1.94 𝜇m +[Ni II] feature (Dhawan et al. 2018; Hoeflich et al. 2021), (3) +and in low S/N MIR spectra (Gerardy et al. 2007; Telesco et al. +2015). However, the JWST spectra of SN 2021afex are the +first to firmly establish the presence of stable Ni in a SNe Ia. +Our simulations of SN 2021aefx suggest that ∼ 0.06 𝑀⊙ of +stable Ni was produced in the explosion. In fact, since the +synthetic Ni lines may be slightly too weak (Figure 9), we +may need slightly more stable Ni than our simulations imply. +Stable Ni is produced in NSE by shifting the ratio 𝑌𝑒 from +≈ 0.5 to a lower value either by electron-capture under high +density burning or in a WD with super-solar metallicities, as +a result of an initial high 22Ne abundance (see, for example, +Brachwitz et al. 2000; Timmes et al. 2003; Thielemann et al. +2018). +Gronow et al. (2021) simulate a variety of He-detonation +explosions at various metallicities, with various He-shell +masses. They find that super-solar metallicities produce EC +elements due to the decrease𝑌𝑒 from the presence of neutron- +rich 22Ne. Gronow et al. (2021) find that WDs with masses of +∼ 1.1 𝑀⊙ and primordial metallicity 3𝑍⊙ produce 0.046 𝑀⊙ +of stable Ni, an the amount of 58Ni sufficient to produce the +strong Ni features observed in SN 2021aefx. However, since +this result is due to the primordial metallicity of the WD, +which reduced the 𝑌𝑒 uniformly throughout the WD — the + +SN 2021aefx: High-density Burning in SNe Ia +15 +full half-width of the [Co III] line and of the Ni features +should be comparable because both 56Ni and 58Ni are formed +in the same NSE region and constant 𝑌𝑒 results in a constant +isotopic ratio. Future MIRI/MRS observations will be able to +resolve the Ni lines and accurately measure its elemental dis- +tribution. In particular, in these sub-𝑀Ch models the [Ni IV] +should be broad as the high ionization stage will occur in +the low-density, high-velocity region of the envelope because +the recombination rates scale with the density squared (Oster- +brock & Ferland 2006). Moreover, the model that produces +the large 58Ni abundance requires of a He shell of 0.02 𝑀⊙. +Finally, if large stable Ni abundances are ubiquitous to all +SNe Ia, within the sub-𝑀Ch paradigm it would require all of +them to have super-solar metallicity, which is unlikely. +Based on recent 3D simulations for solar metallicities, Boos +et al. (2021) showed that a thin He-triggered detonation in a +1.1 𝑀⊙ C/O WD may produce 0.02 𝑀⊙ of stable Ni, an +amount that is insufficient to explain the observed NIR fea- +tures from stable Ni (see Wilk et al. 2020).5 +For both sets of He-det simulations discussed above, the +mass of the outer He layers are inconsistent with recent limits +from other early-time normal SNe Ia spectra that show carbon +in the outer 2 − 5 × 10−3 𝑀⊙ (Yang et al. 2020; Hoeflich et al. +2023). Furthermore, thin He-detonation models have nearly +spherical 56Ni distributions (Fesen et al. 2007; Hoeflich et al. +2023), which contradict the observation of SN 2021aefx. +Lacking advanced He-triggered detonation models, we fo- +cus on spherical explosion models with sub-𝑀Ch cores such +as DET2 (Hoeflich & Khokhlov 1996). This model has a pure +C/O WD without a He surface layer. It originates from a WD +whose mass is 1.2 𝑀⊙, and produces a sufficient amount of EC +material. Strict limits on the WD mass and the density of the +central region can be obtained from both the [Ar III] 8.991 𝜇m +and [Co III] 11.888 𝜇m features. In particular, a tight mass +limit on the WD can be obtained via the width of the flat-top +component of the [Ar III] 8.991 𝜇m feature. The observed +edge of the Ar profiles implies a central hole of Ar between +∼ 0 − 8000 km s−1 (see Figure 4). However, DET2 produces +an inner hole of Ar of 6000 km s−1. This is 2000 km s−1 +lower than observed. To produce a wider flat-topped Ar fea- +ture requires a higher mass WD. In a high mass model (such +as a near 𝑀Ch explosion), the Ar hole extends further out in +velocity space. +We note that, as an upper limit, detonating a WD with +an ejecta mass of 1.38 𝑀⊙ would mostly produce 56Ni and +5 Blondin et al. (2022) claim that sub-𝑀Ch models with 𝑀 > 1 𝑀⊙ can +produce the NIR [Ni II] lines at late times. +They argue that the lack +of [Ni II] in the NIR at earlier times can be simply an ionization effect, +with the mixing of radioactive products into the EC region dramatically +reducing the strength of [Ni II], thus, providing an option for the absence of +the observed NIR [Ni II] feature. This explanation is, however, inconsistent +with the fact that in SN 2021aefx many ionization stages of Ni are observed. +few QSE elements, resulting in a SN that produced too much +56Ni, is too bright at maximum light, and has spectra that +are inconsistent with a typical SN Ia (Hoeflich et al. 1996; +Marquardt et al. 2015). +The velocity extent of the Ar hole would require a C/O WD +of ≈ 1.24 𝑀⊙ from HYDRA simulations. From stellar evo- +lution (Straniero et al. 2016), a maximum mass of a C/O core +of 1.2 𝑀⊙ is produced by stars with a main sequence mass of +≈ 8 𝑀⊙. For more massive progenitors, burning continues +beyond He, resulting in O/Ne/Mg WDs and a core collapse +SN (see, for example Woosley & Baron 1992). Alternatively, +O/Ne/Mg WDs that accrete material from a companion end +their lives in accretion-induced collapse (AIC) to a neutron +star (Woosley & Baron 1992; Wasserburg et al. 1996) be- +cause compression will not reach temperatures in excess of +≈ 3 × 109 K needed to trigger explosive O-burning. In the +case of C/O detonation produced via an external trigger (i.e. +disruption by a black hole; Rosswog et al. 2009b), thermonu- +clear burning would result in a low-velocity explosion with +expansion velocities smaller by a factor of 2 − 3 compared +to typical SNe Ia (Hoeflich 2017). +Thus, such C/O WDs +can only be produced via accretion over a long time (Kip- +penhahn et al. 2013). This is, however, inconsistent with the +progenitor evolution channel commonly assumed for He-shell +detonations (Nomoto 1982; Woosley et al. 1980; Hoeflich & +Khokhlov 1996; Shen et al. 2018). +7. CONCLUSION +The successful launch of JWST heralds a new era in our +understanding of the physics of thermonuclear supernovae. +Late nebular phase MIR studies of SNe Ia are now possible +thanks to JWST’s impressive sensitivity, obtaining spectra +with higher S/N and higher resolution than any prior MIR +observatory capable of observing SNe Ia. +Here, we present a JWST/MIRI LRS spectrum of SN +2021aefx at +323 days after maximum light obtained through +JWST program GO-JWST-2114 (P.I: C. Ashall). We show +how a single spectrum can be used to extract previously un- +available information about SNe Ia. We demonstrate how by +combining JWST data with spectral models the nature of these +important astrophysical objects can be determined. Below, +we highlight our most important results: +• The observed spectrum of SN 2021aefx is linked to the +physics of SNe Ia through the construction of multi- +dimensional radiation hydrodynamical NLTE models. +We show that the spectrum and line profiles can be +understood within the context of a delayed-detonation +𝑀Ch model that produced an asymmetric 56Ni distri- +bution originating from a WD with a central density of +𝜌𝑐 ≈ 1.1 × 109 g cm−3. + +16 +DerKacy, Ashall, Hoeflich, Baron et al. +• These models are used to identify the spectral lines +which comprise the main features seen in the ob- +served spectrum. +The main lines we identify in- +clude: +[Co +III] 11.888 𝜇m, [Ar +III] 8.991 𝜇m, +[Ni +IV] +8.945 +𝜇m, +[Ni +I] +7.507 +𝜇m, +and +[Ni III] 7.349 𝜇m. Weaker identifiable blends include +lines of: [Ar II], [Fe II], [Fe III], [Co II], and [Ni II] +(see § 2). +• The presence of multiple Ni lines in the observed spec- +trum demonstrates that electron capture elements (e.g. +58Ni) are present in the inner region of SN 2021aefx. +Significant amounts of these elements can only be pro- +duced by high-density burning (above 5 × 108 g cm−3). +These densities are found in C/O WDs with masses +above ∼ 1.2 𝑀⊙. Such massive WDs must be produced +via accretion (see § 5). +• We find evidence for no, or very limited, mixing on mi- +croscopic scales between the electron capture elements +and the 56Ni region in the ejecta. In the context of near +𝑀Ch models this suggests a central point of ignition +(see § 6.3.2) +• Both the [Co III] 11.888 𝜇m and [Ar III] 8.991 𝜇m +features show flat-tilted profiles, which vary by 10% +in flux across their peaks (see § 5). +These profiles +are consistent with a central hole in the corresponding +element distributions. The profiles also indicate that +the explosion is seen at an inclination of −30◦ relative +to the point of the deflagration to detonation transition +(see § 6.3.1). +• We demonstrate how a flat-tilted profile can be used as +a tool to determine the electron capture element and Ar +distribution within the ejecta. The length of the flat- +top component corresponds to the Doppler shift of the +inner hole in the element distribution and the measured +velocity extent corresponds to the average projected +expansion velocity of the hole (see § 5.1). +• By combining information on both the strength and +profiles of the Ar and stable Ni features, we show that +SN 2021aefx was most likely produced from a C/O WD +with a mass > 1.2 𝑀⊙. This makes a He-detonation +sub-𝑀Ch explosion an unlikely candidate for this SN. +SN 2021aefx appears to be a normal SN Ia with typ- +ical light curves and spectra. We rule out supersolar +metallicity in sub-𝑀Ch WDs as an alternative option to +produce EC elements in SN 2021aefx, due to the fact +that they produce spherical cores (e.g. +56Ni distribu- +tions) which are not seen in SN 2021aefx, and are also +inconsistent with the carbon-rich surfaces commonly +seen in normal SNe Ia (see § 6.4). +Although the data presented in this work have larger errors +in wavelength calibration than anticipated, most aspects of +the physical interpretation present here are insensitive to this +error. For example, the off-center nature of the DDT is driven +by the shape of the line profiles. Moving forward, improved +wavelength calibration from the JWST pipeline, additional +MIRI/LRS data (from program 2072; P.I: S. Jha), and future +MIRI/MRS data (from program 2114; P.I: C. Ashall) will +allow us to further constrain the physics of SN 2021aefx and +other SNe Ia, and can validate our interpretation. +In par- +ticular, MIRI/MRS observations will improve the precision +of the data by probing the SN ejecta to scales smaller than +∼ 100 km s−1 which is essential. This MRS data will also +extend to longer wavelengths (∼ 20 𝜇m) revealing different +lines and ions, as well as allowing us to identify weaker fea- +tures by resolving many of the blends seen in the LRS spectra. +It will also open a new window to probe for smaller-scale ef- +fects such as mixing and positron transport within the ejecta +at later times. +Overall, this work demonstrates the ability and potential +that JWST MIR spectral observations have to provide previ- +ously inaccessible information to the scientific community. +This new information will allow us to determine the progen- +itor scenario and explosion mechanism(s) of SNe Ia. As the +sample size of MIR spectra grows over the coming years we +will be able to look for diversity within the SNe Ia population. + +SN 2021aefx: High-density Burning in SNe Ia +17 +ACKNOWLEDGMENTS +JD, CA, PH, and EB acknowledge support by NASA grant +JWST-GO-02114.032-A. Support for program #2114 was +provided by NASA through a grant from the Space Tele- +scope Science Institute, which is operated by the Associ- +ation of Universities for Research in Astronomy, Inc., un- +der NASA contract NAS 5-03127. PH acknowledges sup- +port by the National Science Foundation (NSF) through +grant AST-1715133. +EB acknowledges support by NASA +grant 80NSSC20K0538. +The simulations have been per- +formed on the Beowulf system of the Astrophysics Group +at Florida State University. This publication was made pos- +sible through the support of an LSSTC Catalyst Fellowship +to KAB funded through Grant 62192 from the John Tem- +pleton Foundation to LSST Corporation. The opinions ex- +pressed in this publication are those of the author(s) and +do not necessarily reflect the views of LSSTC or the John +Templeton Foundation. +ID acknowledges partial support +by the Spanish project PID2021-123110NB-100 financed +by MCIN/AEI/10.13039/501100011033/FEDER/UE. LG ac- +knowledges financial support from the Spanish Ministerio de +Ciencia e Innovación (MCIN), the Agencia Estatal de In- +vestigación (AEI) 10.13039/501100011033, and the Euro- +pean Social Fund (ESF) "Investing in your future" under the +2019 Ramón y Cajal program RYC2019-027683-I and the +PID2020-115253GA-I00 HOSTFLOWS project, from Cen- +tro Superior de Investigaciones Científicas (CSIC) under the +PIE project 20215AT016, and the program Unidad de Exce- +lencia María de Maeztu CEX2020-001058-M. The research +of YY is supported through a Bengier-Winslow-Robertson +Fellowship. SWJ and LAK acknowledge support by NASA +grant JWST-GO-02072.001 and NASA FINESST fellowship +80NSSC22K1599. This work is based on observations made +with the NASA/ESA/CSA James Webb Space Telescope. The +data were obtained from the Mikulski Archive for Space Tele- +scopes at the Space Telescope Science Institute, which is op- +erated by the Association of Universities for Research in As- +tronomy, Inc., under NASA contract NAS 5-03127 for JWST. +These observations are associated with program #2114. +Facilities: JWST (LRS/MIRI), MAST (JWST) +Software: HYDRA (Höflich 2003, 2009; Hoeflich et al. +2017), OpenDx (an open-sourced visualization package de- +veloped by IBM), SNooPy (Burns et al. 2011, 2014), As- +tropy (Astropy Collaboration et al. 2013, 2018, 2022), NumPy +(Harris et al. 2020), SciPy (Virtanen et al. 2020), Matplotlib +(Hunter 2007). + +18 +DerKacy, Ashall, Hoeflich, Baron et al. +REFERENCES +Allison, J. R., Sadler, E. M., & Meekin, A. M. 2014, MNRAS, 440, +696, doi: 10.1093/mnras/stu289 +Alsabti, A. 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Near-Infrared Model Line Identifications +S +𝜆 [𝜇m ] +Ion +S +𝜆 [𝜇m ] +Ion +S +𝜆 [𝜇m ] +Ion +S +𝜆 [𝜇m ] +Ion +S +𝜆 [𝜇m ] +Ion +∗ ∗ +2.211 +[Fe II] +∗ ∗ ∗ +2.478 +[Fe II] +∗ +2.935 +[Co II] +3.169 +[Fe III] +∗ ∗ ∗ +4.076 +[Fe II] +∗ ∗ ∗ +2.219 +[Fe III] +∗ +2.479 +[Ni II] +∗ +2.954 +[Co I] +3.185 +[Fe III] +∗ +4.077 +[Fe III] +∗ +2.219 +[Fe III] +2.481 +[Co I] +∗ ∗ +2.961 +[Fe II] +∗ +3.187 +[Co II] +∗ ∗ +4.082 +[Fe II] +∗ ∗ ∗ +2.243 +[Fe III] +2.493 +[Fe III] +2.963 +[Fe III] +∗ ∗ ∗ +3.230 +[Fe III] +4.108 +[Co I] +∗ +2.243 +[Fe III] +2.506 +[Co I] +2.965 +[Fe III] +3.230 +[Fe III] +∗ ∗ ∗ +4.115 +[Fe II] +∗ ∗ ∗ +2.244 +[Fe II] +∗ ∗ +2.515 +[Fe II] +2.966 +[Fe III] +∗ +3.239 +[Co II] +∗ +4.307 +[Co II] +∗ ∗ +2.257 +[Fe II] +∗ +2.526 +[Co I] +2.987 +[Fe III] +3.242 +[Fe III] +4.340 +[Co I] +∗ ∗ +2.267 +[Fe II] +∗ +2.531 +[Co II] +3.006 +[Co I] +3.286 +[Co II] +4.357 +[Fe III] +∗ ∗ +2.281 +[Co III] +2.570 +[Fe III] +3.012 +[Co II] +3.332 +[Co I] +4.357 +[Fe III] +∗ +2.282 +[Co II] +2.581 +[Co I] +∗ +3.014 +[Co III] +3.353 +[Co I] +∗ +4.410 +[Co II] +2.284 +[Co I] +∗ +2.601 +[Co II] +∗ +3.014 +[Co II] +∗ +3.394 +[Ni III] +∗ ∗ +4.435 +[Fe II] +2.285 +[Co I] +∗ +2.652 +[Co I] +3.017 +[Fe III] +3.471 +[Co I] +∗ +4.520 +[Ni I] +2.297 +[Co I] +2.686 +[Co I] +3.018 +[Fe III] +∗ +3.492 +[Co III] +∗ ∗ ∗ +4.608 +[Fe II] +∗ ∗ +2.309 +[Ni II] +∗ +2.692 +[Co II] +∗ +3.031 +[Co I] +3.498 +[Fe III] +∗ +4.672 +[Fe II] +2.316 +[Co I] +∗ ∗ +2.717 +[Fe III] +∗ ∗ ∗ +3.044 +[Fe III] +∗ +3.630 +[Co II] +∗ +4.788 +[Ni I] +∗ +2.335 +[Ni II] +2.717 +[Fe III] +3.044 +[Fe III] +∗ +3.633 +[Co I] +4.860 +[Fe III] +2.348 +[Co I] +2.726 +[Co I] +3.046 +[Co I] +3.647 +[Co I] +∗ ∗ ∗ +4.889 +[Fe II] +∗ ∗ ∗ +2.349 +[Fe III] +∗ +2.767 +[Co II] +3.061 +[Co I] +3.655 +[Co I] +5.054 +[Co I] +∗ +2.349 +[Fe III] +2.833 +[Co I] +3.063 +[Fe III] +∗ +3.659 +[Co II] +∗ ∗ +5.062 +[Fe II] +∗ +2.361 +[Ni II] +∗ +2.839 +[Co II] +3.085 +[Fe III] +∗ +3.705 +[Co II] +5.164 +[Co I] +∗ ∗ +2.370 +[Ni II] +∗ +2.848 +[Co II] +3.085 +[Fe III] +3.738 +[Co I] +∗ +5.180 +[Co II] +∗ ∗ ∗ +2.371 +[Fe II] +∗ +2.871 +[Co I] +3.095 +[Fe III] +3.750 +[Co I] +∗ ∗ +5.187 +[Ni II] +2.411 +[Fe III] +∗ ∗ ∗ +2.874 +[Fe III] +3.097 +[Fe III] +∗ +3.752 +[Co II] +5.211 +[Co I] +∗ +2.414 +[Co I] +2.874 +[Fe III] +3.100 +[Fe III] +3.771 +[Co I] +∗ ∗ ∗ +5.340 +[Fe II] +2.447 +[Fe III] +∗ +2.889 +[Co II] +∗ +3.100 +[Co II] +∗ +3.802 +[Ni III] +∗ ∗ +5.674 +[Fe II] +∗ +2.453 +[Fe III] +∗ ∗ ∗ +2.905 +[Fe III] +3.120 +[Co I] +3.823 +[Co I] +∗ ∗ +5.704 +[Co II] +2.453 +[Fe III] +2.905 +[Fe III] +∗ ∗ ∗ +3.120 +[Ni I] +∗ +3.849 +[Co II] +∗ ∗ +5.893 +[Ni I] +∗ ∗ +2.474 +[Co III] +∗ ∗ +2.911 +[Ni II] +3.129 +[Fe III] +3.877 +[Co I] +∗ +5.940 +[Co II] +∗ +2.477 +[Co II] +∗ +2.933 +[Co II] +∗ +3.151 +[Co II] +∗ ∗ +3.952 +[Ni I] +∗ +5.953 +[Ni II] +Note—For each transition, the markers correspond to strong (∗ ∗ ∗), moderate (∗ ∗), weak (∗), and scarcely detectable ( ) on top of the quasi-continuum +formed by a large number of lines. The relative strength S is estimated by the integral over the envelope, +∫ +𝐴𝑖 𝑗𝑛 𝑗 𝑑𝑉 where 𝑛 𝑗 is the particle density +of the upper level. + diff --git a/b9E2T4oBgHgl3EQfFgZ2/content/tmp_files/load_file.txt b/b9E2T4oBgHgl3EQfFgZ2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b7cb9d85133b9dc484ed8eab54ed44eedecbd82 --- /dev/null +++ b/b9E2T4oBgHgl3EQfFgZ2/content/tmp_files/load_file.txt @@ -0,0 +1,2142 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf,len=2141 +page_content='Draft version January 11, 2023 Typeset using LATEX twocolumn style in AASTeX63 JWST Low-Resolution MIRI Spectral Observations of SN 2021aefx: High-density Burning in a Type Ia Supernova J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' M.' metadata={'source': 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Telesco ,23 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Tucker ,24, † S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Valenti ,12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Wang ,25 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Yang ,26, ‡ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' W.' metadata={'source': 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112, D-21029 Hamburg, Germany 5Institute for Astronomy, University of Hawai’i at Manoa, 2680 Woodlawn Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=', Hawai’i, HI 96822, USA 6European Organization for Astronomical Research in the Southern Hemisphere (ESO), Karl-Schwarzschild-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2, 85748 Garching b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' München, Germany 7Gemini Observatory/NSF’s NOIRLab, 670 North A‘ohoku Place, Hilo, HI 96720-2700, USA 8Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA 9George P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Texas A&M University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' College Station,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' TX 77843,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' USA 10Observatories of the Carnegie Institution for Science,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Germany 20Astrophysics Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Liverpool John Moores University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' UK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' and Max-Planck Institute for Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Germany 21Las Campanas Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Carnegie Observatories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Casilla 601,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' La Serena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Chile 22Space Telescope Science Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 3700 San Martin Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' MD 21218-2410,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' USA 23Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' University of Florida,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Gainesville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' FL 32611 USA 24Center for Cosmology and AstroParticle Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 191 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Woodruff Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=', Columbus, OH 43210, USA 25Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA 26Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA 27Department of Physics and Astronomy, Rutgers, the State University of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA (Received xxx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Revised xxx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Accepted xxx) Submitted to ApJL ABSTRACT We present a JWST/MIRI low-resolution mid-infrared (MIR) spectroscopic observation of the normal Type Ia supernova (SN Ia) SN 2021aefx at +323 days past rest-frame 𝐵-band maximum light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The spectrum ranges from 4-14 𝜇m, and shows many unique qualities including a flat-topped [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m profile, a strongly tilted [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m feature, and multiple stable Ni lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These features provide critical information about the physics of the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The observations are compared to synthetic spectra from detailed NLTE multi-dimensional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The results of the best-fitting model are used to identify the components of the Corresponding author: James M DerKacy jmderkacy@vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='03647v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='HE] 9 Jan 2023 ID2 DerKacy, Ashall, Hoeflich, Baron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' spectral blends and provide a quantitative comparison to the explosion physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Emission line profiles and the presence of electron capture (EC) elements are used to constrain the mass of the exploding white dwarf (WD) and the chemical asymmetries in the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We show that the observations of SN 2021aefx are consistent with an off-center delayed-detonation explosion of a near-Chandrasekhar mass (𝑀Ch) WD at a viewing angle of −30◦ relative to the point of the deflagration-to-detonation transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' From the strength of the stable Ni lines we determine that there is little to no mixing in the central regions of the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Based on both the presence of stable Ni and the Ar velocity distributions, we obtain a strict lower limit of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 𝑀⊙ the initial WD, implying that most sub-𝑀Ch explosions models are not viable models for SN 2021aefx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The analysis here shows the crucial importance of MIR spectra for distinguishing between explosion scenarios for SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Keywords: supernovae: general - supernovae: individual (SN 2021aefx), JWST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' INTRODUCTION Type Ia supernovae (SNe Ia) arise from the thermonuclear explosion of at least one carbon/oxygen (C/O) white dwarf (WD) in a binary system (Hoyle & Fowler 1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Despite being the most precise extra-galactic distance indicators in the Universe (Phillips 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Perlmutter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2016, 2018), the exact make-up of SNe Ia progenitor systems and the mechanism of their explosions are still unknown (see Maoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Branch & Wheeler 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Jha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2019, for recent reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' There are multiple progenitor scenarios that may produce SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These include: the single-degenerate (SD) scenario where the companion is a main-sequence star or an evolved, non-degenerate companion like a red giant or He-star (Whelan & Iben 1973);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' the double-degenerate (DD) scenario, where the companion is also a WD (Iben & Tutukov 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Webbink 1984);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' or a triple system where at least two of the bodies are C/O WDs (Thompson 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Kushnir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Ad- ditionally, a wide range of explosion mechanisms also exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Multiple mechanisms originate from the merger of both stars in the progenitor system, including the dynamical merger of two WDs (Benz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' García-Berro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2017), the vi- olent merger of two WDs (Pakmor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2012, 2013), and the collisions of two WDs within triple systems (Rosswog et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2009a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Kushnir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Currently, two of the leading explosion models are of SNe Ia arising from the explosion of a near-𝑀Ch mass WD, and the detonation of a sub-𝑀Ch mass WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In 𝑀Ch explosions H, He, or C material is accreted from a companion star (which can be degenerate or non-degenerate) until the central density in the primary WD is high enough to trigger a thermonuclear runaway (Iben & Tutukov 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Diamond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The flame can then propagate as a de- flagration, detonation or both via a deflagration-to-detonation transition (DDT) (Khokhlov 1991a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hoeflich & Khokhlov 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Gamezo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Poludnenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In con- ∗ LSSTC Catalyst Fellow † CCAPP Fellow ‡ Bengier-Winslow-Robertson Postdoctoral Fellow trast, a sub-𝑀Ch explosion is triggered when a surface He layer detonates and drives a shock-wave inwards, causing a secondary detonation that disrupts the whole WD (Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Woosley & Weaver 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Livne & Arnett 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hoeflich & Khokhlov 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Similar to a 𝑀Ch explosion, a sub-𝑀Ch explosion can occur in both the single and double degenerate scenarios (Piersanti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2003a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Due to the degenerate nature of C/O WDs, the central density (𝜌𝑐) of the star is directly correlated with its mass (Chandrasekhar 1939).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Therefore, one of the key differences between 𝑀Ch and sub-𝑀Ch scenarios is the peak density of the burning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In particular when 𝜌𝑐 > 5 × 108 g cm−3, significant amounts of stable iron group elements (IGEs) such as 58Ni are produced (Seitenzahl & Townsley 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These central densities correspond to WD masses of ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 𝑀⊙, where the thermonuclear runaway must start via compressional heating in the center of the WD (Seitenzahl & Townsley 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Traditionally there have been fewer studies of SNe Ia in the longer near-infrared (NIR) and mid-infrared (MIR) wave- lengths compared to the optical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' However, recent efforts have shown that these longer wavelengths offer additional, and sometimes better, information about the physics of SN explosions (Meikle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Marion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Diamond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Wilk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This is due, in part, to the fact that the location of the photosphere is wavelength-dependent, and that different diagnostic spectral lines are revealed at longer wavelengths (Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1991, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Wheeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Kasen 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Ashall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2019a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Prior to the launch of the James Webb Space Telescope (JWST), there were only seven MIR (𝜆 > 5 𝜇m) spectral observations of SNe Ia across four different objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Three spectrawereobtainedwiththeSpitzer SpaceTelescope(SST);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' one of SN 2003hv at ∼ +375 days (relative to estimated explo- sion), one of SN 2005df at ∼ +135 days (Gerardy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2007), and one of SN 2006ce at +120 days (GO-30292, PI: W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Meikle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Four MIR spectra of SN 2014J were obtained with CanariCam on the 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4-m Gran Telesco- SN 2021aefx: High-density Burning in SNe Ia 3 pio Canarias (GTC) between 57 − 137 days after explosion (Telesco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Despite the small sample size it is apparent that the MIR contains many diagnostics to differ- entiate between leading explosion scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' For example, nebular phase MIR spectral observations, which probe the high-density central layers, can reveal the presence and dis- tribution of stable Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These lines are direct indicators of high-density burning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' With the successful launch of JWST, high-S/N MIR spec- tral observations during the nebular phase of SNe Ia are now possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The first spectrum of a SN Ia obtained with JWST was that of SN 2021aefx at +255 days after maximum light (MJD=59801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Here we present and analyze a spectrum of SN 2021aefx taken +323 days after maximum light (MJD=59871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In contrast to the work of Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2022), who focused primarily on line identifica- tions and determination of observed velocities, we interpret the explosion physics of SN 2021aefx through comparisons to a self-consistent set of non-local thermodynamic equilib- rium (NLTE) radiation hydrodynamic models of SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This allows us to provide a set of line IDs specific to SN 2021aefx in addition to a consistent picture of the explosion based on our newly observed spectrum and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In § 2 we describe our observations, and in § 3 the details of our spectral reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Line identifications from full NLTE models are performed in § 4, while an analysis of their velocities is presented in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' § 6 discusses the details of our chosen NLTE models and a comparison to the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Alternative explosion scenarios are discussed in § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Finally, we summarize our findings in § 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' OBSERVATIONS SN 2021aefx was discovered on 2021 Nov 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3 (MJD=59529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5) by the Distance Less Than 40 Mpc Survey (DLT40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Tartaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2018) and classified as a young SN Ia (Bostroem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hosseinzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' SN 2021aefx was subsequently followed by several groups, including a multi-band optical and spectroscopic follow-up campaign by the Precision Observations of Infant Supernova Explosions Collaboration (POISE, Burns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Ashall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' POISE’s detailed photometric observations re- vealed an early blue excess, which may be explained by a rapid change in the velocities of spectral lines (Ashall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' An analysis of the complete POISE data set reveals the basic light curve properties of SN 2021aefx, including a decline rate of Δm15(B) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='06 mag, and a peak absolute magnitude of 𝑀𝐵 = −19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='49 mag, which places SN 2021aefx in the normal part of the luminosity- width relation (Phillips 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Ashall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=', in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' SN 2021aefx is located 105′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3 south, 37′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 west from the center of its host NGC 1566 at a redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='005 (𝛼 = 04ℎ20𝑚00𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='42, 𝛿 = −54◦56′16′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Allison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' JWST/MIRI Observation Details Parameter Value Acquisition Image Filter F1000W Exp Time [s] 89 Readout Pattern FASTGRPAVG8 SN 2021aefx Spectrum Mode LRS Exp Time [s] 1493 𝑇obs [MJD] 59871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 Epocha [days] 322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='71 Groups per Integration 134 Integrations per Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2 Exposures per Dither 1 Total Dithers 2 Note— a Rest frame days relative to 𝐵-band maximum of MJD = 59547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='25 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Stevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=', in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' NGC 1566 is a face-on spiral galaxy with systemic recessional velocity of 1500 km s−1, and a rotational velocity of 65 ± 60 km s−1 at the location of the SN (Elagali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' All figures showing observed spectra of SN 2021aefx have been corrected for the combined recessional and ratio- nal velocities of 1550 km s−1 at the location of the SN in the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This low rotational velocity implies that any observed off-center lines (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' lines shifted relative to the line-of-sight velocity) are intrinsic to the progenitor system itself, and not attributable to a peculiar velocity within the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We present a MIR observation of SN 2021aefx ob- tained through program GO-JWST-2114 (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Ashall) from ∼ 4 − 14 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The data were obtained using JWST’s Mid- Infrared Instrument (MIRI) in its Low Resolution Spec- troscopy (LRS) configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In this mode, MIRI/LRS ob- tains slit spectroscopy of objects with a spectral resolving power (𝑅 = 𝜆/Δ𝜆) of 𝑅 ∼ 100 at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 𝜇m, varying from 𝑅 ∼ 40 at 5 𝜇m to 𝑅 ∼ 160 at 10 𝜇m (Kendrew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2015, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The instrumental configuration is identical to that of Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The spectral observa- tions were performed with a 2-point dither strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' For each grating setting there were 134 groups per integration, 2 inte- grations per exposure and 1 exposure per dither.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This results in an exposure of 734.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 seconds at each dither position, which are combined for a total exposure time of 1493 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Full details of our observational set-up are found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' DATA REDUCTION 4 DerKacy, Ashall, Hoeflich, Baron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 4 6 8 10 12 14 Rest Wavelength [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 Flux [mJy] [Co II] [Ni II] [Ar II] [Ni I] [Ni III] [Ni IV] [Ni IV] [Ar III] [Co II] [Ni II] [Co III] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' JWST/MIRI LRS spectrum of SN 2021aefx at +323 days relative to 𝐵-band maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The ions responsible for the most prominent features in the spectrum are labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' A full set of line identifications is plotted in Figure 3 and shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The data were obtained on 2022 Oct 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 (MJD=59871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6) and reduced with the JWST calibration pipeline1, version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 (Bushouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Both raw (Stage 1 cal- ibrated) and fully reduced data were retrieved from the Mikulski Archive for Space Telescopes (MAST)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The raw data was processed with a local installation of the ver- sion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 pipeline for comparison to the fully reduced data from MAST, using the spec_mode_stage_2 and spec_mode_stage_3 Jupyter notebooks as templates for the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Both reductions used the most up-to-date wave- length (jwst_miri_specwcs_0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='fits) and flux calibration (jwst_miri_photom_0085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='fits) files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These calibration files produce a wavelength solution accurate to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='05 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='02 𝜇m, varying from short to long wavelengths, and a flux calibra- tion accurate to a ∼ 2 – 5% global offset between 5 – 12 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Kendrew, private communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Furthermore, Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2022) found that the flux calibra- tion of their MIRI spectrum was accurate to 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1 https://jwst-pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='io/en/stable/jwst/introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='html 2 https://mast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='edu/portal/Mashup/Clients/Mast/Portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='html Using the LRS Optimal Spectral Extraction notebook3, the spectra were re-extracted using multiple techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This re- extraction was necessary to properly center the position of the spectrum in the science aperture, as the pipeline-derived aper- ture produced a poor extraction at long wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' After proper re-extraction with the Optimal Extraction notebook, no significant differences were found between the locally re- duced data and the fully calibrated (but un-extracted) data available from MAST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Future updates to the JWST calibra- tion files are expected to further improve the accuracy of the automated extractions (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Kendrew, private communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' DATA COMPARISON & LINE IDENTIFICATIONS Figure 1 presents the spectrum of SN 2021aefx acquired on 2022 Oct 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 (corresponding to +323 days after 𝐵-band maximum light) from 4 − 14 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' At these phases, the ejecta are optically thin and dominated by emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The strongest of these lines are labeled in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Figure 2 shows our spectrum of SN 2021aefx compared to the MIR spectra of SNe 2005df (Gerardy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2007), 2006ce (Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022), 2014J (Telesco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2015), and the earlier spec- 3 https://spacetelescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='io/jdat_notebooks/notebooks/MIRI_LRS_ spectral_extraction/miri_lrs_spectral_extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='html SN 2021aefx: High-density Burning in SNe Ia 5 6 8 10 12 14 Rest Wavelength [µm] 0 1 2 3 4 5 Normalized Flux + Constant SN 2014J (+57) SN 2014J (+81) SN 2014J (+108) SN 2014J (+137) SN 2006ce (+120) SN 2005df (+135) SN 2021aefx (+255) SN 2021aefx (+323) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Comparison of +323 day spectrum of SN 2021aefx to other MIR spectral observations of SNe Ia, including SNe 2005df (Gerardy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2007), 2006ce (Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022), 2014J (Telesco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2015), and the +255 days spectrum of 2021aefx (Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The primary difference between the +255 and +323 day spectra of SN 2021aefx is the increased strength of other features relative to the peak at ∼ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='9 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' trum of SN 2021aefx at +255 days (Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' From Figure 2 it is clear that SN 2021aefx is similar to other previ- ously observed SNe Ia, but the size and sensitivity of JWST produces a high S/N spectrum with a quality that was previ- ously impossible to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Comparing the two JWST spectra of SN 2021aefx, the most noticeable difference is the decrease in the relative strength of the ∼ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='9 𝜇m profile compared to the other features caused by the radioactive decay of 56Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' To assist in line identifications, we use a suite of full NLTE radiation transport models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These models reproduce both the early and late time properties of SN 2021aefx, and an in-depth discussion of the models with respect to the MIR observables can be found in § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' A detailed examination of the SN 2021aefx MIR spectrum reveals four prominent wavelength regions of line formation, which are described individually in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Detailed line identifications in each of these regions are plot- ted in Figure 3, while Table 2 lists the lines that contribute significantly to the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Mid-Infrared Line Identifications from Model 25 S 𝜆 [𝜇m ] Ion S 𝜆 [𝜇m ] Ion ∗ ∗ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='214 [Co II] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='555 [Fe III] ∗ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='273 [Co I] ∗ ∗ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='611 [Fe III] ∗ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='274 [Co II] ∗ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='644 [Co II] ∗ ∗ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='383 [Ar III] ∗ ∗ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='733 [Fe II] ∗ ∗ ∗ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='636 [Ni II] ∗ ∗ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='945 [Ni IV] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='636 [Ni II] ∗ ∗ ∗ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 [Ar III] ∗ ∗ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='920 [Ni II] ∗ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='618 [Ni II] ∗ ∗ ∗ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='985 [Ar II] ∗ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='080 [Ni II] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='985 [Ar II] ∗ ∗ ∗ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='189 [Fe II] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='045 [Co I] ∗ ∗ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='203 [Fe III] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='985 [Ar II] ∗ ∗ ∗ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='523 [Co II] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='103 [Co III] ∗ ∗ ∗ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='682 [Ni II] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='147 [Fe III] ∗ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='002 [Ni III] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='272 [Fe III] ∗ ∗ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='130 [Ni IV] ∗ ∗ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='349 [Ni III] ∗ ∗ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='167 [Co II] ∗ ∗ ∗ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='507 [Ni I] ∗ ∗ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='307 [Ni I] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='773 [Co I] ∗ ∗ ∗ ∗ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 [Co III] ∗ ∗ ∗ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='791 [Fe III] ∗ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='978 [Fe III] ∗ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='044 [Co II] ∗ ∗ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='001 [Ni I] ∗ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='063 [Ni II] ∗ ∗ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='255 [Co I] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='114 [Co I] ∗ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='261 [Mn II] ∗ ∗ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='211 [Fe III] ∗ ∗ ∗ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='642 [Fe II] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='282 [Ni I] ∗ ∗ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='681 [Co III] ∗ ∗ ∗ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='405 [Ni IV] ∗ ∗ ∗ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='729 [Ni II] ∗ ∗ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='489 [Co III] 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='058 [Co I] Note—For each transition, the markers correspond to domi- nant (∗ ∗ ∗ ∗), strong (∗ ∗ ∗), moderate (∗ ∗), weak (∗), and scarcely detectable ( ) on top of the quasi-continuum formed by a large number of lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The relative strength S is estimated by the integral over the envelope, ∫ 𝐴𝑖 𝑗𝑛 𝑗 𝑑𝑉 where 𝑛 𝑗 is the particle density of the upper level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The list is based on the simulations described in § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 𝜇m Region The 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 𝜇m region is dominated by emission lines of stable Ni, the most prominent of which is a blend of [Ni III] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='349 𝜇m and [Ni I] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='507 𝜇m that defines the red edge of the feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The blue edge of this peak is blended with several other weaker lines, creating a series of shoulders, extending from ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 𝜇m to ∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Mov- ing from red to blue, these shoulders are comprised of [Ar II] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='985 𝜇m, [Ni II] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='920 𝜇m, and [Ni II] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='636 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Finally, there is a small bump associated with a combination of [Co II] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='214 𝜇m, [Co I] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='273 𝜇m, and [Co II] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='274 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 𝜇m Region 6 DerKacy, Ashall, Hoeflich, Baron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 Rest Wavelength [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 Normalized Flux [Co II] [Co I] [Co II] [Ni II] [Ni II] [Ar II] [Ni III] [Ni I] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 Rest Wavelength [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 Normalized Flux [Ni IV] [Fe II] [Ni IV] [Ar III] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 Rest Wavelength [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 Normalized Flux [Fe II]+[Fe III] [Co II] [Ni II] [Ni III] [Co II] [Ni IV] [Ni I] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 Rest Wavelength [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 Normalized Flux [Co III] [Fe III] [Ni I] [Co I] [Fe II] [Co III] [Ni II] Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Detailed line identifications in the four prominent feature regions based on the lines from Model 25 (Hoeflich 2017) included in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The color intensity of the vertical lines corresponds to the strength of the spectral lines, with 4-star lines the most intense and 1-star lines being the faintest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Dashed lines correspond to ground state ions, solid lines singly-ionized species, dotted lines doubly-ionized species, and dash-dotted lines triply-ionized species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' SN 2021aefx: High-density Burning in SNe Ia 7 The 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 𝜇m region is dominated by two features whose edges are blended with other weaker lines at ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='7 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The bluer of the two is due to the emission of [Ni IV] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='405 𝜇m, while the redder feature is dominated by the [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The [Ar III] line shows a distinct flat-topped profile, which increases in flux moving from blue to red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Tilted flat- topped profiles are connected to both an ion’s velocity distri- bution in the ejecta and the viewing angle of the explosion (see § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 and § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1, also Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Small contri- butions from the weak [Fe II] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='733 𝜇m and [Ni IV] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='945 𝜇m lines may also add to the observed flux at the 10% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 𝜇m Region The 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 𝜇m region shows a structure reminiscent of the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 𝜇m region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' with one dominant blended fea- ture and a series of smaller bumps and shoulders blended into the wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The strongest peak arises from a blend of [Co II] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='523 𝜇m and [Ni II] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='682 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' A blend of [Fe II] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='189 𝜇m and [Fe III] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='203 𝜇m forms a shoulder that is partially blended into the blue wing of the [Co II]+[Ni II] blend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Blended into the red wing is a series of three other weaker features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The first feature, centered near ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='85 𝜇m is not associated with any strong lines in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The next feature in the series arises from the comparatively weak [Ni III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='002 𝜇m line, while a blend of [Ni IV] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='130 𝜇m and [Co II] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='167 𝜇m forms a shoulder on the red wing of the [Ni III] line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Finally, there may be a small contribution to the red wing of the [Ni IV]+[Co II] shoulder from [Ni I] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='307 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 − 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 𝜇m Region The 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 − 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 𝜇m region contains the only relatively isolated, un-blended feature in the MIR spectrum, the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m resonance line which produces the strongest line in the entire MIR spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Our model shows weak contributions from [Fe III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='978 𝜇m and [Ni I] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='001 𝜇m, however they only produce ∼1% of the flux and do not alter the line profile in a significant man- ner (again see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' A small shoulder at the edge of the red wing of the [Co III] line is attributable to [Co I] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='255 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' A series of peaks between 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5−13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 𝜇m suggests the presence of multiple weak lines, however the low S/N in this region prevents us from unambiguously iden- tifying any lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We tentatively identify the first peak with [Fe II] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='642 𝜇m and [Co III] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='681 𝜇m, and the second peak with [Ni II] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='729 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Our model shows no strong lines in the vicinity of the third and final peak in the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' VELOCITY DISTRIBUTIONS AND LINE PROFILES In this section, we discuss the velocity distributions and line profiles of three important species in the ejecta: Ar, Co, and Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In discussing these velocities and profiles we reiter- ate that the current wavelength calibration of the MIRI/LRS observations is accurate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='05 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='02 𝜇m, with lower errors at longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This corresponds to an error on the order of ∼ 500 km s−1 in the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m line, and ∼ 1400 km s−1 in the [Ni III] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='349 𝜇m line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Future updates to the JWST pipeline calibration files may increase the precision of these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m Ar traces the transition region between incomplete oxy- gen burning and nuclear statistical equilibrium (NSE) in the ejecta;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' thereby providing details about the chemical distribution between the 56Ni and Si-group layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m line profile is plotted in Figure 4 in ve- locity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The profile is flat-topped with an increasing tilt from blue to red wavelengths, which we refer as a “flat-tilted” profile hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Flat-topped profiles are indicative of a cen- tral hole or void in the emitting material — that is, a shell of line emitting material (Beals 1929;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Menzel 1929;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Struve 1931).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' For [Ar III] the flat-top component of the feature starts at ∼ −7000 km s−1 and extends to ∼ 8000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The feature increases in flux by 10% across the profile from the blue to red side, and the flat-topped component of the profile indicates that there is a central hole in the ejecta of ∼ ±8000 km s−1 which does not contain Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This is because Ar is destroyed in high temperature regimes of the NSE where 𝑇 ≥ 6 × 109 K, and there is a lack of strong mixing during the explosion, consistent with explosion models of near 𝑀Ch WDs (see § 6 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m SNe Ia are powered by the nuclear decay chain of 56Ni to 56Co to 56Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Since the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m feature is a resonance line, most of the de-excitation and recombination of Co passes through this transition, making it a direct tracer of the distribution and amount of 56Ni in the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This feature covers a width of ∼ ±10000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' If the shape of the line is assumed to be symmetric, and thus well described by a Gaussian profile, it peaks at 740 ± 200 km s−1 with a FWHM of 4840 ± 170 km s−1 (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Combining the error in the line-of-sight velocity (recessional plus rotational;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' ∼ 60 km s−1) and the estimated error in the wavelength cali- bration of ∼ 500 km s−1 with that of the fit error yields a total estimated error of 544 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The fact that this resonance line is not located at the kinematic center of the explosion indicates that the bulk of the 56Ni is off-center, at the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4𝜎 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m profile also shows hints of a flat-tilted profile, peaking to the red at ∼ 2000 km s−1 (see Figure 4), although the low resolution prevents a definitive identification of this profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Similarly, hints of this flat-tilted peak are also seen in the spectrum at the earlier epoch of SN 2021aefx (see Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' If real, this flat-tilted profile may extend from ∼ −1000 km s−1 to ∼ 2000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 8 DerKacy, Ashall, Hoeflich, Baron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' −10 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='35 Normalized Flux [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 µm −10 0 10 Velocity [103 km s−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 Normalized Flux [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 µm Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Line profiles of the [Ar III] (top) and [Co III] lines (bottom) in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The blue boxed region around 𝑣 = 0 km s−1 in the rest frame denote the 1𝜎 error in the rest wavelength for the given line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Red vertical lines mark the left and right edges of the flat-tilted profiles in both panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Similar to the [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m line, the flat-tilted profile of the [Co III] feature may imply a central hole of 56Ni in the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This hole is smaller than that of Ar, and would only be a few thousand km s−1 across (Telesco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Diamond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Note that unlike Ar, which is pro- duced by nuclear burning that has a steep temperature depen- dence leading to a sharp cutoff in velocity extent and thus flat line profiles, electron capture is nearly temperature in- dependent, so its effects follow the density profile leading to somewhat rounder line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The increase in flux across the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m profile is 10%, the same as that in the [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m feature, implying the distribution of the two elements are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Since Ar is produced at the edge −10 −5 0 5 10 Velocity [103 km s−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 Normalized Flux [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 µm Fit Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='866 𝜇m line compared to a Gaussian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The Gaussian peaks at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='91±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='01 𝜇m, 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='19±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='04 𝜇m, which in velocity space corresponds to a peak at 740 ± 200 km s−1 and 𝜎 = 4840 ± 170 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' of the 56Ni region (see § 6), it is reasonable that Ar and Co have similar changes in flux across their profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In § 6, we discuss the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m line profile in the context of off-center 56Ni distributions in the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' However, in order to confirm that the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m feature is truly asymmetrical and off-center, higher resolution spectra are required (such as those obtainable by the Medium Res- olution Spectrograph (MRS) of JWST/MIRI) and improved wavelength calibrations are also needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Stable Ni Multiple ionization states of Ni have forbidden emission lines which occur in the MIR, making nebular phase MIR spectra an invaluable resource for probing the explosion physics and corresponding nucleosynthesis of SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Since the 56Ni that powers the early light curves of SNe Ia has a half life of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 days, any emission from Ni at these late phases comes from isotopes of stable Ni (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 58Ni) and not from radioactive isotopes like 56Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Figure 6 presents three of these regions in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The left panel shows the [Ni III] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='349 𝜇m line, which appears to be red-shifted in ve- locity space with an apparent maximum around 3000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' However, this feature is blended with [Ni I] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='506 𝜇m, such that the velocity extent of [Ni III] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='349 𝜇m appears to be larger than its true distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The middle panel shows the [Ni III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='405 𝜇m line profile in velocity space, while the right panel depicts the [Ni III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='002 𝜇m feature within a much larger series of blended lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' At higher resolution SN 2021aefx: High-density Burning in SNe Ia 9 −10 −5 0 5 10 Velocity [103 km s−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='7 Normalized Flux [Ni I] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='506 µm [Ni III] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='349 µm −10 −5 0 5 10 Velocity [103 km s−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3 Normalized Flux [Ni IV] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='405 µm −10 0 10 Velocity [103 km s−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 Normalized Flux [Ni II] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='682 µm [Ni IV] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='130 µm [Ni I] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='307 µm [Ni III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='002 µm Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Velocity space profiles of the three spectral regions with prominent Ni lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Vertical lines indicate the ionization and line strength as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The left panel shows the [Ni III] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='349 𝜇m region is contaminated by the [Ni I] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='506 𝜇m feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The right panel, centered on the [Ni III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='002 𝜇m line, shows evidence for multiple stable Ni lines contributing to the series of weak features and shoulders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' these blends, including the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='002 𝜇m line, are likely to be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' NUMERICAL MODELING AND IMPLICATIONS FOR EXPLOSION SCENARIOS To explore the explosion physics of SN 2021aefx we turn to detailed comparisons with NLTE radiation hydrodynam- ical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The goals of these comparisons are: (1) to demonstrate that MIR spectral features and line profiles can be used as a critical tool to determine the explosion physics and progenitor scenario of SNe Ia, (2) to show that JWST has opened up a new frontier in MIR SN science and that there is a need to test and calculate atomic models and processes, including cross sections to improve future models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Specifi- cally, we address how the data allows us to measure the mass of the exploding WD, the chemical asymmetries in the initia- tion of the explosion, and small-scale mixing processes in the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' When taken in total, these measurements allow us to determine the most likely explosion scenario of SN 2021aefx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' As discussed in § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3 and shown by Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2022), SN 2021aefx presents many spectral lines of stable Ni (Fig- ure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This Ni requires high-density burning in the ejecta, above 5 × 108 g cm−3, which must originate from a massive WD, making the explosion either a near-𝑀Ch WD where the explosion is triggered by compressional heating in the center of the explosion, or the detonation of a high-mass, sub-𝑀Ch larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='15–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 𝑀⊙ (Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hoeflich & Khokhlov 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Seitenzahl & Townsley 2017, but see also Blondin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Such massive WDs can only be pro- duced via accretion (Kippenhahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Therefore, we limit our comparisons to models within this region of param- eter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Numerics The simulations employ modules of the HYDrodynami- cal RAdiation (HYDRA) code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' HYDRA solves the time- dependent radiation transport equation (RTE) and positron transport (Penney & Hoeflich 2014), including the rate equa- tions that calculate the nuclear reactions based on a network with 211 isotopes and statistical equations for the atomic level populations, the equation of state, the matter opacities, and the hydrodynamic evolution as applied to SN 2020qxp (Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hristov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' De- tailed atomic models and line lists are based on the database for bound-bound transitions of van Hoof (2018)4, supple- mented by additional forbidden lines from Diamond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2015) and Telesco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' For details on modeling of nebular phase spectra with HYDRA, see Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2021), and for more general discussions on modeling the nebular phase and downward cascading of high-energy parti- cles and photons by Monte Carlo, see also Spencer & Fano (1954), Axelrod (1980), Kozma & Fransson (1992), Fransson (1994), Fransson & Jerkstrand (2015), Botyánszki & Kasen (2017), Wilk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2018), Shingles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2020), and Wilk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The models include transitions for ionization stages I-IV of C, O, Ne, Mg, Si, S, Cl, Ar, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, and Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Though most of the prominent features in the MIR are caused by forbidden lines, the underlying quasi- continuum is formed by allowed lines in the inner layers well above the critical density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' At these phases, the iron-rich layers are still partially optically thick at UV wavelengths, meaning the inclusion of permitted lines is important to fully charac- terize the ionization balance via Rosseland cycles (Mihalas 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' A Delayed-Detonation Model for SN 2021aefx 4 Version v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='00b3, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='uky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='edu/~peter/newpage/ 10 DerKacy, Ashall, Hoeflich, Baron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Model 25 Parameters Parameter Value Mej ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='38 𝑀⊙ 𝜌𝑐 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 × 109 g cm−3 Mtr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='24 𝑀⊙ MDDT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 𝑀⊙ B(WD) 106 G We compare SN 2021aefx to new simulations of off-center 𝑀Ch mass explosion models, based upon the spherical model of the Model 25-series from Hoeflich (2017), as it produces early light-curve properties and a maximum light luminosity very similar to those of SN 2021aefx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These new simu- lations are parameterized explosion models, using a spher- ical delayed-detonation to constrain the global parameters of the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Fine-tuning these models is not nec- essary to achieve the goals of this study as we focus on spectra rather than high-precision photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The model produces Δ𝑚15(𝑉) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='68 mag (for reference, Δ𝑚15(𝑉) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='01 mag for SN 2021aefx, which is within the error of the model), and ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 𝑀⊙ of 56Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The model originates from a C/O WD with a main-sequence progenitor mass of 5 𝑀⊙, solar metallicity, and a central den- sity 𝜌𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 × 109 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We adopt this 𝜌𝑐 due to the line width and shape of the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m line and due to the strength of the stable Ni lines in the MIR spectrum (see § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In this model, burning starts as a deflagration front near the center and transitions to a detonation (Khokhlov 1991b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The deflagration–detonation transition is triggered when the density at the burning front drops below 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5×107 g cm−3, when ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='24 𝑀⊙ of the material has been burned by the deflagration front, and is induced by the mixing of un- burned fuel and hot ashes (Khokhlov 1991b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The model has a magnetic field of B(WD) = 106 G, which has been found in magneto-hydrodynamical simulations, suggesting that tur- bulent magnetic fields are produced during the deflagration phase (Diamond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hristov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The basic model parameters are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Off-center 56Ni and Abundance Distributions To investigate the line profiles and asymmetries, we con- sider the Model 25-series which includes off-center DDTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' For the construction of the off-center DDT we follow the description of Livne (1999) that has also been employed by Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2006), Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2015), Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2021), and Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The DDT is triggered at 𝑀DDT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Note that due to the buoyancy of flame fronts in the explosion, the DDT can be triggered at a dif- ferent mass coordinate relative to the total integrated mass of the deflagration burning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This leads to asymmetric abundance distributions of all elements produced during the detonation phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2 in Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In principle, the use of multiple resolved line profiles allows us to determine the value of 𝑀DDT as well as the viewing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In the case of SN 2021aefx we use the two strongest features: [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m and [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' As shown in Figure 4 we see a consistent tilt in the [Ar III] and [Co III] lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We can determine the viewing angle from the tilt of these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The value of 𝑀DDT determined here was also consistent with the spectrophotmetric observations of the normal SN Ia 2019np (Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Most normal SNe Ia have very similar polarization properties (Cikota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Overall Abundance Distribution The angle-averaged abundance structure and the 56Ni dis- tribution of Model 25 are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In the model, the region of high electron capture is spherical because we as- sume central ignition, no fragmentation during the 56Ni decay over the first week after the explosion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2015), and that Rayleigh-Taylor instabilities are largely suppressed by high magnetic fields (Hristov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The most no- table results in the abundance distribution are: (1) ∼ 6 × 10−2 of 58Ni is produced in the center of the ejecta, (2) the velocity extent of the central hole in 56Ni is ∼ 3200 km s−1, (3) the velocity extent of the 56Ni region produced in NSE ranges from ∼ 3200 – 10000 km s−1, and (4) the size of the shell of the Ar region covering a range of ∼ 8000 – 15000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We note that simulating the point of the DDT in multi-dimensions does not lead to a strong rarefaction wave (Gamezo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2015) as seen in all spherical delayed-detonation models (Khokhlov 1991b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The off-center DDT at a point in an already-expanding medium results in a run-time effect which yields an asym- metric distribution of burning products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The material closer to the DDT burns under higher density than the opposite side because the front reaches the corresponding layer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 − 1 s later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The result is a bulge of all elements that undergo only Si and O burning including Ca, Ar, and 56Ni (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' For a more complete depiction of this, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 7 of Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These asymmetries are aligned along the axis defined by the center and the DDT ignition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Spectral Modeling 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Determining the Inclination Angle We begin our discussion of Model 25’s fit to the obser- vations by illustrating its ability to determine the inclination angle of the explosion relative to our line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Remember that to first order, the [Co III] profile can be fit with a Gaus- sian of a half-width ≈ 4800 km s−1, emission wings ranging SN 2021aefx: High-density Burning in SNe Ia 11 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (Left:) The chemical composition of our best fit model, Model 25 from Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2017) and Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The model has a chemically stratified ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' EC elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 58Ni with M(58Ni) ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='9 × 10−2 𝑀⊙) are located in the center of the ejecta, followed by 56Ni further out in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The Ar distribution goes between 8000 – 15000 km s−1 and the lightest elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' O and C) are located in the outermost layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' For illustration, the thin red line at expansion velocities larger than 3200 km s−1 shows the EC distribution after microscopic mixing applied (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (Right:) The distribution of the IGEs of the off-center delayed detonation Model 25 at a point (black dot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The bulk of the 56Ni is in a ring-like structure between 3000 – 9500 km s−1 as well as a bulge produced at the point of the delayed detonation transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Depending upon the viewing angle, differently shaped line profiles will be produced in the [Co III] feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These profiles are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' from −10000 – +10000 km s−1, and an offset from the rest wavelength of +740 km s−1 (see § 5 and Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This is consistent with the overall 56Ni distribution seen in the model (see Figure 7), but we note that assuming an emission feature is a Gaussian makes an implicit assumption about the under- lying chemical distribution of an element within the ejecta, and should be used with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' As previously discussed, the host galaxy is seen face on and has a very small projected rotation (65 ± 60 km s−1), implying that host rotation plays a minor role in this offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This leaves the peculiar motion of the progenitor system and the orbital velocity of the progen- itor as remaining potential sources of this offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' However, if these were the dominant factors, one would expect a con- sistent velocity offset in all of the spectral lines, contrary to observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' On the other hand, the observed flux in the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m line center changes by ∼ 10% across the peak of the feature (see § 5), consistent with expectations of flux arising from the asymmetric ejecta of an off-center DDT model when viewed from a specific angle (Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' As previously shown in § 5, the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m feature appears to show a flat-tilted profile, where the velocity extent of the central tilted region corresponds to the region in velocity space of partial burning in quasi-statistical equi- librium (QSE) (≈ 1800 km s−1 across in the angle averaged spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This flat-tilted profile is seen in both the +255 and +323 day JWST spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In Model 25, the inner size of the electron capture region and the distribution of 56Ni produce different line profiles when viewed at different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Three specific viewing angles, −90◦, −30◦ and +30◦ are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' From the bottom left panel, we see that the ob- servations are well matched by a viewing angle of ≈ −30◦, including replicating the ∼ 10% change in flux across the peak seen in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' While the high signal-to-noise (S/N ≈ 100) of both JWST spectra of SN 2021aefx suggests that the flat-tilted profile is real and significant, future planned observations with JWST/MIRI MRS (JWST-GO-2114, PI: Ashall) will better resolve the line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Overall MIR Spectra Having determined the inclination angle, we now compare our full model spectrum to our observations, as seen in Fig- ure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Model spectra are shown with and without mixing of electron capture elements on the scale of the pressure scale height of the WD (Höflich & Stein 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We examine mi- croscopic mixing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' smaller than the mean free path of the positrons) in the center of the explosion to constrain the position (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' central, off-center, or multi-spot) of the ther- monuclear runaway ignition (Niemeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Höflich & Stein 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Calder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Livne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Röpke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The most dominant lines produced by the models are shown in Table 2, and have been successfully identified in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' For NIR lines outside the observed range, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Identification of weaker lines within the spectrum will be possible after the acquisition of MIRI/MRS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The model reproduces the observations overall, including all four regions of prominent spectral lines and does especially well in reproducing the blends of the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 𝜇m and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 𝜇m regions in addition to the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 36 C 40Ca 0 44Ti 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='8 Si 56Ni S Ne EC Mg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 10 15 20 25 5 [1000 km/sec]1 X, +30° .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 30° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='06-12 DerKacy, Ashall, Hoeflich, Baron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' −10 −5 0 5 10 Velocity [103 km s−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 Normalized Flux +255 d +323 d −10 −5 0 5 10 Velocity [103 km s−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 Normalized Flux +323 d +30◦ −30◦ −90◦ −10 −5 0 5 10 Velocity [103 km s−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 Normalized Flux +323 d −30◦ −10 −5 0 5 10 Velocity [103 km s−1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 Normalized Flux +255 d +323 d +30◦ −30◦ −90◦ Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Top Left: Comparison of the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m line profile at +255 (dashed grey) and +323 days (solid black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Top Right: Dependence of the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m line profile as a function of inclination in comparison with the +323 day spectrum (solid black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Note that the profile and the red-shift of the observed peak are consistent with the off-center DDT model seen at −30◦ (bottom left) and the −30◦ model is also the best fit to both observations (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' SN 2021aefx: High-density Burning in SNe Ia 13 6 7 8 9 10 11 12 13 14 Rest Wavelength [µm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 Normalized Flux SN 2021aefx (+255 d) SN 2021aefx (+323 d) Model 25 w/ EC mixing Model 25 w/ no EC mixing Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Comparison of the synthetic MIR spectrum of the off-center Model 25 seen from −30◦ without (blue) and with (red) mixing of the EC elements (see Figure 6) and the JWST/MIRI LRS spectrum of SN 2021aefx at +255 (dashed grey) and +323 (solid black) days relative to 𝐵-band maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The angle-averaged spectra would look similar, but they would show a flat-topped rather than a flat-tilted [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Though [Ni II] lines are present in both the synthetic spectra, the sensitivity to microscopic mixing should be noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In particular, the [Ni II] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='6 𝜇m line shows a strong variation with mixing (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' While the exact contribution of each ion may vary with the underlying explosion model, the synthetic spectra have been obtained without further tuning and in general are in good agreement with the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The similarity between the mixed and unmixed model shows the stability of the synthetic spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Most of the Ni features are in blends with other iron-group elements of similar strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In light of uncertainties in the atomic models and cross sections, the photons at a given wavelength may couple to elements other than Ni (through fluorescence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Morrison & Sartori 1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Thus, many of the line IDs of weaker features in low resolution spectra are model dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The easiest way to separate the elements is by comparing the mixed and unmixed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In the unmixed models, the electron capture elements are effectively shielded from non-thermal excitations from radioactive decay, thus the electron capture features will be weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Features dominated by Ni show variations in the region between 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 𝜇m and weak variation at longer wavelengths, for example at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' One major effect of mixing can be seen in the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 – 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4 𝜇m region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In the unmixed models, EC and radioactive elements are mostly separate, leading to weak spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This due to the locally trapped positrons dominating the excita- tion in the EC region, with 𝛾-rays responsible for ∼ 10% of the excited ions (Penney & Hoeflich 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In turn, micro- scopic mixing produces narrow features, for example, [Ni II] at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='636 𝜇m has a half-width of ≈ 4000 km s−1 determined by the numerical resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In contrast, the observed broad features suggest a central ignition of the WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The broad feature at ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='0 𝜇m dominated by Ar is a strong diagnostic of the point of the transition from the deflagra- tion to the detonation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Ar is located in a shell with a large central hole because it is destroyed in moderately high den- sity burning environments (𝜌 ≳ 1 − 2 × 107 g cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Our model (Figure 8) is consistent with the estimates for the minimum Ar velocity, strongly suggesting a high mass WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m feature shows the same slope as the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m, providing evidence of an off-center DDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Line profiles are strongly affected by the ionization balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Typically for normal SNe Ia at this phase, iron group elements are dominated by doubly ionized ions, and the ionization frac- tion decreases with increasing density because the recombi- 14 DerKacy, Ashall, Hoeflich, Baron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' nation rate scales with the square of the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Only in the center do we see effects at the 10% level of singly ionized iron group elements that produce a strong resonance [Ni I] feature at ≈ 3 𝜇m (Kwok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Fisher 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In the line forming regions, the ionization balance hardly changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Therefore, our results are insensitive to the differences in the ionization structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' More detailed discussions are given in Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2021), Wilk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2020), and § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In the synthetic spectrum, the feature at ∼ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='9 𝜇m agrees well with the observations, it is dominated by [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m and has minor line blends of [Ni I] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='001 𝜇m and [Co I] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='255 𝜇m in the red wing at the 1% level relative to the [Co III] peak (see the line strengths given in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' However, our models tend to show features from singly ion- ized elements that are too strong by about 10 − 20% as can be seen from the ratio of the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m and the [Co II] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='523 𝜇m plus [Ni II] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='682 𝜇m blend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The discrepancies in the ionization balance are not unexpected, due to uncer- tainties in the atomic data and the ∼ 2 − 5% accuracy of the flux calibration of the observed spectrum (see § 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Uncertain ionization and excitation by non-thermal leptons and uncer- tainties in the recombination rates lead to ionization balance uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Though we treat the cascades in energy within our Monte-Carlo scheme, missing and uncertain atomic lev- els are likely responsible for some of the discrepancies (Wilk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Shingles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2020, and see Appendix A of Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Finally, we discuss other observables obtainable at a higher spectral resolution that may support our interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Off- center delayed-detonation models predict an offset between electron capture elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=', 58Ni) produced during the deflagration phase and elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=', 56Ni, Fe, Co, Si, S, Ar) synthesized during the detonation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The former are created in a subsonic deflagration resulting in a slow pre-expansion phase of the WD with an almost spherical density structure, whereas the latter are formed in a weak detonation with a burning speed close to the speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This offset may be seen with higher-resolution spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Note that the unresolved small flux variations near 11 𝜇m and in the wavelength range 12 − 14 𝜇m seen in MIR spectra are at a 1𝜎 level which, if confirmed by MRS spectra, have im- portant implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The computational results indicate that these small variations signal the presence of a caustic struc- ture in density and abundance of the inner electron-capture core as has been observed by Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2007) and Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' At this epoch, positrons remain local for the high 𝐵-field required (Penney & Hoeflich 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Mera Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hristov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Alternative Explosion Scenarios A detailed discussion of explosion scenarios and progenitor systems of SNe Ia is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' For reviews, we refer to Alsabti & Murdin (2017) and Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The total mass of the exploding WD is one of the parameters separating different explosion scenarios such as He-triggered detonations of sub-𝑀Ch, dynamical mergers, violent mergers, collisions of two WDs in a triple system, and 𝑀Ch explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Dynamical mergers go through a loosely bound hydrostatic WD state and are unlikely to synthesize EC elements because the peak density during merging is too low (Benz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' García-Berro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Collisions of two WDs may result in high density burning, high masses, and may produce EC elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' but also produce large polarization signatures and a 90◦ flip in the polarization angle (Höflich 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Bulla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2016), both of which are not observed in any SNe Ia (Cikota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Similarly, violent mergers would be expected to yield high continuum polarization (∼ 1%) at ∼ 10 days before the 𝐵−band maximum light (Bulla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2016), which is also not seen in observations (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Patra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Therefore, we focus on sub-𝑀Ch He-triggered detonations of C/O WDs as an alternative viable candidate to produce SN 2021aefx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' For normal SNe Ia sub-𝑀Ch models have WD masses of 1 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='05 𝑀⊙ and for bright SNe Ia they have WD masses up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 𝑀⊙ (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Blondin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Stable Ni has been tentatively suggested to have been seen in observations of several SN Ia based upon: (1) heavily blended optical features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Mazzali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2020), (2) the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='94 𝜇m [Ni II] feature (Dhawan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2021), (3) and in low S/N MIR spectra (Gerardy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Telesco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' However, the JWST spectra of SN 2021afex are the first to firmly establish the presence of stable Ni in a SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Our simulations of SN 2021aefx suggest that ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='06 𝑀⊙ of stable Ni was produced in the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In fact, since the synthetic Ni lines may be slightly too weak (Figure 9), we may need slightly more stable Ni than our simulations imply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Stable Ni is produced in NSE by shifting the ratio 𝑌𝑒 from ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 to a lower value either by electron-capture under high density burning or in a WD with super-solar metallicities, as a result of an initial high 22Ne abundance (see, for example, Brachwitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Timmes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Thielemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Gronow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2021) simulate a variety of He-detonation explosions at various metallicities, with various He-shell masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' They find that super-solar metallicities produce EC elements due to the decrease𝑌𝑒 from the presence of neutron- rich 22Ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Gronow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2021) find that WDs with masses of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 𝑀⊙ and primordial metallicity 3𝑍⊙ produce 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='046 𝑀⊙ of stable Ni, an the amount of 58Ni sufficient to produce the strong Ni features observed in SN 2021aefx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' However, since this result is due to the primordial metallicity of the WD, which reduced the 𝑌𝑒 uniformly throughout the WD — the SN 2021aefx: High-density Burning in SNe Ia 15 full half-width of the [Co III] line and of the Ni features should be comparable because both 56Ni and 58Ni are formed in the same NSE region and constant 𝑌𝑒 results in a constant isotopic ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Future MIRI/MRS observations will be able to resolve the Ni lines and accurately measure its elemental dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In particular, in these sub-𝑀Ch models the [Ni IV] should be broad as the high ionization stage will occur in the low-density, high-velocity region of the envelope because the recombination rates scale with the density squared (Oster- brock & Ferland 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Moreover, the model that produces the large 58Ni abundance requires of a He shell of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='02 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Finally, if large stable Ni abundances are ubiquitous to all SNe Ia, within the sub-𝑀Ch paradigm it would require all of them to have super-solar metallicity, which is unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Based on recent 3D simulations for solar metallicities, Boos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2021) showed that a thin He-triggered detonation in a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 𝑀⊙ C/O WD may produce 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='02 𝑀⊙ of stable Ni, an amount that is insufficient to explain the observed NIR fea- tures from stable Ni (see Wilk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='5 For both sets of He-det simulations discussed above, the mass of the outer He layers are inconsistent with recent limits from other early-time normal SNe Ia spectra that show carbon in the outer 2 − 5 × 10−3 𝑀⊙ (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Furthermore, thin He-detonation models have nearly spherical 56Ni distributions (Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2023), which contradict the observation of SN 2021aefx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Lacking advanced He-triggered detonation models, we fo- cus on spherical explosion models with sub-𝑀Ch cores such as DET2 (Hoeflich & Khokhlov 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This model has a pure C/O WD without a He surface layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' It originates from a WD whose mass is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 𝑀⊙, and produces a sufficient amount of EC material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Strict limits on the WD mass and the density of the central region can be obtained from both the [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m and [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In particular, a tight mass limit on the WD can be obtained via the width of the flat-top component of the [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The observed edge of the Ar profiles implies a central hole of Ar between ∼ 0 − 8000 km s−1 (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' However, DET2 produces an inner hole of Ar of 6000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This is 2000 km s−1 lower than observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' To produce a wider flat-topped Ar fea- ture requires a higher mass WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In a high mass model (such as a near 𝑀Ch explosion), the Ar hole extends further out in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We note that, as an upper limit, detonating a WD with an ejecta mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='38 𝑀⊙ would mostly produce 56Ni and 5 Blondin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' (2022) claim that sub-𝑀Ch models with 𝑀 > 1 𝑀⊙ can produce the NIR [Ni II] lines at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' They argue that the lack of [Ni II] in the NIR at earlier times can be simply an ionization effect, with the mixing of radioactive products into the EC region dramatically reducing the strength of [Ni II], thus, providing an option for the absence of the observed NIR [Ni II] feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This explanation is, however, inconsistent with the fact that in SN 2021aefx many ionization stages of Ni are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' few QSE elements, resulting in a SN that produced too much 56Ni, is too bright at maximum light, and has spectra that are inconsistent with a typical SN Ia (Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Marquardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The velocity extent of the Ar hole would require a C/O WD of ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='24 𝑀⊙ from HYDRA simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' From stellar evo- lution (Straniero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2016), a maximum mass of a C/O core of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 𝑀⊙ is produced by stars with a main sequence mass of ≈ 8 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' For more massive progenitors, burning continues beyond He, resulting in O/Ne/Mg WDs and a core collapse SN (see, for example Woosley & Baron 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Alternatively, O/Ne/Mg WDs that accrete material from a companion end their lives in accretion-induced collapse (AIC) to a neutron star (Woosley & Baron 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Wasserburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1996) be- cause compression will not reach temperatures in excess of ≈ 3 × 109 K needed to trigger explosive O-burning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In the case of C/O detonation produced via an external trigger (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' disruption by a black hole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Rosswog et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2009b), thermonu- clear burning would result in a low-velocity explosion with expansion velocities smaller by a factor of 2 − 3 compared to typical SNe Ia (Hoeflich 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Thus, such C/O WDs can only be produced via accretion over a long time (Kip- penhahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This is, however, inconsistent with the progenitor evolution channel commonly assumed for He-shell detonations (Nomoto 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Woosley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hoeflich & Khokhlov 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' CONCLUSION The successful launch of JWST heralds a new era in our understanding of the physics of thermonuclear supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Late nebular phase MIR studies of SNe Ia are now possible thanks to JWST’s impressive sensitivity, obtaining spectra with higher S/N and higher resolution than any prior MIR observatory capable of observing SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Here, we present a JWST/MIRI LRS spectrum of SN 2021aefx at +323 days after maximum light obtained through JWST program GO-JWST-2114 (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='I: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Ashall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We show how a single spectrum can be used to extract previously un- available information about SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We demonstrate how by combining JWST data with spectral models the nature of these important astrophysical objects can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Below, we highlight our most important results: The observed spectrum of SN 2021aefx is linked to the physics of SNe Ia through the construction of multi- dimensional radiation hydrodynamical NLTE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We show that the spectrum and line profiles can be understood within the context of a delayed-detonation 𝑀Ch model that produced an asymmetric 56Ni distri- bution originating from a WD with a central density of 𝜌𝑐 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1 × 109 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 16 DerKacy, Ashall, Hoeflich, Baron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These models are used to identify the spectral lines which comprise the main features seen in the ob- served spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The main lines we identify in- clude: [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m, [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m, [Ni IV] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='945 𝜇m, [Ni I] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='507 𝜇m, and [Ni III] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='349 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Weaker identifiable blends include lines of: [Ar II], [Fe II], [Fe III], [Co II], and [Ni II] (see § 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The presence of multiple Ni lines in the observed spec- trum demonstrates that electron capture elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 58Ni) are present in the inner region of SN 2021aefx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Significant amounts of these elements can only be pro- duced by high-density burning (above 5 × 108 g cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These densities are found in C/O WDs with masses above ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Such massive WDs must be produced via accretion (see § 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We find evidence for no, or very limited, mixing on mi- croscopic scales between the electron capture elements and the 56Ni region in the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In the context of near 𝑀Ch models this suggests a central point of ignition (see § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2) Both the [Co III] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='888 𝜇m and [Ar III] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='991 𝜇m features show flat-tilted profiles, which vary by 10% in flux across their peaks (see § 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These profiles are consistent with a central hole in the corresponding element distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The profiles also indicate that the explosion is seen at an inclination of −30◦ relative to the point of the deflagration to detonation transition (see § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We demonstrate how a flat-tilted profile can be used as a tool to determine the electron capture element and Ar distribution within the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The length of the flat- top component corresponds to the Doppler shift of the inner hole in the element distribution and the measured velocity extent corresponds to the average projected expansion velocity of the hole (see § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' By combining information on both the strength and profiles of the Ar and stable Ni features, we show that SN 2021aefx was most likely produced from a C/O WD with a mass > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='2 𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This makes a He-detonation sub-𝑀Ch explosion an unlikely candidate for this SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' SN 2021aefx appears to be a normal SN Ia with typ- ical light curves and spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' We rule out supersolar metallicity in sub-𝑀Ch WDs as an alternative option to produce EC elements in SN 2021aefx, due to the fact that they produce spherical cores (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 56Ni distribu- tions) which are not seen in SN 2021aefx, and are also inconsistent with the carbon-rich surfaces commonly seen in normal SNe Ia (see § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Although the data presented in this work have larger errors in wavelength calibration than anticipated, most aspects of the physical interpretation present here are insensitive to this error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' For example, the off-center nature of the DDT is driven by the shape of the line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Moving forward, improved wavelength calibration from the JWST pipeline, additional MIRI/LRS data (from program 2072;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='I: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Jha), and future MIRI/MRS data (from program 2114;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='I: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Ashall) will allow us to further constrain the physics of SN 2021aefx and other SNe Ia, and can validate our interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' In par- ticular, MIRI/MRS observations will improve the precision of the data by probing the SN ejecta to scales smaller than ∼ 100 km s−1 which is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This MRS data will also extend to longer wavelengths (∼ 20 𝜇m) revealing different lines and ions, as well as allowing us to identify weaker fea- tures by resolving many of the blends seen in the LRS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' It will also open a new window to probe for smaller-scale ef- fects such as mixing and positron transport within the ejecta at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Overall, this work demonstrates the ability and potential that JWST MIR spectral observations have to provide previ- ously inaccessible information to the scientific community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This new information will allow us to determine the progen- itor scenario and explosion mechanism(s) of SNe Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' As the sample size of MIR spectra grows over the coming years we will be able to look for diversity within the SNe Ia population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' SN 2021aefx: High-density Burning in SNe Ia 17 ACKNOWLEDGMENTS JD, CA, PH, and EB acknowledge support by NASA grant JWST-GO-02114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='032-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Support for program #2114 was provided by NASA through a grant from the Space Tele- scope Science Institute, which is operated by the Associ- ation of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=', un- der NASA contract NAS 5-03127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' PH acknowledges sup- port by the National Science Foundation (NSF) through grant AST-1715133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' EB acknowledges support by NASA grant 80NSSC20K0538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The simulations have been per- formed on the Beowulf system of the Astrophysics Group at Florida State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This publication was made pos- sible through the support of an LSSTC Catalyst Fellowship to KAB funded through Grant 62192 from the John Tem- pleton Foundation to LSST Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The opinions ex- pressed in this publication are those of the author(s) and do not necessarily reflect the views of LSSTC or the John Templeton Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' ID acknowledges partial support by the Spanish project PID2021-123110NB-100 financed by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='13039/501100011033/FEDER/UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' LG ac- knowledges financial support from the Spanish Ministerio de Ciencia e Innovación (MCIN), the Agencia Estatal de In- vestigación (AEI) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='13039/501100011033, and the Euro- pean Social Fund (ESF) "Investing in your future" under the 2019 Ramón y Cajal program RYC2019-027683-I and the PID2020-115253GA-I00 HOSTFLOWS project, from Cen- tro Superior de Investigaciones Científicas (CSIC) under the PIE project 20215AT016, and the program Unidad de Exce- lencia María de Maeztu CEX2020-001058-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The research of YY is supported through a Bengier-Winslow-Robertson Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' SWJ and LAK acknowledge support by NASA grant JWST-GO-02072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='001 and NASA FINESST fellowship 80NSSC22K1599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' The data were obtained from the Mikulski Archive for Space Tele- scopes at the Space Telescope Science Institute, which is op- erated by the Association of Universities for Research in As- tronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=', under NASA contract NAS 5-03127 for JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' These observations are associated with program #2114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Facilities: JWST (LRS/MIRI), MAST (JWST) Software: HYDRA (Höflich 2003, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 1980, in Texas Workshop on Type I Supernovae, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Wheeler, 96–112 Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=', Hoeflich, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=', Baade, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' 2020, ApJ, 902, 46, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content='3847/1538-4357/aba759 SN 2021aefx: High-density Burning in SNe Ia 21 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' NEAR-INFRARED LINE IDENTIFICATIONS FROM MODEL 25 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfFgZ2/content/2301.03647v1.pdf'} +page_content=' Near-Infrared Model Line Identifications S 𝜆 [𝜇m ] Ion S 𝜆 [𝜇m ] Ion S 𝜆 [𝜇m ] Ion S 𝜆 [𝜇m ] Ion S 𝜆 [𝜇m ] Ion ∗ ∗ 2.' metadata={'source': 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Not. R. Astron. Soc. 000, 1–12 (?) +Printed 23 January 2023 +(MN LATEX style file v2.2) +Evolutionary implications of a magnetar interpretation for +GLEAM-X J162759.5–523504.3 +Arthur G. Suvorov1,2⋆ and Andrew Melatos3,4 +1Manly Astrophysics, 15/41-42 East Esplanade, Manly, NSW 2095, Australia +2Theoretical Astrophysics, Eberhard Karls University of T¨ubingen, T¨ubingen, D-72076, Germany +3School of Physics, University of Melbourne, Parkville VIC 3010, Australia +4ARC Centre of Excellence for Gravitational Wave Discovery (OzGrav), Hawthorn VIC 3122, Australia +Accepted ?. Received ?; in original form ? +ABSTRACT +The radio pulsar GLEAM-X J162759.5–523504.3 has an extremely long spin pe- +riod (P += 1091.17 s), and yet seemingly continues to spin down rapidly ( ˙P +< +1.2 × 10−9 ss−1). The magnetic field strength that is implied, if the source is a neu- +tron star undergoing magnetic dipole braking, could exceed 1016 G. This object may +therefore be the most magnetised neutron star observed to date. In this paper, a crit- +ical analysis of a magnetar interpretation for the source is provided. (i) A minimum +polar magnetic field strength of B ∼ 5 × 1015 G appears to be necessary for the star +to activate as a radio pulsar, based on conventional ‘death valley’ assumptions. (ii) +Back-extrapolation from magnetic braking and Hall-plastic-Ohm decay suggests that +a large angular momentum reservoir was available at birth to support intense field am- +plification. (iii) The observational absence of X-rays constrains the star’s field strength +and age, as the competition between heating from field decay and Urca cooling im- +plies a surface luminosity as a function of time. If the object is an isolated, young +(∼ 10 kyr) magnetar with a present-day field strength of B ≳ 1016 G, the upper limit +(≈ 1030 erg s−1) set on its thermal luminosity suggests it is cooling via a direct Urca +mechanism. +Key words: +stars: magnetars, magnetic fields, pulsars: GLEAM-X J162759.5– +523504.3 +1 +INTRODUCTION +Hurley-Walker et al. (2022) recently reported that observa- +tions, made between January and March of 2018 with the +Murchison Widefield Array (MWA) in the 72 − 231 MHz +band, revealed the presence of a pulsating Galactic source, +since named GLEAM-X J162759.5–523504.3 (henceforth +GLEAM-X J1627), which is 88 ± 1% linearly polarised. +Barycentric correction and alignment of the pulses estab- +lished a periodicity with a pulsar-like regularity, P += +1091.1690(5) s, and further that the source is slowing down +at a best-fit rate of ˙P = 6×10−10 s s−1. (Though we note the +data can only confidently assert that | ˙P| < 1.2×10−9 ss−1). +The characteristic (polar) magnetic field strength relevant +for a neutron star, B ≈ 6.4 × 1019� +P ˙P G (e.g., Ruder- +man & Sutherland 1975), is arguably in excess of 1016G. +Furthermore, since the pulse structure of the source varies +on ∼hour-long timescales in a way that is similar to what +⋆ arthur.suvorov@manlyastrophysics.org +is seen from known radio magnetars, Hurley-Walker et al. +(2022) offered the tantalising conclusion that the source is an +ultra-long period magnetar (see also Ronchi et al. 2022; Ek¸si +& S¸a¸smaz 2022; Gen¸cali, Ertan & Alpar 2022; Katz 2022). +This would likely make GLEAM-X J1627 the most magne- +tised neutron star observed to date1 (Olausen & Kaspi 2014; +Coti Zelati et al. 2018). +Followup searches by the Swift X-ray Telescope (XRT) +found no evidence for thermal or soft X-rays (Hurley- +Walker et al. 2022). Upper limits to the photon count +were placed which, depending on the spectral fit, imply +an upper limit to the flux. The strongest limit is placed +at 1.9 × 10−13 erg s−1cm−2 for the absorbed flux in the +0.3–10keV band, with a marginally lower value (≈ 1.5 × +10−13 erg s−1cm−2) applying for a blackbody fit at kT ∼ +1 A list of known magnetars and their properties is main- +tained +at +http://www.physics.mcgill.ca/~pulsar/magnetar/ +main.html; see also the Magnetar Outburst Online Catalogue +http://magnetars.ice.csic.es. +© ? RAS +arXiv:2301.08541v1 [astro-ph.HE] 20 Jan 2023 + +2 +A. G. Suvorov & A. Melatos +0.1 keV. Based on the greatest distance allowed by the +dispersion measure, dmax = 1.8 kpc, this gives LX ⩽ 7 × +1031 erg s−1 (Hurley-Walker et al. 2022). A deeper search +using the Chandra X-ray Observatory was carried out by +Rea et al. (2022), who placed the even stricter upper limit +LX ⩽ 2 × 1030 erg s−1, with the exact bound depending on +assumptions about the spectral shape. The latter upper limit +would generally be expected of persistent, thermal emis- +sions from a ≳ Myr-old magnetar (Thompson & Duncan +1996; Turolla et al. 2011; Olausen & Kaspi 2014). Should +GLEAM-X J1627 be a magnetar, its existence as a radio, +but not an X-ray, source has a number of interesting impli- +cations for emission physics and magnetic field evolution in +neutron stars, some of which we explore in this work. +It is generally put forth that electron-positron pair pro- +duction, likely occurring in magnetospheric ‘gaps’, is a nec- +essary ingredient to spark the radio emissions seen from pul- +sars (Goldreich & Julian 1969; Sturrock 1971; Ruderman & +Sutherland 1975) (though cf. Melrose, Rafat & Mastrano +2021). Depending on the topological properties of the stel- +lar magnetic field, a variety of different ‘death lines’, defined +through the threshold to generate the requisite pairs, com- +prise an overall ‘death valley’ (Chen & Ruderman 1993; Hi- +bschman & Arons 2001). Requiring that GLEAM-X J1627 +reside outside of the valley allows us to assess the validity of +a number of evolutionary scenarios. Having some idea about +what surface field structures are permissible for the object +‘today’, we can back-extrapolate from analytic or numeri- +cal simulations of Hall-plastic-Ohm decay in stellar crusts +(Lander & Gourgouliatos 2019; Gourgouliatos, De Grandis +& Igoshev 2022; Kojima, Kisaka & Fujisawa 2022) to see +what magnetic conditions at birth are indicated. +An intense magnetic field within the stellar core is ex- +pected to lead to ambipolar heating (Goldreich & Reiseneg- +ger 1992; Aguilera, Pons & Miralles 2008), as charged con- +stituents (e.g., protons) experience Lorentz forces that the +uncharged components (e.g., neutrons) do not, leading to a +kind of collisional friction that gradually heats up the star +at the expense of the magnetic energy. From models of core- +crust heat transfer (Potekhin et al. 2003; Turolla et al. 2011; +Vigan`o et al. 2013; Anzuini et al. 2022a), we can estimate +the surface luminosity implied by the competition between +ambipolar diffusion, mechanical dissipation, Joule heating, +and particle backflow against neutrino cooling (Beloborodov +& Li 2016). The absence of thermal X-rays may then hint at +an upper limit to the magnetic field strength, which can be +compared with the requirements set by the radio activation. +In this paper, we revisit the magnetar interpretation of +GLEAM-X J1627 in the context of death valley physics (Sec. +2.1), Hall-plastic-Ohm evolutions (Sec. 2.2), braking mecha- +nisms (Sec. 2.3), field amplifications at birth (Sec. 2.4), and +ambipolar heating (Sec. 3). The conclusions are summarised +in Sec. 4, emphasising that they depend on details of the +model and cannot be asserted strongly based on the limited +data at hand. Quantities written with numerical subscripts +are logarithmically normalised, e.g., B16 = B/1016 for field +strength B measured in G. +2 +MAGNETAR NATURE OF GLEAM-X J1627 +AND RADIO OBSERVATIONS +Between January and March 20182, 71 pulses from GLEAM- +X J1627 were detected by the MWA which, after alignment +and barycentric correction, revealed a Galactic source puls- +ing with period P = 1091.17 s. The best-fit value for the +period derivative is +˙P = 6 × 10−10 s s−1, though Hurley- +Walker et al. (2022) noted that their analysis cannot exclude +even larger values ˙P < 1.2 × 10−9 s s−1. For a neutron star +moment of inertia I0 ∼ 1045 g cm2, the spin-down luminos- +ity associated with the object is then ˙Esd ≈ 4π2I0 ˙P/P 3 ∼ +1028 erg s−1, which is several orders of magnitude lower than +the observed radio luminosity Lν ≈ 4 × 1031 erg s−1, esti- +mated assuming a best-fit distance of d = 1.3 ± 0.5 kpc. +This puzzling feature, which is unique to GLEAM-X J1627, +has prompted interest in a white dwarf interpretation for +the source, essentially to boost I0 (Loeb & Maoz 2022; Katz +2022). However, Erkut (2022) argue that the beaming angle +assumptions made by Hurley-Walker et al. (2022) may be in- +appropriate for an object with such a long spin period (see +also Szary et al. 2014; Szary, Melikidze & Gil 2015), and the +radio luminosity may in fact be closer to ≈ 3 × 1026 erg s−1 +– much lower than ˙Esd. +In the standard picture of magnetic-dipole braking (cf. +Sec. 2.3), the characteristic magnetic field strength for a +neutron star reads Bp ≈ 6.4 × 1019� +P ˙P G. The P- ˙P upper +limits therefore hint towards a huge magnetic energy. This +observation, combined with the high degree of linear polari- +sation in the pulses, led Hurley-Walker et al. (2022) to sug- +gest that GLEAM-X J1627 may be a magnetar. In this Sec- +tion, we examine this suggestion in the theoretical context +of radio emission mechanisms (Sec. 2.1), crustal magnetic +field evolution (Sec. 2.2), braking physics (Sec. 2.3), and nu- +merical simulations of birth properties (Sec. 2.4). The Swift +XRT observations of GLEAM-X J1627 are then discussed +in Sec. 3. +2.1 +Death valley +A neutron star crust provides a reservoir of free electrons +that are continuously accelerated into the magnetosphere by +induction-generated electric fields as the star spins. In ‘gap’ +regions where the Goldreich & Julian (1969) charge den- +sity is comparatively low, the electric field along magnetic +field lines can be sufficiently intense that photons emitted by +accelerated charges possess the requisite energy to pair pro- +duce. It is generally put forth that e+e− production is an es- +sential ingredient in the powering of coherent radio emissions +from neutron stars (Sturrock 1971; Ruderman & Sutherland +1975; Chen & Ruderman 1993; Hibschman & Arons 2001) +(cf. Melrose, Rafat & Mastrano 2021). Secondary charges +2 Hurley-Walker et al. (2022) note that while the MWA, over the +course of its eight years of operation, has accumulated around +≲ 200 hours of observing time within 15◦ of GLEAM-X J1627, +the data span many different array configurations, frequencies, +and observing modes. It is therefore difficult to formally deduce +the source duty cycle. Uncertainties notwithstanding, they argue +that a ∼ 2% duty cycle is likely, similar to the ∼ 5% cycle of the +radio-loud magnetar XTE J1810–197 (Eie et al. 2021). +© ? RAS, MNRAS 000, 1–12 + +A magnetar interpretation for GLEAM-X J1627 +3 +generated by curvature- or inverse-Compton-produced pho- +tons can themselves be accelerated and emit photons3, cul- +minating in a pair cascade. Beam instabilities, where free +energy associated with streaming motions is transferred to +plasma waves, then lead to radio emission (though again +cf. Melrose, Rafat & Mastrano 2021, who argue this picture +requires revision). +In this scenario, there is a maximum potential drop, +∆Vmax, which can be produced in the magnetosphere, viz. +∆Vmax ≈ 2π2BdR3 +⋆ +c2P 2 +, +(1) +for stellar radius R⋆, speed of light c, and polar dipole +strength Bd. This maximum must exceed that which is re- +quired for the pair production mechanism to activate. In a +curvature radiation and polar-gap scenario, this entails (Ru- +derman & Sutherland 1975) +�e∆Vmax +mec2 +�3 +ℏH +2mecR2c +Bp +BQED ≳ 1 +15, +(2) +where BQED = m2 +ec3/eℏ ≈ 4.4×1013 G is the Schwinger field +for electron mass and charge me and e, respectively, ℏ is the +reduced Planck constant, and H denotes the gap thickness. +Note that the numerical factor 1/15 in (2) depends weakly +on the angle made between the direction of photon prop- +agation and B; see the discussion around equation (19) in +Ruderman & Sutherland (1975) for more details. Moreover, +the polar field strength Bp may exceed the dipole value, Bd, +which is the relevant quantity at large radii. The curvature +radius of a magnetic field line, Rc, scales inversely with the +multipole order. Depending on the assumptions one places +on the magnetic configuration, most notably H, Rc, and +Bp/Bd, a variety of possible ‘death lines’ arise. +There are two main radii characteristic to the magne- +tosphere of an isolated object, being the stellar radius and +the light-cylinder radius, RLC = cP/2π. Chen & Ruderman +(1993) posit that depending on the magnetospheric twist, H +and Rc can assume a variety of values that are built from +these two radii, such as (R⋆RLC)1/2, R⋆(R⋆/RLC)1/2, and so +on. For polar-gaps, the thickness H may also scale with the +dimensionless ratio β = Bp/Bd, as multipoles can dominate +over the dipole component near the stellar surface. In real- +ity, a force-free magnetospheric model, though possibly with +some displacement currents, is necessary to determine the +gap size and curvature radius self-consistently for some field +geometry. Nevertheless, we can explore the various extrema +by taking the approximate scalings considered by Chen & +Ruderman (1993) and others. +Following Chen & Ruderman (1993) (though see also +Hibschman & Arons 2001; Szary, Melikidze & Gil 2015), +there are four types of polar-gap scenario that we consider: +(a) Pure central dipole. This is the simplest such model, +where +Bp += +Bd, +Rc += +(R⋆RLC)1/2 +and +H += +R⋆(R⋆/RLC)1/2. +The +required +field +strength +for +pair- +production is Bd,min = 2.2 × 1012(P 15/8R−19/8 +6 +) G. +3 Above hot polar caps with temperatures exceeding ∼ 106 K, +collisional interactions between photons could potentially also +produce abundant pairs through the Breit & Wheeler (1934) in- +teraction γγ → e+ + e− (Jones 2022). +Table 1. Minimum polar dipole field strengths required for +GLEAM-X J1627, assuming a death line according to the charac- +terisation given in the main text. The final column shows Bd,min +for a canonical radius R6 = 1, with the bracketed number corre- +sponding to a larger radius of 13 km. The second column indicates +the strength of the surface field relative to the dipole component, +which influences the cap thickness, H, and curvature radius, Rc. +Magnetic geometry +Bp/Bd +Bmin +d,16 (R6 = 1.3) +(a) Pure dipole +1 +101 (58.8) +(b) Twisted dipole +1 +2.30 (1.32) +(c) Starspot +2 +2.10 (1.21) +5 +1.88 (1.08) +10 +1.72 (0.99) +(d) Twisted multipoles +2 +0.279 (0.165) +5 +0.222 (0.131) +10 +0.187 (0.111) +(b) Twisted dipole. As above, though instead Rc = R⋆. We +find Bd,min = 2.7 × 1011(P 13/8R−17/8 +6 +) G. +(c) Starspot +configuration. +Numerical +simulations +of +crustal Hall drift tend to find that concentrated ‘mag- +netic +spots’ +emerge +near +the +polar-cap +(e.g., +Vigan`o +et al. 2013; Suvorov, Mastrano & Geppert 2016), where +Bp +≫ +Bd. +Taking +Rc += +R⋆ +and +a +reduced +cap +size H += β−1/2R⋆(R⋆/RLC)1/2 yields Bd,min += 2.7 × +1011(β−1/8P 13/8R−17/8 +6 +) G. +(d) Twisted multipoles. Similar to case (c), though with +maximum pitch angle between the magnetic field and the +direction of emitted photons, sin θ ≈ H/Rc (and thus +H +≈ Rc += R⋆). Effectively, one assumes the magne- +tosphere is so twisted that a curvature-radiation photon, +emitted almost parallel to B, crosses another part of the +cap’s open field line bundle at a large angle. This gives +Bd,min = 9.2 × 1010(β−1/4P 3/2R−2 +6 ) G. +Table 1 displays the minimum polar dipole strength +Bd,min, in the context of the death lines (a)–(d) described +above, for a range of surface-to-dipole ratios, Bp/Bd, and a +canonical radius R⋆ = 10 km, such that GLEAM-X J1627 +can activate as a radio pulsar. Taking instead a stellar ra- +dius of 13 km allows for an easier switch on, as shown by the +numbers in parentheses. Rows show different values for the +relative strength of multipoles, which influence the curva- +ture radius and gap thickness, as described above. Although +lines (b) through (d) imply high multipole orders (ℓ ≳ 102) +when interpreted via the numerical simulations of Asseo & +Khechinashvili (2002), for instance, magnetohydrodynamic +evolutions in proto-magnetars indicate that the generation +of high-order multipoles in the interior is a generic quality +of strongly-magnetised systems (Kiuchi et al. 2018; Lander +et al. 2021). Lander et al. (2021) found that truncating their +numerical output to harmonic expansions with ℓmax < 32 led +to sizeable inaccuracies in the inferred field strength (see also +Sec. 2.2). Fallback accretion onto the proto-star, or later in +life, can also twist field lines near the stellar surface (Melatos +& Priymak 2014; Suvorov & Melatos 2020). +We emphasise that the above models, while phenomeno- +logical, represent the extrema that could be expected for a +given activation mechanism. It is unlikely that any given +© ? RAS, MNRAS 000, 1–12 + +4 +A. G. Suvorov & A. Melatos +line applies to the entirety of the neutron star population at +all times. For example, magnetospheric twists (lines b and +d) and starspots (line c) are dynamical in nature and sub- +ject to diffusion, implying that the radius of curvature is +a function of time. Regarding magnetars in particular, Be- +loborodov (2009) suggests that their radio activation may +be related to quake activity in the crust, where overstressed +zones fracture and flow plastically, dragging the magnetic +footpoints with them and pumping a current (‘j-bundle’) +into the magnetosphere (see also Beloborodov & Thomp- +son 2007). Overshearing events may also be expected from +spindown (Baym & Pines 1971), which tends to be faster +in magnetars. Magnetospheric twists can survive on ≳ year- +long timescales (Parfrey, Beloborodov & Hui 2013), until the +field lines relax to the pre-twist (or some other) equilibrium. +During a twisting episode one may expect that either lines +(b) or (d) apply, after which a dipole or starspot configura- +tion is reinstated (lines a or c) depending on the surface-field +multipolarity. This may help to explain why certain magne- +tars are radio loud while others are not, depending on the +waiting time between twist injections (see also Morozova, +Ahmedov & Zanotti 2012). +Detailed simulations of pair cascades in magnetar envi- +ronments were carried out by Medin & Lai (2010) to ex- +amine whether the polar gap story, discussed above, ap- +plies in super-strong fields (see also Thompson & Duncan +1996; Baring & Harding 1998; Beloborodov & Thompson +2007). Although they find the cascade proceeds differently +for fields above and below BQED, mostly because of the sup- +pression of synchrotron emission in strong fields, the over- +all multiplicity λ is relatively insensitive to B. The energy +spectrum of electron-initiated cascades depends mostly on +the polar-cap voltage, and hence the spin period, and not +B alone. The critical multiplicity, necessary for radio acti- +vation, that they find in the strong-field case is λdeath ≈ +1.5 × 107(Bp/1012G)−1/6(Rc/108cm)2/3 (see their Sec. 4.2). +For untwisted dipoles this implies Bd,min ∝ P 2, similar to +line (a) though marginally steeper. For highly-twisted fields +with Rc ∼ R, one recovers lines qualitatively similar to ei- +ther (b) or (d) from their results, depending on the field +topology. Therefore, although the overall slope of death lines +in B-P space may vary depending on how the cascade pro- +ceeds, the death valley defined as the area spanned by lines +(a) through (d) is a reasonable approximation for the valley +extrema, even for magnetars4. +Figure 1 shows death lines in the context of the wider +neutron star population. Although noting the caveats dis- +cussed above, we see that all known, pulsating objects lie +above the overall valley, with the possible exception of +GLEAM-X J1627 (shown by a black star). The vertical axis +shows the dipolar field strength of known objects, which is +relatively uncertain: one requires a braking model to esti- +mate this quantity. We assume that the standard magneto- +dipole picture [equation (5) with n = 3] applies, though +4 Given the high degree of nulling and that only bright and vari- +able single pulses were detected from GLEAM-X J1627, it may be +that the radio activity is not attributable to traditional cascades. +Different death lines altogether may apply, such as the fast radio +burst death lines described by Wadiasingh et al. (2020), which +can be satisfied with somewhat weaker fields (see their equation +9). +allow for uncertainties in the inclination angle, π/4 ⩽ α ⩽ +π/2, and the stellar radius, 10 ⩽ R⋆/km ⩽ 13. Note that +polarisation data from radio pulsars indicate that α spans +an even larger range (e.g., Rankin 1990). Therefore, indi- +vidual death lines have some width, and pulsar positions on +the diagram have some uncertainty (see Tab. 1). Although +the mean of line (a) cuts right through the middle of the +population, inclinations tending towards alignment weaken +the intrinsic torque, and thus predict a larger B for a given +˙P. It was argued by Contopoulos & Spitkovsky (2006) that +radio pulsars may evolve towards an aligned configuration +(α → 0), and hence even line (a) could be sufficient in most +cases. Evolution towards alignment is also observed in 3D +magnetospheric simulations (Philippov, Tchekhovskoy & Li +2014) (though cf. Lander & Jones 2018). For many systems +however it is likely that dynamical phenomena, such as mag- +netospheric twist injections via magnetically- (Beloborodov +2009) or spindown-induced (Baym & Pines 1971) quakes, +or starspot formation (Zhang, Gil & Dyks 2007), play a +role in pair cascade phenomena. Lines (b) through (d) may +therefore only apply sporadically over ∼ year-long timescales +(Parfrey, Beloborodov & Hui 2013). +In the context of Fig. 1, we see that the pure dipole +[model (a)] is unable to explain the radio switch-on of +GLEAM-X J1627 unless the polar field takes on super- +virial values, Bp ≳ 1018 G. This is in contrast with all +(other known) radio-loud magnetars, namely PSR J1745– +2900, PSR J1622–4950, XTE J1810–197, 1E 1547.0–5408, +Swift J1818.0-1607, PSR J1119–6127 and SGR 1935+2154 +(red stars). This line fails to explain the pulsar population +at large though, cutting through the middle of the diagram. +Only in the case of a highly-twisted configuration [model +(d)] can the local field of GLEAM-X J1627 assume values +Bp ≲ 1016 G. For these models, the minimum required for +the surface field is still greater than the maximum polar field +amongst all other 31 known magnetars, with the runner up +being SGR 1806–20 (Olausen & Kaspi 2014), an extraor- +dinarily bright and young (< kyr) burster, which boasts a +polar field strength Bp ≈ 4 × 1015 G. +The uncertainty implied by the final column of Tab. 1 +is a lower limit, as consideration of outer gap models (Chen +& Ruderman 1993), thermionic emissions (Szary, Melikidze +& Gil 2015), and general-relativistic corrections (Hibschman +& Arons 2001) can also adjust the voltage drop. Outer-gap +models, however, tend to fare worse. For example, for the +partially-inclined, outer magnetosphere accelerator model +[Eq. (27) of Chen & Ruderman (1993)], the relevant death +line, 5 log Bp − 12 log P ≈ 69.5, demands a magnetic field +for GLEAM-X J1627 that exceeds the virial limit. Szary, +Melikidze & Gil (2015) argued, in the context of a partially- +screened gap, that the polar cap must be below some crit- +ical B-dependent temperature, else thermionic emissions +effectively screen the acceleration potential (such consid- +erations are pertinent to the observational absence of X- +ray emissions; see Sec. 3). Similarly, the general-relativistic +Lense-Thirring corrections discussed by Hibschman & Arons +(2001) may be important for GLEAM-X J1627, despite its +long spin period, because the Goldreich & Julian (1969) +plasma density and the Lense-Thirring frequency both scale +linearly with the rotation rate. Na¨ıvely applying the ‘low- +altitude’ estimate for the pair multiplicity computed by Hib- +schman & Arons (2001), which includes Lense-Thirring pre- +© ? RAS, MNRAS 000, 1–12 + +A magnetar interpretation for GLEAM-X J1627 +5 +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ ⋄ +⋄ +⋄ +⋄⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ ⋄ +⋄ +⋄⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ +⋄ 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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  + + + + + + + +  + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  + + + + + + + + + + + + + + + + + + + + +★★ +★★★ +★ +★ ★ +★ +★ - Radio Magnetars +⋄ - Radio Pulsars +★ - GLEAM-X J1627 + - (R-)Quiet Magnetars + - Binary member +(d) +(c) +(b) +(a) +0.001 +0.010 +0.100 +1 +10 +100 +1000 +108 +1010 +1012 +1014 +1016 +Spin period P (s) +Polar dipole strength Bd(G) +Figure 1. Bd − P diagram overlaid with death ‘lines’ (a)–(d), as shown by the coloured curves (see plot legends). Each curve comes +with some thickness because we allow for uncertainties in the stellar radius, 10 ⩽ R⋆/km ⩽ 13, and the polar-to-dipole field strength +ratio, 1 ⩽ β ⩽ 10. Overlaid are known objects, with available ˙P measurements, from the ATNF pulsar catalogue (http://www.atnf. +csiro.au/research/pulsar/psrcat; Manchester et al. 2005) with ‘ordinary’ radio pulsars shown by black diamonds, those in binaries +with black squares (for which B-field estimates are even more uncertain as they depend on accretion assumptions), radio-loud magnetars +with red stars, and radio-quiet magnetars with blue circles. The (vacuum dipole) upper limit for GLEAM-X J1627 is indicated with a +black star. The Bd values for other pulsars come from the standard dipole-braking formula (5) with n = 3 for a range of obliquities +(π/4 ⩽ α ⩽ π/2); these variations, together with those on R⋆ and timing errors on ˙P, imply some uncertainty on the dipole strength. For +magnetars (except J1119), mean values from the McGill catalogue are used (Olausen & Kaspi 2014). In principle, all objects lie above +the overall valley, defined as the area between the top of line (a) and the bottom of line (d), with the possible exception of GLEAM-X +J1627, unless it has a highly-twisted magnetosphere (Rc ∼ R⋆) with a polar dipole field strength exceeding ∼ 1015G. +cession and is appropriate when Bp/(1012 G) ≳ P/(1 s) [see +their Eq. (69)], we obtain a minimum polar field strength +Bp,min ≈ 2 × 1016 G, comparable to lines (b) and (c). The +Hibschman & Arons (2001) models however invoke cap tem- +peratures set by backflowing positrons, which may be unre- +alistically low for GLEAM-X J1627 because internal heating +driven by field decay is likely to be non-negligible (see Sec. +3); thermal transport simulations would be necessary to self- +consistently assess the valley structure in this case. +2.2 +Hall-plastic-Ohm decay +A simplified picture of the neutron star crust is that of a +rigid, ion lattice strewn with mobile electrons. The latter +carry a current as they flow relative to the ions, gradually +advecting the field lines that thread the crust. This pro- +cess of Hall drift, while conserving magnetic energy, can act +to accelerate Ohmic decay through a sequence of cascades +to smaller-scale magnetic structures, possibly aided further +by thermoelectric effects (see Gourgouliatos, De Grandis +& Igoshev 2022, for a review). The Hall timescale obeys +τHall ∝ B−1, and thus magnetar crusts are particularly +prone to field decay. Depending on the initial conditions +however, the system may enter into an ‘attractor’ state +where the Hall term vanishes (Gourgouliatos & Cumming +2014). Although we will not consider this complication fur- +ther, a Hall-stalled evolution may help GLEAM-X J1627 to +maintain a strong field while cooling quickly as it ages. +As magnetic gradients form, Maxwell stresses are ex- +erted on the crust. For magnetar-like field strengths B ≳ +1015 G, the crust may not be sufficiently malleable to ab- +sorb these stresses, and rather a crustal failure may occur +(Duncan 1998; Lander et al. 2015). Crustquakes are popu- +lar models for the progenitors of magnetar outbursts, such +as giant flares (e.g., G¨oˇg¨u¸s et al. 2000) or fast radio bursts +(e.g., Suvorov & Kokkotas 2019). Once the crust experi- +ences a failure however, it is unlikely to ‘heal’ immediately +and rather may enter a state of azimuthal shearing termed +plastic flow (Beloborodov & Levin 2014; Lander & Gourgou- +liatos 2019; Kojima, Kisaka & Fujisawa 2022). Plastic flow +is generally a dissipative process, and thus depending on the +‘plastic viscosity’, the Hall effect may be enhanced, implying +© ? RAS, MNRAS 000, 1–12 + +6 +A. G. Suvorov & A. Melatos +that numerical Hall-Ohm (as opposed to Hall-plastic-Ohm) +investigations underestimate the degree of field decay. On +the other hand, plastic flows can move against the existing +flow of the electron fluid (Gourgouliatos & Lander 2021), +and thus inhibit magnetic dissipation by counteracting the +formation of the small-scale (i.e., highly multipolar) mag- +netic substructures most susceptible to Ohmic decay. The +density-dependent, and hence radially-stratified, nature of +the electron fluid flow also facilitates the growth of a toroidal +field, making an investigation of a realistic Hall-plastic-Ohm +system a challenging task. +The evolution of the crustal magnetic field B is de- +scribed by the induction equation +0 =∂B +∂t + ∇ × +� +c +4πene (∇ × B) × B +− vpl × B + c2 +4πσ ∇ × B +� +, +(3) +for electron number density 1034 ≲ ne/cm−3 ≲ 1036 and +conductivity 1016 ≲ σ/s−1 ≲ 1024, where vpl denotes the +plastic flow velocity. The lower limit for the electrical con- +ductivity applies to the crust-magnetosphere interface, while +the latter is appropriate for the inner crust (Akg¨un et al. +2018). A proper description for vpl, including a determina- +tion of characteristic plastic speeds, requires an additional +equation of motion, typically set by the requirement that +a Stokes flow is induced in regions of excess stress (deter- +mined, e.g., by the von Mises criterion; Lander et al. 2015). +Following Aguilera, Pons & Miralles (2008), we construct an +approximate model by replacing the gradient operator with +the inverse of a relevant lengthscale, L, yielding (see also +Lander 2022) +dB +dt = − B +B0 +B +τHall + vplB +L +− +B +τOhm , +(4) +with τHall = 4πeneL2/cB0 and τOhm = 4πσL2/c2 read off +from (3), with small-scale structures dominating the choice +of L. Equation (4), which is subject to the initial condi- +tion B(0) = B0, reduces to the phenomenological Hall-Ohm +model of Aguilera, Pons & Miralles (2008) when vpl = 0. +In the simulations of Lander & Gourgouliatos (2019), it +was found that larger B0 values lead to swifter plastic flows, +and more precisely that doubling B0 leads to an approxi- +mately 3-fold increase in vpl. For magnetar-level fields and +low plastic viscosities, these authors (see also Gourgouliatos +& Lander 2021; Gourgouliatos, De Grandis & Igoshev 2022) +found that vpl can approach a few hundred cm per year +in regions where field lines are particularly tangled. How- +ever, slower plastic speeds emerge in the bulk of the crust, +and no flow at all occurs in unstressed regions. As we have +washed out all spatial dependencies in building relation (4), +the flow is nominally non-zero everywhere, rather than only +in regions localised around failures. We thus consider instead +spatially-averaged speeds of vpl ≲ 40 cm yr−1 for cases with +ultra-strong fields. Note that if vpl is negative (i.e., if one +takes ∇ → −1/L rather than ∇ → 1/L), plastic flow in- +stead accelerates field decay; such cases have been observed +in the studies cited above, depending on the plastic viscosity. +Figure 2 shows solutions to equation (4) for several ini- +tial field strengths 1 ⩽ B16 ⩽ 50, in both the Hall-Ohm +(Aguilera, Pons & Miralles 2008, solid curves) and Hall- +plastic-Ohm (dashed curves) cases, where the plastic flow +B0=1016G +B0=51016G +B0=1017G +B0=51017G +vpl=0.5cm/yr +vpl=3.8cm/yr +vpl=7.5cm/yr +vpl=38cm/yr +(d): Bp,min=41015G +0.01 +0.10 +1 +10 +100 +1014 +1015 +1016 +1017 +Time (kyr) +Bp (G) +Figure 2. Evolution of the polar field strength, Bp(t), given as a +solution to equation (4) for both Hall-Ohm (vpl = 0, solid curves) +and Hall-plastic-Ohm (vpl ̸= 0, dashed curves) evolutions, for sev- +eral birth field strengths and plastic velocities (see colour-coded +legends). The dotted, horizontal line shows the minimum field +strength set by the type (d) death lines with Bp/Bd ∼ 2 consid- +ered in Sec. 2.1. +velocity is chosen to scale with B in the manner described +above. To provide an optimistic but realistic5 scenario, we +set L = 105 cm, ne = 1036, and σ = 1024 s−1; smaller val- +ues lead to faster decays. When vpl ⩽ 0, the field enters a +state of rapid decay after ∼ 1 kyr, reducing by an order of +magnitude after only ≈ 2 kyr in the ultra-strong case with +B0 = 5 × 1017 G. The dotted line illustrates the minimum +field strength required to fulfil the death valley requirements +discussed in Sec. 2.1. We remind the reader that even if the +dipole field is of order Bd ∼ 1015 G, the surface field implied +by the death valley constraints is of order ≳ 4 × 1015 G; see +Tab. 1. +Demanding that Bp ≳ 4×1015 G at present implies that +the system can be at most ∼ 20 kyr old independently of the +birth field strength if plastic flow is ignored, because one has +τHall ∝ B−1 +0 . Such a conclusion may be in tension with the +observed spin period of GLEAM-X J1627 (see Sec. 2.3), un- +less the star underwent a period of extreme spin-down early +in its life (from, e.g., propellering fallback material shortly +after birth, as discussed by Ronchi et al. 2022; Gen¸cali, Er- +tan & Alpar 2022). Including a sufficiently rapid plastic flow, +vpl ≳ 30 cm yr−1, however stalls the impact of the Hall ef- +fect (Gourgouliatos & Lander 2021), allowing the field to +decay only on the true Ohmic timescale, τOhm ≫ 102 kyr. +In this case, field strengths of order ≳ 5 × 1015 G can be +maintained over relatively long timescales if B0 ∼ 1017 G. +2.3 +Braking mechanism +Neutron stars in isolation spin down gradually as electro- +magnetic and gravitational torques are applied, the mag- +nitude of which can be phenomenologically quantified in +terms of a braking index, n. For a centered dipole that never +5 Note that, because |∇B|/B ∝ ℓ−1 for a general ℓ-pole, if one +were to assume a purely dipole field for all t, a longer lengthscale +L ≲ R⋆ could be justified, which would extend the Hall time. Such +an assumption would, however, be inconsistent with the twisted +surface configurations studied for death valleys in Sec. 2.1. +© ? RAS, MNRAS 000, 1–12 + +A magnetar interpretation for GLEAM-X J1627 +7 +decays one has n = 3, while for a general ℓ-pole we have +n = 2ℓ + 1. Leading-order contributions from gravitational +radiation, being quadrupolar, also give n = 5, though with +a different prefactor. Values n < 3 are also possible for an +oblique and/or precessing rotator (Melatos 1997, 1999), or in +cases where particle outflows dominate the spin-down torque +(Harding, Contopoulos & Kazanas 1999; Thompson et al. +2000). It is therefore useful to consider an evolution with an +arbitrary braking index. +The spin evolution of an inclined rotator in vacuum can +be described by (e.g., Manchester & Taylor 1977) +˙P = (2π)n−1 B2 +p sin2 αP 2−nR3+n +⋆ +6I0cn +, +(5) +for moment of inertia I0 and magnetic inclination angle α. +The quantity n in (5) represents the observational brak- +ing index only if Bp is constant, though in the absence +of ¨P data we treat it as phenomenological. We adopt the +general-relativistic Tolman VII equation of state, for which +I0 = 0.38M⋆R2 +⋆ (Lattimer & Prakash 2001) for stellar mass +M⋆. Equation (5) provides two useful pieces of information. +Firstly, the present-day observations of P and +˙P provide +an estimate for Bp for a given braking index. Secondly, by +solving equation (5) for some (time-dependent) choices of +n and Bp, one can infer the age of the system. Given that +we anticipate the object was born rapidly rotating so as to +explain its large field strength (see Sec. 2.4), its present- +day period must far exceed its birth period P0, though age +(τ) estimates from equation (5) are insensitive to P(0) for +P(0) ≪ P(τ). Note that magnetospheric (Spitkovsky 2006; +Philippov, Tchekhovskoy & Li 2014), spheroidal, general rel- +ativistic, or offset corrections (P´etri 2022) can be accounted +for in the above to adjust the effective Bp value; one ob- +tains a hybrid Spitkovsky (2006) formula, for example, by +replacing B2 +p sin2 α in expression (5) with B2 +p(1 + sin2 α). +We solve equation (5) simultaneously with the volume- +averaged induction equation (4) for several values of ˙P(τ) ⩽ +1.2 × 10−9ss−1 (Hurley-Walker et al. 2022) assuming an or- +thogonal rotator, α = π/2. We fix n by demanding Bp(τ) = +5 × 1015 G, as it is difficult to explain the present-day radio +switch on if the field is weaker (see Fig. 1, keeping in mind +the caveats noted in Sec. 2.1). The three a priori free pa- +rameters, namely B0, τ, and n, are uniquely determined by +the specified values of ˙P(τ), Bp(τ), and P(τ). Solutions are +built through a shooting method: a set of initial conditions +are iteratively determined such that there exists an age τ +for which the aforementioned conditions are met. +Figure 3 shows the evolutions of the spin period (top +panel) and the polar magnetic field (bottom) for cases where +plastic flow is ignored. Even for a relatively large range of the +present-day period derivative, 6.0 × 10−11 ⩽ ˙P(τ)/(ss−1) ⩽ +1.2×10−9, the evolutions proceed in a similar manner. This +occurs because we require that the present-day B field is still +strong, Bp(τ) = 5 × 1015G, so as to accommodate the death +valley minima discussed in Sec. 2.1. In the run with ˙P(τ) = +6.0 × 10−11 ss−1, for example, the birth field strength must +exceed 1017G so that it can survive long enough (until τ = +12kyr) to ensure that the present-day switch-on minimum is +met, which implies greater spindown during early times t ≪ +τ. As such, even if a factor ≳ 10 weaker ˙P(τ) is assumed than +the best-fit value reported by Hurley-Walker et al. (2022), +predictions for the age are quantitatively similar in cases +n=2.65, P = 1.210-9ss-1 +n=2.69, P = 6.010-10ss-1 +n=2.84, P = 6.010-11ss-1 +τ=12kyr +τ=9kyr +τ=7kyr +PGLEAM-X J1627 = 1091 s +0.01 +0.10 +1 +10 +100 +50 +100 +500 +1000 +Time (kyr) +Spin period (s) +n=2.65, P = 1.210-9ss-1 +n=2.69, P = 6.010-10ss-1 +n=2.84, P = 6.010-11ss-1 +τ=12kyr +τ=9kyr +τ=7kyr +B(τ)=51015G +0.01 +0.10 +1 +10 +100 +5 ×1014 +1 ×1015 +5 ×1015 +1 ×1016 +5 ×1016 +1 ×1017 +Time (kyr) +Polar field strength (G) +Figure 3. Solutions to equation (5) for the rotational period +P(t) (top panel), assuming a time-dependent magnetic field Bp(t) +whose evolution is governed by (4) (bottom panel), for a variety +of +˙P(τ) values (see plot legends). The plastic velocity is set to +zero in these examples. The age, braking index, and birth field +strengths are set by the conditions that P(τ) = 1091.17s and +Bp(τ) = 5 × 1015G for some given value of ˙P(τ). +where plastic flow and other torques are inactive (cf. Ronchi +et al. 2022; Gen¸cali, Ertan & Alpar 2022). +By contrast, evolutions carried out for vpl ̸= 0 are shown +in Fig. 4. In these cases, ˙P(τ) makes a significant difference +for the age prediction: for ˙P(τ) = 1.2 × 10−9 ss−1 we find +τ = 8kyr, while for the smaller value ˙P(τ) = 6.0×10−11 ss−1 +the age prediction increases to τ = 47kyr. This is because +plastic flow stalls field decay (see Fig. 2), allowing the star +to match Bp(τ) = 5×1015G without having to be born with +a field exceeding 1017G. In this way, spindown is slower in +the early stages and the star can be older. Increasing the +plastic velocity can increase the age further; in the limit +that vpl → ∞ (or τOhm → ∞) the field does not decay at +all, and the age is simply given by the characteristic value +τ ∝ P(τ)/ ˙P(τ), which can be arbitrarily large if ˙P tends +towards zero. +2.4 +Birth conditions: field amplification +To a large degree, it remains an open question as to how +magnetars acquire their intense fields, especially large-scale +dipoles. The saturation amplitude of the core field in the case +of dynamo activity shortly after birth could reach ≲ 1016 G +for convective heat fluxes of order ≳ 1039 erg cm−2s−1 +© ? RAS, MNRAS 000, 1–12 + +8 +A. G. Suvorov & A. Melatos +n=2.65, P = 1.210-9ss-1 +n=2.69, P = 6.010-10ss-1 +n=2.84, P = 6.010-11ss-1 +τ=8kyr +τ=10kyr +τ=47kyr +PGLEAM-X J1627 = 1091 s +0.01 +0.10 +1 +10 +100 +50 +100 +500 +1000 +Time (kyr) +Spin period (s) +n=2.65, P = 1.210-9ss-1 +n=2.69, P = 6.010-10ss-1 +n=2.84, P = 6.010-11ss-1 +τ=47kyr +τ=10kyr +τ=8kyr +B(τ)=51015G +0.01 +0.10 +1 +10 +100 +1 ×1015 +5 ×1015 +1 ×1016 +5 ×1016 +1 ×1017 +Time (kyr) +Polar field strength (G) +Figure 4. Similar to Fig. 3, though with a non-zero plastic ve- +locity whose value is set by the scaling discussed in Sec. 2.2, i.e., +doubling B relative to some fixed value implies a 3-fold increase +in vpl, where we set vpl = 0.5 cm yr−1 for B0 = 1016G. +(Thompson & Duncan 1993). Provided that an ‘inverse cas- +cade’ can operate, where energy from turbulent patches is +transferred into a large-scale dipole (cf. Guilet et al. 2017; +Raynaud et al. 2020), birth fields of this magnitude are suf- +ficient for all of the known Galactic magnetars. +Mechanisms beyond dynamo activity can amplify a +magnetic field. In particular, the Kelvin-Helmholtz and +magneto-rotational instabilities can potentially lead to satu- +ration magnetic energies of order Umag ∼ 1051 erg provided +that the star is born with a (sub-)millisecond period (Kiuchi +et al. 2018; Ciolfi 2020a,b; Shibata, Fujibayashi & Sekiguchi +2021). We stress however that numerical studies reporting +such large magnetic energies do so in the context of merger +remnants, which generally possess more angular momentum +and seed magnetic fluxes than stars born from core-collapse. +Regardless, magnetic energies of this order imply an up- +per limit to the volume-averaged magnetic field strength at +birth, viz. +⟨B⟩max ≈ 7.7 × 1016 +� +Umag +1051 erg +�1/2 � +R⋆ +10 km +�−3/2 +G. (6) +If the core field is at least as strong as the surface one +(see also Sec. 3), expression (6) implies that Bp ≲ ⟨B⟩max. +The numerical simulations referenced above therefore sug- +gest it is difficult to justify values exceeding Bp(t = 0) ∼ +1017 G (though cf. Suvorov & Glampedakis 2022), even if +toroidal fields (Glampedakis & Lasky 2015) or intense mag- +netic spots (Vigan`o et al. 2013; Suvorov, Mastrano & Gep- +pert 2016) are localised in the crust. +3 +IS THE MAGNETAR HYPOTHESIS +EXCLUDED BY THE ABSENCE OF +X-RAYS? +Follow-up searches were carried out with the Swift X- +ray Telescope for 2 ks. An upper limit of FX < 1.9 × +10−13 erg s−1cm−2, is found for the flux in the 0.3–10keV +band, with FX < 1.5 × 10−13 erg s−1cm−2 applying instead +for a blackbody fit6 at kT ∼ 0.1 keV. Based on the greatest +distance allowed by the dispersion measure, dmax = 1.8 kpc, +this gives LX ⩽ 7 × 1031 erg s−1 for the X-ray luminosity +(Hurley-Walker et al. 2022). A followup search conducted by +Rea et al. (2022) implies an even tighter upper-limit for this +dmax, LX ⩽ 2×1030 erg s−1. In this section, we review mod- +els of thermal regulation in magnetars as a means to predict +the surface temperature as a function of field strength (Sec. +3.1), which is quantitatively applied to GLEAM-X J1627 in +Sec. 3.2. +3.1 +Heating and cooling +The absence of thermal X-rays in particular poses a chal- +lenge to the magnetar interpretation of GLEAM-X J1627: +the magnetised electron-proton plasma in the core experi- +ences friction with the approximately static neutron fluid, +gradually heating up the star while depleting magnetic en- +ergy (Goldreich & Reisenegger 1992). +Ambipolar heating, which sets a floor value to the tem- +perature for a given age, is counteracted by neutrino cooling +(Turolla et al. 2011; Ho, Glampedakis & Andersson 2012; +Vigan`o et al. 2013; Anzuini & Melatos 2021; Anzuini et al. +2022a). There is, therefore, a quasi-static balance tempera- +ture, Tbal, set by matching the (time-dependent) heating and +cooling rates, which generally must be several times 108 K +to explain observations from active magnetars (Thompson +& Duncan 1996; Beloborodov & Li 2016). +Performing a volume average, the core temperature evo- +lution can be approximately described by the first law of +thermodynamics, +CV dTcore +dt += ˙QB − ˙Qν, +(7) +for +heat +capacity +CV +≈ +2 +× +1020(Tcore/109K)(ρ/ρnuc)1/3 erg K−1cm−3 (Beloborodov & +Li 2016). Here, ˙QB and ˙Qν are the heating and cooling rates +provided by magnetic field decay and neutrino emission, +respectively. The quantity ρnuc is the nuclear saturation +density, which may be exceeded in the core of a particularly +heavy neutron star or if the equation of state (EOS) is +soft. For the Akmal, Pandharipande & Ravenhall (1998) +6 Note that, in general, power-law components and not just one +or more blackbodies are also needed to fit magnetar spectra, see +Table 2 in Coti Zelati et al. (2018). For 4U 0142+61, for example, +blackbody emissions represent ∼ 25% of the total X-ray power +(Rea et al. 2007). Spectral complications can be accounted for +crudely in the models here via the efficiency parameter ϵ intro- +duced below. +© ? RAS, MNRAS 000, 1–12 + +A magnetar interpretation for GLEAM-X J1627 +9 +EOS [which passes constraints from GW170817 (Abbott +et al. 2018) and can accommodate the heaviest pulsar +observed to date, PSR J0740+6620, with M = 2.08+0.07 +−0.07M⊙ +(Fonseca et al. 2021)], a star of mass 1.39M⊙ has a central +density ρc = 9 × 1014 g cm−3 ≈ 3.2ρnuc. This increases to +ρc = 1.1 × 1015 g cm−3 for a 1.66M⊙ star. +Following Beloborodov & Li (2016) and others, the +two main cooling mechanisms we consider are the modified +(mUrca) and the fast, direct Urca (dUrca) mechanisms; the +former is thought to be the dominant neutrino mechanism in +(non-superfluid) nucleon matter (ρ ≲ 2ρnuc; Yakovlev et al. +2002), while the latter may activate in the core of particular +dense stars (ρ ∼ 4ρnuc; Lattimer et al. 1991). The presence +of hyperons may reduce fast cooling thresholds (Anzuini & +Melatos 2021; Anzuini et al. 2022a). +The mUrca cooling rate can be approximated by +(Friman & Maxwell 1979) +˙QM +ν ≈ 7×1020 +� Tcore +109K +�8 � ρ +ρnuc +�2/3 +RM erg s−1cm−3, (8) +where RM ⩽ 1 is a suppression factor relevant if either pro- +tons or neutrons are superfluid, whereupon the breaking of +Cooper pairs instead becomes the dominant cooling mech- +anism at densities ρ ∼ ρnuc (e.g., Page et al. 2009). We +henceforth ignore such complications in our phenomenologi- +cal heating model (7), though these should be considered in +realistic magnetothermal modelling if the core temperature +drops below the superfluidity onset value 1 ≲ Tcrit/108K ≲ +10 (Potekhin, Pons & Page 2015). The dUrca cooling rate +is given by (Lattimer et al. 1991) +˙QD +ν ≈ 1027 +� Tcore +109K +�6 +erg s−1cm−3, +(9) +which exceeds (8) by several orders of magnitude for tem- +peratures in the range of interest. +The rate of heating, provided by ambipolar diffusion, +can be estimated through (Beloborodov & Li 2016) +˙QB ≈ τpn +ρp +� B2 +4πL +�2 +, +(10) +for core field strength B which varies over lengthscale L, +where 1/τpn denotes the rate of p-n collisions per proton (ig- +noring core exotica), given by (Yakovlev & Shalybkov 1990) +τ −1 +pn ≈ 4.7 × 1018 +� Tcore +109K +�9 � ρ +ρnuc +�−1/3 +s−1. +(11) +In the simplified model (7), a magnetar, born with tem- +perature T0 ≲ 1011 K, reaches a quasi-static balance temper- +ature Tbal (i.e., dT/dt = 0) after ≳ 10 years (even less with +dUrca), where the temperature remains until field decay sets +in (∼kyr for B ∼ 1016G). Assuming a present-day core field +of ∼ 1016G, these balance temperatures read +T M +bal ≈ 8 × 108 +�B2 +16 +L5 +�1/5 � ρ +ρnuc +�−7/30 +K, +(12) +for mUrca, and +T D +bal ≈ 1.3 × 108 +�B2 +16 +L5 +�1/4 � ρ +ρnuc +�−1/12 +K, +(13) +for dUrca. +Core-crust thermal transport depends primarily on the +chemical composition of the stellar envelope and the mag- +netic stratification, which influence the photon opacity (Tsu- +ruta et al. 1972). Given a surface temperature Ts, the flux +Fs = σSBT 4 +s , +(14) +for +Stefan-Boltzmann +constant +σSB +≈ +5.67 +× +10−5erg cm−2 s−1K−4, defines a surface luminosity +Ls = 4πR2 +⋆ +� 1 +0 +d (cos θ) Fs. +(15) +The co-latitude (θ) dependence in (15) comes through the +angle between the magnetic field, assumed dipolar (see be- +low), and the surface normal (see Beloborodov & Li 2016, for +more details), which affects the thermal conductivity tensor. +For a slow source (i.e., ignoring rotational corrections to the +metric tensor), the redshifted luminosity seen by an observer +at infinity is then L∞ +s = Ls(1 − 2GM⋆/c2R⋆). +3.2 +Magneto-thermal modelling +Numerical simulations for the core-surface temperature re- +lationship were carried out by Potekhin et al. (2003). Us- +ing their analytic fits (which are too long to repeat here; +see their Appendix A), we calculate the luminosity an ob- +server expects to see from GLEAM-X J1627 as a function +of the core field strength, assuming the system is in thermal +quasi-equilibrium with balance temperature (12) (mUrca) or +(13) (dUrca) and that there are no other heat sources. For +young (≪ kyr) stars or ones where Joule heating, mechani- +cal heating, or positron backflow from the magnetosphere is +also significant, higher temperatures are expected. We fur- +ther assume an iron envelope, as a crust composed of lighter +elements (e.g., accreted materials) conducts heat more ef- +ficiently and predicts a higher Ts for a given B. Landau +quantization, which we also ignore, similarly leads to higher +temperatures, because electrons are forced to move along +the field lines, thereby suppressing their ability to transfer +heat radially. +Figure 5 shows the balance temperature (12) (red +curves; left axis) as a function of the core field strength, +where we consider core densities of ρ = ρnuc (upper curves) +and ρ = 4ρnuc (lower curves) and the mUrca cooling rate +(8). Figure 6 is similar, though instead with the dUrca +rate (9); note the different scales. To provide an optimistic +outlook, we take L = R⋆ so that the magnetic energy is +predominantly concentrated in low multipoles (cf. Footnote +5). The right axes (blue curves) show the surface luminos- +ity (15) witnessed by an observer at infinity. These fig- +ures illustrate that there is generally an upper limit for +the core field strength implied by the absence of X-rays. +For example, even if we assume a tiny X-ray efficiency +of ϵ ≲ 0.1% (i.e., LX ≲ 10−3L∞ +s ), the Chandra observa- +tions of GLEAM-X J1627, which translate into an upper- +limit of LX ∼ 1030 erg s−1 (Rea et al. 2022), require core +field strengths of B ≲ 3.1 × 1014 G for ρ = ρnuc and +B ≲ 5.5 × 1014 G for ρ = 4ρnuc, as shown by the dashed, +vertical lines in Fig. 5. Larger, percent-level efficiencies place +even tighter constraints. The corresponding limits for dUrca +are much less restrictive, viz. B ≲ 5.2×1015 G for ρ = 4ρnuc +for ϵ = 0.1%, or B ≲ 1.8 × 1014 G for ϵ = 10%. +In +the +magnetothermal +evolutions +carried +out +by +© ? RAS, MNRAS 000, 1–12 + +10 +A. G. Suvorov & A. Melatos +Figure 5. Quasi-static core temperatures (red curves) set by bal- +ancing mUrca cooling and ambipolar heating [expression (12); +left axis], as a function of the magnetic field strength, for two +different densities, ρ = ρnuc (upper curve) and ρ = 4ρnuc (lower +curve). The right-axis (blue curves) shows the predicted, redshift- +corrected surface luminosity (15). An upper limit to L∞ +s +implies +an upper limit to the internal B field, thereby issuing a con- +straint on GLEAM-X J1627, for which LX,max ≈ 1030 erg s−1. +The solid, horizontal line corresponds to this upper limit for a +conversion efficiency of ϵ ≲ 0.1%, i.e., LX ≲ 10−3L∞ +s , which +translates into upper limits for B (dashed, vertical lines) for a +given core density. +Figure 6. Similar to Fig. 5 though with direct Urca cooling (9) +and also a greater efficiency, ϵ = 10% (lower, solid line). +Anzuini et al. (2022b), it was shown that the surface lumi- +nosity of a 1.8M⊙ magnetar (B ≳ 1015 G) with Joule heating +only dips below ≳ 1033 erg s−1 at times t ≲ Myr post-birth, +even if there are hyperons and fast cooling mechanisms are +active (see Figures 1 and 2 therein). This estimate, which +is a factor ∼ 5 more restrictive than the most optimistic, +ambipolar model used here (see Fig. 6), is at odds with the +minima required by the radio activation mechanisms (see +Fig. 1). This casts doubt on a magnetar interpretation for +the source, unless the thermal luminosity is much higher, +or the system is old (Ronchi et al. 2022; Gen¸cali, Ertan & +Alpar 2022). We emphasise however that a realistic investi- +gation for GLEAM-X J1627 requires a proper magnetother- +mal evolution in the presence of an ultra-strong field, which +is difficult (though see Rea et al. 2022). +We close by noting that the magnetothermal study of +Perna & Pons (2011) found that the waiting time distribu- +tion for flares from young (≲ kyr) magnetars peaks at ∼ 1 yr, +and thus the absence of any flare phenomena in the ∼ 2 ks +window, where the source was observed with Swift, is not +entirely surprising. For a source that is several kyr old, the +peak of the waiting time distribution shifts to ≳ 3 yr. Fur- +thermore, bursts may be missed if beamed away from Earth, +making it less clear how long one might need to observe be- +fore expecting a flare. However, a plastically-flowing crust +could be even hotter than one that never breaks because +thermoplastic waves can dissipate magnetic energy, the ef- +fects of which resemble deflagration fronts in combustion +(Beloborodov & Li 2016). Targeted searches would be use- +ful in this direction to shed light on the matter. +4 +CONCLUSIONS +The source GLEAM-X J1627 was recently discovered by +Hurley-Walker et al. (2022), who reported an extremely long +spin period (P = 1091.17s) together with a possibly large +period derivative (| ˙P| < 1.2×10−9 ss−1). The magnetic field +strength implied, assuming a neutron star undergoing mag- +netic dipole braking (though see Loeb & Maoz 2022; Katz +2022, for a white dwarf interpretation), comfortably exceeds +1016 G when using the best-fit value ˙P = 6.0 × 10−10 ss−1 +(Ruderman & Sutherland 1975). In this paper, a critical +examination of the magnetar interpretation is carried out, +though under the proviso that model-based specifics are in- +escapable and conclusions cannot be asserted strongly based +on the limited data at hand. +Magnetospheric gap models require a minimum mag- +netic field strength, for a given period, for the switch-on of +the star as a radio pulsar (Goldreich & Julian 1969; Stur- +rock 1971; Hibschman & Arons 2001; Medin & Lai 2010). +For canonical stellar parameters, we find in Sec. 2.1 that +minimum fields of order ∼ 1016 G appear to be necessary, +even when assuming a high degree of multipolarity, long- +lived twists in the magnetosphere (Beloborodov 2009), and +small curvature radii Rc ∼ R (Medin & Lai 2010). If the star +has a large radius, R⋆ ≳ 13 km, this requirement may drop +to Bp,min ≲ 5 × 1015 G. Standard electromagnetic braking +theory suggests that the star is between ∼ 10 and 50 kyrs +old, depending on the historical braking index and field evo- +lution model (though cf. Ronchi et al. 2022; Gen¸cali, Ertan +& Alpar 2022). Assuming ages much larger than 10 kyr and +a present-day ∼ 5 × 1015 G polar field, a Hall-Ohm back- +extrapolation implies a birth strength of ≲ 1017 G, and fur- +ther that field decay was stalled to some degree, possibly by +plastic opposition of the electron fluid motion in the crust +(Lander & Gourgouliatos 2019; Gourgouliatos, De Grandis +& Igoshev 2022). This points towards there having been a +large angular momentum reservoir at birth to support in- +tense field amplification via some combination of dynamo +activity, Kelvin-Helmholtz action, or magneto-rotational in- +stabilities (Ciolfi 2020a,b). +A simple magneto-thermal model is employed in Sec. 3 +to show that the competition between heating induced by +field decay and neutrino cooling implies a particular sur- +face luminosity, depending on assumptions on the thermal +conductivity, stellar composition, and Urca channel. The +lack of strong thermal emissions from the source (LX ≲ +1030 erg s−1; Rea et al. 2022) is difficult to reconcile with the +radio requirements, unless fast cooling mechanisms are in +© ? RAS, MNRAS 000, 1–12 + +1035 +5.5×1014G +5 +4 + 1034 +V +0 +B +B +(erg/s) +3 +8 +erg/s +1033 +8 +2 +1032 +0 +1031 +0.001 +0.010 +0.100 +1 +B16 (G)0.8 +1035 +0.6 +1034 +L=1033erg/s +(erg/s) +1033 +G +0.4 +G +G +0 +5 +8 +1032 +4 +0.2 +V +1031 +L=1031erg/s +0.0 +1030 +0.01 +0.05 +0.10 +0.50 +B16 (G)A magnetar interpretation for GLEAM-X J1627 +11 +operation (Lattimer et al. 1991). A heavier star with larger +moment of inertia, which is generally easier to cool quickly +(Anzuini et al. 2022a), could also help to alleviate the ten- +sion between the available spin-down power and observed +radio luminosity; see Sec. 2. +Another clue about the nature of GLEAM-X J1627 +comes from the transient character of its radio pulsations. +Hurley-Walker et al. (2022) noted that the source (visibly) +pulsated for only 3 months and then abruptly turned off, +indicating an overall duty cycle of only ∼ 2% within the ob- +servational monitoring window (see also Footnote 2). This +could occur, if the source hovers near the death line, with +magnetohydrodynamic evolutions triggering its descent into +the graveyard around March of 2018. For example, a crustal +fracture may have injected twist into the magnetosphere +prior to the object’s discovery, allowing it to temporarily +access line (d); see Fig. 1. In the case of the so-called ro- +tating radio transients (RRATs), which similarly display +high degrees of nulling, it was suggested by Zhang, Gil & +Dyks (2007) that concentrated starspots may emerge near +the poles, sporadically allowing the host star to rise above +the death line (see also Sec. 2.1, Suvorov, Mastrano & Gep- +pert 2016, and references therein). +Magnetar-like X-ray bursts are known to suppress radio +pulsations in many neutron stars; bursts observed in PSR +J1119–6127 by XMM-Newton and NuSTAR were coincident +with the shut-off of the source as a radio pulsar (Archibald +et al. 2017), for example [see also Coti Zelati et al. (2018) for +a discussion on other sources]. If GLEAM-X J1627 is regu- +larly bursting, as would be expected if Bp ≳ 1016 G and the +crust frequently succumbs to Maxwell stresses, this could +also explain the high degree of nulling. The absence of any +X-ray activity (Hurley-Walker et al. 2022) casts doubt how- +ever on this interpretation, though geometric factors related +to beaming and directionality may explain this. Finally, the +population study recently conducted by Sheikh & MacDon- +ald (2021) indicates that there is a (weak) correlation be- +tween the spin period and nulling fraction in radio pulsars. +The high nulling fraction and long spin period of GLEAM-X +J1627 fits within this scenario. Regardless, further monitor- +ing of the source in both the radio and X-ray bands will help +to unveil its magnetar nature or otherwise. +If indeed GLEAM-X J1627 boasts a polar field strength +greater than 1016 G, as suggested by its place in the P– ˙P dia- +gram and the death valley considerations (Sec. 2.1), it would +have been an ample source of gravitational waves when +born. Even in the absence of a toroidal field, the quadrupo- +lar ellipticity of the source could easily reach ∼ 10−4 (e.g., +Haskell et al. 2008; Mastrano et al. 2011). The source, lo- +cated ∼ 1.3 kpc from Earth (Hurley-Walker et al. 2022), +would have been visible to the advanced Laser Interferom- +eter Gravitational-Wave Observatory (aLIGO) for P ≪ 1s. +From the braking analysis given in Sec. 2.3, if the birth pe- +riod was at most a few ms (as argued in Sec. 2.4), the source +would have been sufficiently bright in gravitational waves +to enable detection for ∼ years. The existence of GLEAM- +X J1627 therefore adds further incentive to perform blind, +gravitational-wave searches for magnetar-like sources. +ACKNOWLEDGEMENTS +We extend our thanks to Filippo Anzuini for discussions +about direct Urca processes and the heating effects of +Landau quantization in magnetars. AGS thanks Kostas +Glampedakis for pointing out the possibility of a Hall attrac- +tor state. The research leading to these results has received +funding from the European Union’s Horizon 2020 Pro- +gramme under the AHEAD2020 project (grant n. 871158). +The constructive criticisms of the anonymous referee, which +led to a richer study, are gratefully acknowledged. +DATA AVAILABILITY STATEMENT +Observational data used in this paper are quoted from the +cited works. Additional data generated from computations +can be made available upon reasonable request. +REFERENCES +Abbott B. P. et al., 2018, Phys. Rev. Lett., 121, 161101 +Aguilera D. N., Pons J. A., Miralles J. 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RAS, MNRAS 000, 1–12 + diff --git a/dNFAT4oBgHgl3EQfYx1Q/content/tmp_files/load_file.txt b/dNFAT4oBgHgl3EQfYx1Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e6433f3e73423a6ed791029fa9d881b4d97c0d5 --- /dev/null +++ b/dNFAT4oBgHgl3EQfYx1Q/content/tmp_files/load_file.txt @@ -0,0 +1,3641 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf,len=3640 +page_content='Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 000, 1–12 (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=') Printed 23 January 2023 (MN LATEX style file v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2) Evolutionary implications of a magnetar interpretation for GLEAM-X J162759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5–523504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3 Arthur G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Suvorov1,2⋆ and Andrew Melatos3,4 1Manly Astrophysics, 15/41-42 East Esplanade, Manly, NSW 2095, Australia 2Theoretical Astrophysics, Eberhard Karls University of T¨ubingen, T¨ubingen, D-72076, Germany 3School of Physics, University of Melbourne, Parkville VIC 3010, Australia 4ARC Centre of Excellence for Gravitational Wave Discovery (OzGrav), Hawthorn VIC 3122, Australia Accepted ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='. Received ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' in original form ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' ABSTRACT The radio pulsar GLEAM-X J162759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5–523504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3 has an extremely long spin pe- riod (P = 1091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='17 s), and yet seemingly continues to spin down rapidly ( ˙P < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2 × 10−9 ss−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The magnetic field strength that is implied, if the source is a neu- tron star undergoing magnetic dipole braking, could exceed 1016 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This object may therefore be the most magnetised neutron star observed to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In this paper, a crit- ical analysis of a magnetar interpretation for the source is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (i) A minimum polar magnetic field strength of B ∼ 5 × 1015 G appears to be necessary for the star to activate as a radio pulsar, based on conventional ‘death valley’ assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (ii) Back-extrapolation from magnetic braking and Hall-plastic-Ohm decay suggests that a large angular momentum reservoir was available at birth to support intense field am- plification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (iii) The observational absence of X-rays constrains the star’s field strength and age, as the competition between heating from field decay and Urca cooling im- plies a surface luminosity as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' If the object is an isolated, young (∼ 10 kyr) magnetar with a present-day field strength of B ≳ 1016 G, the upper limit (≈ 1030 erg s−1) set on its thermal luminosity suggests it is cooling via a direct Urca mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Key words: stars: magnetars, magnetic fields, pulsars: GLEAM-X J162759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5– 523504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3 1 INTRODUCTION Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2022) recently reported that observa- tions, made between January and March of 2018 with the Murchison Widefield Array (MWA) in the 72 − 231 MHz band, revealed the presence of a pulsating Galactic source, since named GLEAM-X J162759.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5–523504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3 (henceforth GLEAM-X J1627), which is 88 ± 1% linearly polarised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Barycentric correction and alignment of the pulses estab- lished a periodicity with a pulsar-like regularity, P = 1091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1690(5) s, and further that the source is slowing down at a best-fit rate of ˙P = 6×10−10 s s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (Though we note the data can only confidently assert that | ˙P| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2×10−9 ss−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The characteristic (polar) magnetic field strength relevant for a neutron star, B ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='4 × 1019� P ˙P G (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', Ruder- man & Sutherland 1975), is arguably in excess of 1016G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Furthermore, since the pulse structure of the source varies on ∼hour-long timescales in a way that is similar to what ⋆ arthur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='suvorov@manlyastrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='org is seen from known radio magnetars, Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2022) offered the tantalising conclusion that the source is an ultra-long period magnetar (see also Ronchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Ek¸si & S¸a¸smaz 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Gen¸cali, Ertan & Alpar 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Katz 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This would likely make GLEAM-X J1627 the most magne- tised neutron star observed to date1 (Olausen & Kaspi 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Coti Zelati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Followup searches by the Swift X-ray Telescope (XRT) found no evidence for thermal or soft X-rays (Hurley- Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Upper limits to the photon count were placed which, depending on the spectral fit, imply an upper limit to the flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The strongest limit is placed at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='9 × 10−13 erg s−1cm−2 for the absorbed flux in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3–10keV band, with a marginally lower value (≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5 × 10−13 erg s−1cm−2) applying for a blackbody fit at kT ∼ 1 A list of known magnetars and their properties is main- tained at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='ca/~pulsar/magnetar/ main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='html;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' see also the Magnetar Outburst Online Catalogue http://magnetars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' © ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' RAS arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='08541v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='HE] 20 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Suvorov & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Melatos 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Based on the greatest distance allowed by the dispersion measure, dmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='8 kpc, this gives LX ⩽ 7 × 1031 erg s−1 (Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' A deeper search using the Chandra X-ray Observatory was carried out by Rea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2022), who placed the even stricter upper limit LX ⩽ 2 × 1030 erg s−1, with the exact bound depending on assumptions about the spectral shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The latter upper limit would generally be expected of persistent, thermal emis- sions from a ≳ Myr-old magnetar (Thompson & Duncan 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Turolla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Olausen & Kaspi 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Should GLEAM-X J1627 be a magnetar, its existence as a radio, but not an X-ray, source has a number of interesting impli- cations for emission physics and magnetic field evolution in neutron stars, some of which we explore in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' It is generally put forth that electron-positron pair pro- duction, likely occurring in magnetospheric ‘gaps’, is a nec- essary ingredient to spark the radio emissions seen from pul- sars (Goldreich & Julian 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Sturrock 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Ruderman & Sutherland 1975) (though cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Melrose, Rafat & Mastrano 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Depending on the topological properties of the stel- lar magnetic field, a variety of different ‘death lines’, defined through the threshold to generate the requisite pairs, com- prise an overall ‘death valley’ (Chen & Ruderman 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Hi- bschman & Arons 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Requiring that GLEAM-X J1627 reside outside of the valley allows us to assess the validity of a number of evolutionary scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Having some idea about what surface field structures are permissible for the object ‘today’, we can back-extrapolate from analytic or numeri- cal simulations of Hall-plastic-Ohm decay in stellar crusts (Lander & Gourgouliatos 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Gourgouliatos, De Grandis & Igoshev 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Kojima, Kisaka & Fujisawa 2022) to see what magnetic conditions at birth are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' An intense magnetic field within the stellar core is ex- pected to lead to ambipolar heating (Goldreich & Reiseneg- ger 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Aguilera, Pons & Miralles 2008), as charged con- stituents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', protons) experience Lorentz forces that the uncharged components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', neutrons) do not, leading to a kind of collisional friction that gradually heats up the star at the expense of the magnetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' From models of core- crust heat transfer (Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Turolla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Vigan`o et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Anzuini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022a), we can estimate the surface luminosity implied by the competition between ambipolar diffusion, mechanical dissipation, Joule heating, and particle backflow against neutrino cooling (Beloborodov & Li 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The absence of thermal X-rays may then hint at an upper limit to the magnetic field strength, which can be compared with the requirements set by the radio activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In this paper, we revisit the magnetar interpretation of GLEAM-X J1627 in the context of death valley physics (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1), Hall-plastic-Ohm evolutions (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2), braking mecha- nisms (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3), field amplifications at birth (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='4), and ambipolar heating (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The conclusions are summarised in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 4, emphasising that they depend on details of the model and cannot be asserted strongly based on the limited data at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Quantities written with numerical subscripts are logarithmically normalised, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', B16 = B/1016 for field strength B measured in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2 MAGNETAR NATURE OF GLEAM-X J1627 AND RADIO OBSERVATIONS Between January and March 20182, 71 pulses from GLEAM- X J1627 were detected by the MWA which, after alignment and barycentric correction, revealed a Galactic source puls- ing with period P = 1091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='17 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The best-fit value for the period derivative is ˙P = 6 × 10−10 s s−1, though Hurley- Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2022) noted that their analysis cannot exclude even larger values ˙P < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2 × 10−9 s s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For a neutron star moment of inertia I0 ∼ 1045 g cm2, the spin-down luminos- ity associated with the object is then ˙Esd ≈ 4π2I0 ˙P/P 3 ∼ 1028 erg s−1, which is several orders of magnitude lower than the observed radio luminosity Lν ≈ 4 × 1031 erg s−1, esti- mated assuming a best-fit distance of d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This puzzling feature, which is unique to GLEAM-X J1627, has prompted interest in a white dwarf interpretation for the source, essentially to boost I0 (Loeb & Maoz 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Katz 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' However, Erkut (2022) argue that the beaming angle assumptions made by Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2022) may be in- appropriate for an object with such a long spin period (see also Szary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Szary, Melikidze & Gil 2015), and the radio luminosity may in fact be closer to ≈ 3 × 1026 erg s−1 – much lower than ˙Esd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In the standard picture of magnetic-dipole braking (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3), the characteristic magnetic field strength for a neutron star reads Bp ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='4 × 1019� P ˙P G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The P- ˙P upper limits therefore hint towards a huge magnetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This observation, combined with the high degree of linear polari- sation in the pulses, led Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2022) to sug- gest that GLEAM-X J1627 may be a magnetar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In this Sec- tion, we examine this suggestion in the theoretical context of radio emission mechanisms (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1), crustal magnetic field evolution (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2), braking physics (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3), and nu- merical simulations of birth properties (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The Swift XRT observations of GLEAM-X J1627 are then discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1 Death valley A neutron star crust provides a reservoir of free electrons that are continuously accelerated into the magnetosphere by induction-generated electric fields as the star spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In ‘gap’ regions where the Goldreich & Julian (1969) charge den- sity is comparatively low, the electric field along magnetic field lines can be sufficiently intense that photons emitted by accelerated charges possess the requisite energy to pair pro- duce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' It is generally put forth that e+e− production is an es- sential ingredient in the powering of coherent radio emissions from neutron stars (Sturrock 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Ruderman & Sutherland 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Chen & Ruderman 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Hibschman & Arons 2001) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Melrose, Rafat & Mastrano 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Secondary charges 2 Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2022) note that while the MWA, over the course of its eight years of operation, has accumulated around ≲ 200 hours of observing time within 15◦ of GLEAM-X J1627, the data span many different array configurations, frequencies, and observing modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' It is therefore difficult to formally deduce the source duty cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Uncertainties notwithstanding, they argue that a ∼ 2% duty cycle is likely, similar to the ∼ 5% cycle of the radio-loud magnetar XTE J1810–197 (Eie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' © ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' RAS, MNRAS 000, 1–12 A magnetar interpretation for GLEAM-X J1627 3 generated by curvature- or inverse-Compton-produced pho- tons can themselves be accelerated and emit photons3, cul- minating in a pair cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Beam instabilities, where free energy associated with streaming motions is transferred to plasma waves, then lead to radio emission (though again cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Melrose, Rafat & Mastrano 2021, who argue this picture requires revision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In this scenario, there is a maximum potential drop, ∆Vmax, which can be produced in the magnetosphere, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' ∆Vmax ≈ 2π2BdR3 ⋆ c2P 2 , (1) for stellar radius R⋆, speed of light c, and polar dipole strength Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This maximum must exceed that which is re- quired for the pair production mechanism to activate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In a curvature radiation and polar-gap scenario, this entails (Ru- derman & Sutherland 1975) �e∆Vmax mec2 �3 ℏH 2mecR2c Bp BQED ≳ 1 15, (2) where BQED = m2 ec3/eℏ ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='4×1013 G is the Schwinger field for electron mass and charge me and e, respectively, ℏ is the reduced Planck constant, and H denotes the gap thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Note that the numerical factor 1/15 in (2) depends weakly on the angle made between the direction of photon prop- agation and B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' see the discussion around equation (19) in Ruderman & Sutherland (1975) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Moreover, the polar field strength Bp may exceed the dipole value, Bd, which is the relevant quantity at large radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The curvature radius of a magnetic field line, Rc, scales inversely with the multipole order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Depending on the assumptions one places on the magnetic configuration, most notably H, Rc, and Bp/Bd, a variety of possible ‘death lines’ arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' There are two main radii characteristic to the magne- tosphere of an isolated object, being the stellar radius and the light-cylinder radius, RLC = cP/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Chen & Ruderman (1993) posit that depending on the magnetospheric twist, H and Rc can assume a variety of values that are built from these two radii, such as (R⋆RLC)1/2, R⋆(R⋆/RLC)1/2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For polar-gaps, the thickness H may also scale with the dimensionless ratio β = Bp/Bd, as multipoles can dominate over the dipole component near the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In real- ity, a force-free magnetospheric model, though possibly with some displacement currents, is necessary to determine the gap size and curvature radius self-consistently for some field geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Nevertheless, we can explore the various extrema by taking the approximate scalings considered by Chen & Ruderman (1993) and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Following Chen & Ruderman (1993) (though see also Hibschman & Arons 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Szary, Melikidze & Gil 2015), there are four types of polar-gap scenario that we consider: (a) Pure central dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This is the simplest such model, where Bp = Bd, Rc = (R⋆RLC)1/2 and H = R⋆(R⋆/RLC)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The required field strength for pair- production is Bd,min = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2 × 1012(P 15/8R−19/8 6 ) G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3 Above hot polar caps with temperatures exceeding ∼ 106 K, collisional interactions between photons could potentially also produce abundant pairs through the Breit & Wheeler (1934) in- teraction γγ → e+ + e− (Jones 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Minimum polar dipole field strengths required for GLEAM-X J1627, assuming a death line according to the charac- terisation given in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The final column shows Bd,min for a canonical radius R6 = 1, with the bracketed number corre- sponding to a larger radius of 13 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The second column indicates the strength of the surface field relative to the dipole component, which influences the cap thickness, H, and curvature radius, Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Magnetic geometry Bp/Bd Bmin d,16 (R6 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3) (a) Pure dipole 1 101 (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='8) (b) Twisted dipole 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='30 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='32) (c) Starspot 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='10 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='21) 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='88 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='08) 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='72 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='99) (d) Twisted multipoles 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='279 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='165) 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='222 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='131) 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='187 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='111) (b) Twisted dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' As above, though instead Rc = R⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We find Bd,min = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='7 × 1011(P 13/8R−17/8 6 ) G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (c) Starspot configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Numerical simulations of crustal Hall drift tend to find that concentrated ‘mag- netic spots’ emerge near the polar-cap (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', Vigan`o et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Suvorov, Mastrano & Geppert 2016), where Bp ≫ Bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Taking Rc = R⋆ and a reduced cap size H = β−1/2R⋆(R⋆/RLC)1/2 yields Bd,min = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='7 × 1011(β−1/8P 13/8R−17/8 6 ) G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (d) Twisted multipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Similar to case (c), though with maximum pitch angle between the magnetic field and the direction of emitted photons, sin θ ≈ H/Rc (and thus H ≈ Rc = R⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Effectively, one assumes the magne- tosphere is so twisted that a curvature-radiation photon, emitted almost parallel to B, crosses another part of the cap’s open field line bundle at a large angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This gives Bd,min = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2 × 1010(β−1/4P 3/2R−2 6 ) G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Table 1 displays the minimum polar dipole strength Bd,min, in the context of the death lines (a)–(d) described above, for a range of surface-to-dipole ratios, Bp/Bd, and a canonical radius R⋆ = 10 km, such that GLEAM-X J1627 can activate as a radio pulsar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Taking instead a stellar ra- dius of 13 km allows for an easier switch on, as shown by the numbers in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Rows show different values for the relative strength of multipoles, which influence the curva- ture radius and gap thickness, as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Although lines (b) through (d) imply high multipole orders (ℓ ≳ 102) when interpreted via the numerical simulations of Asseo & Khechinashvili (2002), for instance, magnetohydrodynamic evolutions in proto-magnetars indicate that the generation of high-order multipoles in the interior is a generic quality of strongly-magnetised systems (Kiuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Lander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Lander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2021) found that truncating their numerical output to harmonic expansions with ℓmax < 32 led to sizeable inaccuracies in the inferred field strength (see also Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Fallback accretion onto the proto-star, or later in life, can also twist field lines near the stellar surface (Melatos & Priymak 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Suvorov & Melatos 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We emphasise that the above models, while phenomeno- logical, represent the extrema that could be expected for a given activation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' It is unlikely that any given © ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' RAS, MNRAS 000, 1–12 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Suvorov & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Melatos line applies to the entirety of the neutron star population at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For example, magnetospheric twists (lines b and d) and starspots (line c) are dynamical in nature and sub- ject to diffusion, implying that the radius of curvature is a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Regarding magnetars in particular, Be- loborodov (2009) suggests that their radio activation may be related to quake activity in the crust, where overstressed zones fracture and flow plastically, dragging the magnetic footpoints with them and pumping a current (‘j-bundle’) into the magnetosphere (see also Beloborodov & Thomp- son 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Overshearing events may also be expected from spindown (Baym & Pines 1971), which tends to be faster in magnetars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Magnetospheric twists can survive on ≳ year- long timescales (Parfrey, Beloborodov & Hui 2013), until the field lines relax to the pre-twist (or some other) equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' During a twisting episode one may expect that either lines (b) or (d) apply, after which a dipole or starspot configura- tion is reinstated (lines a or c) depending on the surface-field multipolarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This may help to explain why certain magne- tars are radio loud while others are not, depending on the waiting time between twist injections (see also Morozova, Ahmedov & Zanotti 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Detailed simulations of pair cascades in magnetar envi- ronments were carried out by Medin & Lai (2010) to ex- amine whether the polar gap story, discussed above, ap- plies in super-strong fields (see also Thompson & Duncan 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Baring & Harding 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Beloborodov & Thompson 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Although they find the cascade proceeds differently for fields above and below BQED, mostly because of the sup- pression of synchrotron emission in strong fields, the over- all multiplicity λ is relatively insensitive to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The energy spectrum of electron-initiated cascades depends mostly on the polar-cap voltage, and hence the spin period, and not B alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The critical multiplicity, necessary for radio acti- vation, that they find in the strong-field case is λdeath ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5 × 107(Bp/1012G)−1/6(Rc/108cm)2/3 (see their Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For untwisted dipoles this implies Bd,min ∝ P 2, similar to line (a) though marginally steeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For highly-twisted fields with Rc ∼ R, one recovers lines qualitatively similar to ei- ther (b) or (d) from their results, depending on the field topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Therefore, although the overall slope of death lines in B-P space may vary depending on how the cascade pro- ceeds, the death valley defined as the area spanned by lines (a) through (d) is a reasonable approximation for the valley extrema, even for magnetars4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Figure 1 shows death lines in the context of the wider neutron star population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Although noting the caveats dis- cussed above, we see that all known, pulsating objects lie above the overall valley, with the possible exception of GLEAM-X J1627 (shown by a black star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The vertical axis shows the dipolar field strength of known objects, which is relatively uncertain: one requires a braking model to esti- mate this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We assume that the standard magneto- dipole picture [equation (5) with n = 3] applies, though 4 Given the high degree of nulling and that only bright and vari- able single pulses were detected from GLEAM-X J1627, it may be that the radio activity is not attributable to traditional cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Different death lines altogether may apply, such as the fast radio burst death lines described by Wadiasingh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2020), which can be satisfied with somewhat weaker fields (see their equation 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' allow for uncertainties in the inclination angle, π/4 ⩽ α ⩽ π/2, and the stellar radius, 10 ⩽ R⋆/km ⩽ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Note that polarisation data from radio pulsars indicate that α spans an even larger range (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', Rankin 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Therefore, indi- vidual death lines have some width, and pulsar positions on the diagram have some uncertainty (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Although the mean of line (a) cuts right through the middle of the population, inclinations tending towards alignment weaken the intrinsic torque, and thus predict a larger B for a given ˙P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' It was argued by Contopoulos & Spitkovsky (2006) that radio pulsars may evolve towards an aligned configuration (α → 0), and hence even line (a) could be sufficient in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Evolution towards alignment is also observed in 3D magnetospheric simulations (Philippov, Tchekhovskoy & Li 2014) (though cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Lander & Jones 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For many systems however it is likely that dynamical phenomena, such as mag- netospheric twist injections via magnetically- (Beloborodov 2009) or spindown-induced (Baym & Pines 1971) quakes, or starspot formation (Zhang, Gil & Dyks 2007), play a role in pair cascade phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Lines (b) through (d) may therefore only apply sporadically over ∼ year-long timescales (Parfrey, Beloborodov & Hui 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In the context of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 1, we see that the pure dipole [model (a)] is unable to explain the radio switch-on of GLEAM-X J1627 unless the polar field takes on super- virial values, Bp ≳ 1018 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This is in contrast with all (other known) radio-loud magnetars, namely PSR J1745– 2900, PSR J1622–4950, XTE J1810–197, 1E 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0–5408, Swift J1818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0-1607, PSR J1119–6127 and SGR 1935+2154 (red stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This line fails to explain the pulsar population at large though, cutting through the middle of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Only in the case of a highly-twisted configuration [model (d)] can the local field of GLEAM-X J1627 assume values Bp ≲ 1016 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For these models, the minimum required for the surface field is still greater than the maximum polar field amongst all other 31 known magnetars, with the runner up being SGR 1806–20 (Olausen & Kaspi 2014), an extraor- dinarily bright and young (< kyr) burster, which boasts a polar field strength Bp ≈ 4 × 1015 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The uncertainty implied by the final column of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 1 is a lower limit, as consideration of outer gap models (Chen & Ruderman 1993), thermionic emissions (Szary, Melikidze & Gil 2015), and general-relativistic corrections (Hibschman & Arons 2001) can also adjust the voltage drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Outer-gap models, however, tend to fare worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For example, for the partially-inclined, outer magnetosphere accelerator model [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (27) of Chen & Ruderman (1993)], the relevant death line, 5 log Bp − 12 log P ≈ 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5, demands a magnetic field for GLEAM-X J1627 that exceeds the virial limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Szary, Melikidze & Gil (2015) argued, in the context of a partially- screened gap, that the polar cap must be below some crit- ical B-dependent temperature, else thermionic emissions effectively screen the acceleration potential (such consid- erations are pertinent to the observational absence of X- ray emissions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Similarly, the general-relativistic Lense-Thirring corrections discussed by Hibschman & Arons (2001) may be important for GLEAM-X J1627, despite its long spin period, because the Goldreich & Julian (1969) plasma density and the Lense-Thirring frequency both scale linearly with the rotation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Na¨ıvely applying the ‘low- altitude’ estimate for the pair multiplicity computed by Hib- schman & Arons (2001), which includes Lense-Thirring pre- © ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' RAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 1–12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='A magnetar interpretation for GLEAM-X J1627 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='★★ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='★★★ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='★ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='★ ★ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='★ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='★ - Radio Magnetars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='⋄ - Radio Pulsars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='★ - GLEAM-X J1627 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='\uf750 - (R-)Quiet Magnetars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='\uf520 - Binary member ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='100 1 10 100 1000 108 1010 1012 1014 1016 Spin period P (s) Polar dipole strength Bd(G) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Bd − P diagram overlaid with death ‘lines’ (a)–(d), as shown by the coloured curves (see plot legends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Each curve comes with some thickness because we allow for uncertainties in the stellar radius, 10 ⩽ R⋆/km ⩽ 13, and the polar-to-dipole field strength ratio, 1 ⩽ β ⩽ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Overlaid are known objects, with available ˙P measurements, from the ATNF pulsar catalogue (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='atnf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='au/research/pulsar/psrcat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Manchester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2005) with ‘ordinary’ radio pulsars shown by black diamonds, those in binaries with black squares (for which B-field estimates are even more uncertain as they depend on accretion assumptions), radio-loud magnetars with red stars, and radio-quiet magnetars with blue circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The (vacuum dipole) upper limit for GLEAM-X J1627 is indicated with a black star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The Bd values for other pulsars come from the standard dipole-braking formula (5) with n = 3 for a range of obliquities (π/4 ⩽ α ⩽ π/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' these variations, together with those on R⋆ and timing errors on ˙P, imply some uncertainty on the dipole strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For magnetars (except J1119), mean values from the McGill catalogue are used (Olausen & Kaspi 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In principle, all objects lie above the overall valley, defined as the area between the top of line (a) and the bottom of line (d), with the possible exception of GLEAM-X J1627, unless it has a highly-twisted magnetosphere (Rc ∼ R⋆) with a polar dipole field strength exceeding ∼ 1015G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' cession and is appropriate when Bp/(1012 G) ≳ P/(1 s) [see their Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (69)], we obtain a minimum polar field strength Bp,min ≈ 2 × 1016 G, comparable to lines (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The Hibschman & Arons (2001) models however invoke cap tem- peratures set by backflowing positrons, which may be unre- alistically low for GLEAM-X J1627 because internal heating driven by field decay is likely to be non-negligible (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' thermal transport simulations would be necessary to self- consistently assess the valley structure in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2 Hall-plastic-Ohm decay A simplified picture of the neutron star crust is that of a rigid, ion lattice strewn with mobile electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The latter carry a current as they flow relative to the ions, gradually advecting the field lines that thread the crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This pro- cess of Hall drift, while conserving magnetic energy, can act to accelerate Ohmic decay through a sequence of cascades to smaller-scale magnetic structures, possibly aided further by thermoelectric effects (see Gourgouliatos, De Grandis & Igoshev 2022, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The Hall timescale obeys τHall ∝ B−1, and thus magnetar crusts are particularly prone to field decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Depending on the initial conditions however, the system may enter into an ‘attractor’ state where the Hall term vanishes (Gourgouliatos & Cumming 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Although we will not consider this complication fur- ther, a Hall-stalled evolution may help GLEAM-X J1627 to maintain a strong field while cooling quickly as it ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' As magnetic gradients form, Maxwell stresses are ex- erted on the crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For magnetar-like field strengths B ≳ 1015 G, the crust may not be sufficiently malleable to ab- sorb these stresses, and rather a crustal failure may occur (Duncan 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Lander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Crustquakes are popu- lar models for the progenitors of magnetar outbursts, such as giant flares (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', G¨oˇg¨u¸s et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2000) or fast radio bursts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', Suvorov & Kokkotas 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Once the crust experi- ences a failure however, it is unlikely to ‘heal’ immediately and rather may enter a state of azimuthal shearing termed plastic flow (Beloborodov & Levin 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Lander & Gourgou- liatos 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Kojima, Kisaka & Fujisawa 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Plastic flow is generally a dissipative process, and thus depending on the ‘plastic viscosity’, the Hall effect may be enhanced, implying © ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' RAS, MNRAS 000, 1–12 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Suvorov & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Melatos that numerical Hall-Ohm (as opposed to Hall-plastic-Ohm) investigations underestimate the degree of field decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' On the other hand, plastic flows can move against the existing flow of the electron fluid (Gourgouliatos & Lander 2021), and thus inhibit magnetic dissipation by counteracting the formation of the small-scale (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', highly multipolar) mag- netic substructures most susceptible to Ohmic decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The density-dependent, and hence radially-stratified, nature of the electron fluid flow also facilitates the growth of a toroidal field, making an investigation of a realistic Hall-plastic-Ohm system a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The evolution of the crustal magnetic field B is de- scribed by the induction equation 0 =∂B ∂t + ∇ × � c 4πene (∇ × B) × B − vpl × B + c2 4πσ ∇ × B � , (3) for electron number density 1034 ≲ ne/cm−3 ≲ 1036 and conductivity 1016 ≲ σ/s−1 ≲ 1024, where vpl denotes the plastic flow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The lower limit for the electrical con- ductivity applies to the crust-magnetosphere interface, while the latter is appropriate for the inner crust (Akg¨un et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' A proper description for vpl, including a determina- tion of characteristic plastic speeds, requires an additional equation of motion, typically set by the requirement that a Stokes flow is induced in regions of excess stress (deter- mined, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', by the von Mises criterion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Lander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Following Aguilera, Pons & Miralles (2008), we construct an approximate model by replacing the gradient operator with the inverse of a relevant lengthscale, L, yielding (see also Lander 2022) dB dt = − B B0 B τHall + vplB L − B τOhm , (4) with τHall = 4πeneL2/cB0 and τOhm = 4πσL2/c2 read off from (3), with small-scale structures dominating the choice of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Equation (4), which is subject to the initial condi- tion B(0) = B0, reduces to the phenomenological Hall-Ohm model of Aguilera, Pons & Miralles (2008) when vpl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In the simulations of Lander & Gourgouliatos (2019), it was found that larger B0 values lead to swifter plastic flows, and more precisely that doubling B0 leads to an approxi- mately 3-fold increase in vpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For magnetar-level fields and low plastic viscosities, these authors (see also Gourgouliatos & Lander 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Gourgouliatos, De Grandis & Igoshev 2022) found that vpl can approach a few hundred cm per year in regions where field lines are particularly tangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' How- ever, slower plastic speeds emerge in the bulk of the crust, and no flow at all occurs in unstressed regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' As we have washed out all spatial dependencies in building relation (4), the flow is nominally non-zero everywhere, rather than only in regions localised around failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We thus consider instead spatially-averaged speeds of vpl ≲ 40 cm yr−1 for cases with ultra-strong fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Note that if vpl is negative (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', if one takes ∇ → −1/L rather than ∇ → 1/L), plastic flow in- stead accelerates field decay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' such cases have been observed in the studies cited above, depending on the plastic viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Figure 2 shows solutions to equation (4) for several ini- tial field strengths 1 ⩽ B16 ⩽ 50, in both the Hall-Ohm (Aguilera, Pons & Miralles 2008, solid curves) and Hall- plastic-Ohm (dashed curves) cases, where the plastic flow B0=1016G B0=5\uf4a01016G B0=1017G B0=5\uf4a01017G vpl=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5cm/yr vpl=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='8cm/yr vpl=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5cm/yr vpl=38cm/yr (d): Bp,min=4\uf4a01015G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='10 1 10 100 1014 1015 1016 1017 Time (kyr) Bp (G) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Evolution of the polar field strength, Bp(t), given as a solution to equation (4) for both Hall-Ohm (vpl = 0, solid curves) and Hall-plastic-Ohm (vpl ̸= 0, dashed curves) evolutions, for sev- eral birth field strengths and plastic velocities (see colour-coded legends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The dotted, horizontal line shows the minimum field strength set by the type (d) death lines with Bp/Bd ∼ 2 consid- ered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' velocity is chosen to scale with B in the manner described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' To provide an optimistic but realistic5 scenario, we set L = 105 cm, ne = 1036, and σ = 1024 s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' smaller val- ues lead to faster decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' When vpl ⩽ 0, the field enters a state of rapid decay after ∼ 1 kyr, reducing by an order of magnitude after only ≈ 2 kyr in the ultra-strong case with B0 = 5 × 1017 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The dotted line illustrates the minimum field strength required to fulfil the death valley requirements discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We remind the reader that even if the dipole field is of order Bd ∼ 1015 G, the surface field implied by the death valley constraints is of order ≳ 4 × 1015 G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Demanding that Bp ≳ 4×1015 G at present implies that the system can be at most ∼ 20 kyr old independently of the birth field strength if plastic flow is ignored, because one has τHall ∝ B−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Such a conclusion may be in tension with the observed spin period of GLEAM-X J1627 (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3), un- less the star underwent a period of extreme spin-down early in its life (from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', propellering fallback material shortly after birth, as discussed by Ronchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Gen¸cali, Er- tan & Alpar 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Including a sufficiently rapid plastic flow, vpl ≳ 30 cm yr−1, however stalls the impact of the Hall ef- fect (Gourgouliatos & Lander 2021), allowing the field to decay only on the true Ohmic timescale, τOhm ≫ 102 kyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In this case, field strengths of order ≳ 5 × 1015 G can be maintained over relatively long timescales if B0 ∼ 1017 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3 Braking mechanism Neutron stars in isolation spin down gradually as electro- magnetic and gravitational torques are applied, the mag- nitude of which can be phenomenologically quantified in terms of a braking index, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For a centered dipole that never 5 Note that, because |∇B|/B ∝ ℓ−1 for a general ℓ-pole, if one were to assume a purely dipole field for all t, a longer lengthscale L ≲ R⋆ could be justified, which would extend the Hall time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Such an assumption would, however, be inconsistent with the twisted surface configurations studied for death valleys in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' © ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' RAS, MNRAS 000, 1–12 A magnetar interpretation for GLEAM-X J1627 7 decays one has n = 3, while for a general ℓ-pole we have n = 2ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Leading-order contributions from gravitational radiation, being quadrupolar, also give n = 5, though with a different prefactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Values n < 3 are also possible for an oblique and/or precessing rotator (Melatos 1997, 1999), or in cases where particle outflows dominate the spin-down torque (Harding, Contopoulos & Kazanas 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' It is therefore useful to consider an evolution with an arbitrary braking index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The spin evolution of an inclined rotator in vacuum can be described by (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', Manchester & Taylor 1977) ˙P = (2π)n−1 B2 p sin2 αP 2−nR3+n ⋆ 6I0cn , (5) for moment of inertia I0 and magnetic inclination angle α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The quantity n in (5) represents the observational brak- ing index only if Bp is constant, though in the absence of ¨P data we treat it as phenomenological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We adopt the general-relativistic Tolman VII equation of state, for which I0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='38M⋆R2 ⋆ (Lattimer & Prakash 2001) for stellar mass M⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Equation (5) provides two useful pieces of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Firstly, the present-day observations of P and ˙P provide an estimate for Bp for a given braking index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Secondly, by solving equation (5) for some (time-dependent) choices of n and Bp, one can infer the age of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Given that we anticipate the object was born rapidly rotating so as to explain its large field strength (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='4), its present- day period must far exceed its birth period P0, though age (τ) estimates from equation (5) are insensitive to P(0) for P(0) ≪ P(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Note that magnetospheric (Spitkovsky 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Philippov, Tchekhovskoy & Li 2014), spheroidal, general rel- ativistic, or offset corrections (P´etri 2022) can be accounted for in the above to adjust the effective Bp value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' one ob- tains a hybrid Spitkovsky (2006) formula, for example, by replacing B2 p sin2 α in expression (5) with B2 p(1 + sin2 α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We solve equation (5) simultaneously with the volume- averaged induction equation (4) for several values of ˙P(τ) ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2 × 10−9ss−1 (Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022) assuming an or- thogonal rotator, α = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We fix n by demanding Bp(τ) = 5 × 1015 G, as it is difficult to explain the present-day radio switch on if the field is weaker (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 1, keeping in mind the caveats noted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The three a priori free pa- rameters, namely B0, τ, and n, are uniquely determined by the specified values of ˙P(τ), Bp(τ), and P(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Solutions are built through a shooting method: a set of initial conditions are iteratively determined such that there exists an age τ for which the aforementioned conditions are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Figure 3 shows the evolutions of the spin period (top panel) and the polar magnetic field (bottom) for cases where plastic flow is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Even for a relatively large range of the present-day period derivative, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0 × 10−11 ⩽ ˙P(τ)/(ss−1) ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2×10−9, the evolutions proceed in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This occurs because we require that the present-day B field is still strong, Bp(τ) = 5 × 1015G, so as to accommodate the death valley minima discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In the run with ˙P(τ) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0 × 10−11 ss−1, for example, the birth field strength must exceed 1017G so that it can survive long enough (until τ = 12kyr) to ensure that the present-day switch-on minimum is met, which implies greater spindown during early times t ≪ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' As such, even if a factor ≳ 10 weaker ˙P(τ) is assumed than the best-fit value reported by Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2022), predictions for the age are quantitatively similar in cases n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='65, P\uf110 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2\uf4a010-9ss-1 n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='69, P\uf110 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0\uf4a010-10ss-1 n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='84, P\uf110 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0\uf4a010-11ss-1 τ=12kyr τ=9kyr τ=7kyr PGLEAM-X J1627 = 1091 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='10 1 10 100 50 100 500 1000 Time (kyr) Spin period (s) n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='65, P\uf110 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2\uf4a010-9ss-1 n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='69, P\uf110 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0\uf4a010-10ss-1 n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='84, P\uf110 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0\uf4a010-11ss-1 τ=12kyr τ=9kyr τ=7kyr B(τ)=5\uf4a01015G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='10 1 10 100 5 ×1014 1 ×1015 5 ×1015 1 ×1016 5 ×1016 1 ×1017 Time (kyr) Polar field strength (G) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Solutions to equation (5) for the rotational period P(t) (top panel), assuming a time-dependent magnetic field Bp(t) whose evolution is governed by (4) (bottom panel), for a variety of ˙P(τ) values (see plot legends).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The plastic velocity is set to zero in these examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The age, braking index, and birth field strengths are set by the conditions that P(τ) = 1091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='17s and Bp(τ) = 5 × 1015G for some given value of ˙P(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' where plastic flow and other torques are inactive (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Ronchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Gen¸cali, Ertan & Alpar 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' By contrast, evolutions carried out for vpl ̸= 0 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In these cases, ˙P(τ) makes a significant difference for the age prediction: for ˙P(τ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2 × 10−9 ss−1 we find τ = 8kyr, while for the smaller value ˙P(τ) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0×10−11 ss−1 the age prediction increases to τ = 47kyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This is because plastic flow stalls field decay (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2), allowing the star to match Bp(τ) = 5×1015G without having to be born with a field exceeding 1017G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In this way, spindown is slower in the early stages and the star can be older.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Increasing the plastic velocity can increase the age further;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' in the limit that vpl → ∞ (or τOhm → ∞) the field does not decay at all, and the age is simply given by the characteristic value τ ∝ P(τ)/ ˙P(τ), which can be arbitrarily large if ˙P tends towards zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='4 Birth conditions: field amplification To a large degree, it remains an open question as to how magnetars acquire their intense fields, especially large-scale dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The saturation amplitude of the core field in the case of dynamo activity shortly after birth could reach ≲ 1016 G for convective heat fluxes of order ≳ 1039 erg cm−2s−1 © ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' RAS, MNRAS 000, 1–12 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Suvorov & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Melatos n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='65, P\uf110 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2\uf4a010-9ss-1 n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='69, P\uf110 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0\uf4a010-10ss-1 n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='84, P\uf110 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0\uf4a010-11ss-1 τ=8kyr τ=10kyr τ=47kyr PGLEAM-X J1627 = 1091 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='10 1 10 100 50 100 500 1000 Time (kyr) Spin period (s) n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='65, P\uf110 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2\uf4a010-9ss-1 n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='69, P\uf110 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0\uf4a010-10ss-1 n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='84, P\uf110 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0\uf4a010-11ss-1 τ=47kyr τ=10kyr τ=8kyr B(τ)=5\uf4a01015G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='10 1 10 100 1 ×1015 5 ×1015 1 ×1016 5 ×1016 1 ×1017 Time (kyr) Polar field strength (G) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3, though with a non-zero plastic ve- locity whose value is set by the scaling discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', doubling B relative to some fixed value implies a 3-fold increase in vpl, where we set vpl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5 cm yr−1 for B0 = 1016G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (Thompson & Duncan 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Provided that an ‘inverse cas- cade’ can operate, where energy from turbulent patches is transferred into a large-scale dipole (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Guilet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Raynaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2020), birth fields of this magnitude are suf- ficient for all of the known Galactic magnetars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Mechanisms beyond dynamo activity can amplify a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In particular, the Kelvin-Helmholtz and magneto-rotational instabilities can potentially lead to satu- ration magnetic energies of order Umag ∼ 1051 erg provided that the star is born with a (sub-)millisecond period (Kiuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Ciolfi 2020a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Shibata, Fujibayashi & Sekiguchi 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We stress however that numerical studies reporting such large magnetic energies do so in the context of merger remnants, which generally possess more angular momentum and seed magnetic fluxes than stars born from core-collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Regardless, magnetic energies of this order imply an up- per limit to the volume-averaged magnetic field strength at birth, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' ⟨B⟩max ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='7 × 1016 � Umag 1051 erg �1/2 � R⋆ 10 km �−3/2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (6) If the core field is at least as strong as the surface one (see also Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3), expression (6) implies that Bp ≲ ⟨B⟩max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The numerical simulations referenced above therefore sug- gest it is difficult to justify values exceeding Bp(t = 0) ∼ 1017 G (though cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Suvorov & Glampedakis 2022), even if toroidal fields (Glampedakis & Lasky 2015) or intense mag- netic spots (Vigan`o et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Suvorov, Mastrano & Gep- pert 2016) are localised in the crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3 IS THE MAGNETAR HYPOTHESIS EXCLUDED BY THE ABSENCE OF X-RAYS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Follow-up searches were carried out with the Swift X- ray Telescope for 2 ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' An upper limit of FX < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='9 × 10−13 erg s−1cm−2, is found for the flux in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3–10keV band, with FX < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5 × 10−13 erg s−1cm−2 applying instead for a blackbody fit6 at kT ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Based on the greatest distance allowed by the dispersion measure, dmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='8 kpc, this gives LX ⩽ 7 × 1031 erg s−1 for the X-ray luminosity (Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' A followup search conducted by Rea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2022) implies an even tighter upper-limit for this dmax, LX ⩽ 2×1030 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In this section, we review mod- els of thermal regulation in magnetars as a means to predict the surface temperature as a function of field strength (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1), which is quantitatively applied to GLEAM-X J1627 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1 Heating and cooling The absence of thermal X-rays in particular poses a chal- lenge to the magnetar interpretation of GLEAM-X J1627: the magnetised electron-proton plasma in the core experi- ences friction with the approximately static neutron fluid, gradually heating up the star while depleting magnetic en- ergy (Goldreich & Reisenegger 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Ambipolar heating, which sets a floor value to the tem- perature for a given age, is counteracted by neutrino cooling (Turolla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Ho, Glampedakis & Andersson 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Vigan`o et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Anzuini & Melatos 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Anzuini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' There is, therefore, a quasi-static balance tempera- ture, Tbal, set by matching the (time-dependent) heating and cooling rates, which generally must be several times 108 K to explain observations from active magnetars (Thompson & Duncan 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Beloborodov & Li 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Performing a volume average, the core temperature evo- lution can be approximately described by the first law of thermodynamics, CV dTcore dt = ˙QB − ˙Qν, (7) for heat capacity CV ≈ 2 × 1020(Tcore/109K)(ρ/ρnuc)1/3 erg K−1cm−3 (Beloborodov & Li 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Here, ˙QB and ˙Qν are the heating and cooling rates provided by magnetic field decay and neutrino emission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The quantity ρnuc is the nuclear saturation density, which may be exceeded in the core of a particularly heavy neutron star or if the equation of state (EOS) is soft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For the Akmal, Pandharipande & Ravenhall (1998) 6 Note that, in general, power-law components and not just one or more blackbodies are also needed to fit magnetar spectra, see Table 2 in Coti Zelati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For 4U 0142+61, for example, blackbody emissions represent ∼ 25% of the total X-ray power (Rea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Spectral complications can be accounted for crudely in the models here via the efficiency parameter ϵ intro- duced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' © ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' RAS, MNRAS 000, 1–12 A magnetar interpretation for GLEAM-X J1627 9 EOS [which passes constraints from GW170817 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2018) and can accommodate the heaviest pulsar observed to date, PSR J0740+6620, with M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='08+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='07M⊙ (Fonseca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2021)], a star of mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='39M⊙ has a central density ρc = 9 × 1014 g cm−3 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2ρnuc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This increases to ρc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1 × 1015 g cm−3 for a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='66M⊙ star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Following Beloborodov & Li (2016) and others, the two main cooling mechanisms we consider are the modified (mUrca) and the fast, direct Urca (dUrca) mechanisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' the former is thought to be the dominant neutrino mechanism in (non-superfluid) nucleon matter (ρ ≲ 2ρnuc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Yakovlev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2002), while the latter may activate in the core of particular dense stars (ρ ∼ 4ρnuc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Lattimer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The presence of hyperons may reduce fast cooling thresholds (Anzuini & Melatos 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Anzuini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The mUrca cooling rate can be approximated by (Friman & Maxwell 1979) ˙QM ν ≈ 7×1020 � Tcore 109K �8 � ρ ρnuc �2/3 RM erg s−1cm−3, (8) where RM ⩽ 1 is a suppression factor relevant if either pro- tons or neutrons are superfluid, whereupon the breaking of Cooper pairs instead becomes the dominant cooling mech- anism at densities ρ ∼ ρnuc (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We henceforth ignore such complications in our phenomenologi- cal heating model (7), though these should be considered in realistic magnetothermal modelling if the core temperature drops below the superfluidity onset value 1 ≲ Tcrit/108K ≲ 10 (Potekhin, Pons & Page 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The dUrca cooling rate is given by (Lattimer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 1991) ˙QD ν ≈ 1027 � Tcore 109K �6 erg s−1cm−3, (9) which exceeds (8) by several orders of magnitude for tem- peratures in the range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The rate of heating, provided by ambipolar diffusion, can be estimated through (Beloborodov & Li 2016) ˙QB ≈ τpn ρp � B2 4πL �2 , (10) for core field strength B which varies over lengthscale L, where 1/τpn denotes the rate of p-n collisions per proton (ig- noring core exotica), given by (Yakovlev & Shalybkov 1990) τ −1 pn ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='7 × 1018 � Tcore 109K �9 � ρ ρnuc �−1/3 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (11) In the simplified model (7), a magnetar, born with tem- perature T0 ≲ 1011 K, reaches a quasi-static balance temper- ature Tbal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', dT/dt = 0) after ≳ 10 years (even less with dUrca), where the temperature remains until field decay sets in (∼kyr for B ∼ 1016G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Assuming a present-day core field of ∼ 1016G, these balance temperatures read T M bal ≈ 8 × 108 �B2 16 L5 �1/5 � ρ ρnuc �−7/30 K, (12) for mUrca, and T D bal ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3 × 108 �B2 16 L5 �1/4 � ρ ρnuc �−1/12 K, (13) for dUrca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Core-crust thermal transport depends primarily on the chemical composition of the stellar envelope and the mag- netic stratification, which influence the photon opacity (Tsu- ruta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Given a surface temperature Ts, the flux Fs = σSBT 4 s , (14) for Stefan-Boltzmann constant σSB ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='67 × 10−5erg cm−2 s−1K−4, defines a surface luminosity Ls = 4πR2 ⋆ � 1 0 d (cos θ) Fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (15) The co-latitude (θ) dependence in (15) comes through the angle between the magnetic field, assumed dipolar (see be- low), and the surface normal (see Beloborodov & Li 2016, for more details), which affects the thermal conductivity tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For a slow source (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', ignoring rotational corrections to the metric tensor), the redshifted luminosity seen by an observer at infinity is then L∞ s = Ls(1 − 2GM⋆/c2R⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2 Magneto-thermal modelling Numerical simulations for the core-surface temperature re- lationship were carried out by Potekhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Us- ing their analytic fits (which are too long to repeat here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' see their Appendix A), we calculate the luminosity an ob- server expects to see from GLEAM-X J1627 as a function of the core field strength, assuming the system is in thermal quasi-equilibrium with balance temperature (12) (mUrca) or (13) (dUrca) and that there are no other heat sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For young (≪ kyr) stars or ones where Joule heating, mechani- cal heating, or positron backflow from the magnetosphere is also significant, higher temperatures are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We fur- ther assume an iron envelope, as a crust composed of lighter elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', accreted materials) conducts heat more ef- ficiently and predicts a higher Ts for a given B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Landau quantization, which we also ignore, similarly leads to higher temperatures, because electrons are forced to move along the field lines, thereby suppressing their ability to transfer heat radially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Figure 5 shows the balance temperature (12) (red curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' left axis) as a function of the core field strength, where we consider core densities of ρ = ρnuc (upper curves) and ρ = 4ρnuc (lower curves) and the mUrca cooling rate (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Figure 6 is similar, though instead with the dUrca rate (9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' note the different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' To provide an optimistic outlook, we take L = R⋆ so that the magnetic energy is predominantly concentrated in low multipoles (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Footnote 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The right axes (blue curves) show the surface luminos- ity (15) witnessed by an observer at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' These fig- ures illustrate that there is generally an upper limit for the core field strength implied by the absence of X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For example, even if we assume a tiny X-ray efficiency of ϵ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', LX ≲ 10−3L∞ s ), the Chandra observa- tions of GLEAM-X J1627, which translate into an upper- limit of LX ∼ 1030 erg s−1 (Rea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022), require core field strengths of B ≲ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1 × 1014 G for ρ = ρnuc and B ≲ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5 × 1014 G for ρ = 4ρnuc, as shown by the dashed, vertical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Larger, percent-level efficiencies place even tighter constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The corresponding limits for dUrca are much less restrictive, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' B ≲ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2×1015 G for ρ = 4ρnuc for ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1%, or B ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='8 × 1014 G for ϵ = 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In the magnetothermal evolutions carried out by © ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' RAS, MNRAS 000, 1–12 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Suvorov & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Melatos Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Quasi-static core temperatures (red curves) set by bal- ancing mUrca cooling and ambipolar heating [expression (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' left axis], as a function of the magnetic field strength, for two different densities, ρ = ρnuc (upper curve) and ρ = 4ρnuc (lower curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The right-axis (blue curves) shows the predicted, redshift- corrected surface luminosity (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' An upper limit to L∞ s implies an upper limit to the internal B field, thereby issuing a con- straint on GLEAM-X J1627, for which LX,max ≈ 1030 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The solid, horizontal line corresponds to this upper limit for a conversion efficiency of ϵ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', LX ≲ 10−3L∞ s , which translates into upper limits for B (dashed, vertical lines) for a given core density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 5 though with direct Urca cooling (9) and also a greater efficiency, ϵ = 10% (lower, solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Anzuini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2022b), it was shown that the surface lumi- nosity of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='8M⊙ magnetar (B ≳ 1015 G) with Joule heating only dips below ≳ 1033 erg s−1 at times t ≲ Myr post-birth, even if there are hyperons and fast cooling mechanisms are active (see Figures 1 and 2 therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This estimate, which is a factor ∼ 5 more restrictive than the most optimistic, ambipolar model used here (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 6), is at odds with the minima required by the radio activation mechanisms (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This casts doubt on a magnetar interpretation for the source, unless the thermal luminosity is much higher, or the system is old (Ronchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Gen¸cali, Ertan & Alpar 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We emphasise however that a realistic investi- gation for GLEAM-X J1627 requires a proper magnetother- mal evolution in the presence of an ultra-strong field, which is difficult (though see Rea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' We close by noting that the magnetothermal study of Perna & Pons (2011) found that the waiting time distribu- tion for flares from young (≲ kyr) magnetars peaks at ∼ 1 yr, and thus the absence of any flare phenomena in the ∼ 2 ks window, where the source was observed with Swift, is not entirely surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For a source that is several kyr old, the peak of the waiting time distribution shifts to ≳ 3 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Fur- thermore, bursts may be missed if beamed away from Earth, making it less clear how long one might need to observe be- fore expecting a flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' However, a plastically-flowing crust could be even hotter than one that never breaks because thermoplastic waves can dissipate magnetic energy, the ef- fects of which resemble deflagration fronts in combustion (Beloborodov & Li 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Targeted searches would be use- ful in this direction to shed light on the matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 4 CONCLUSIONS The source GLEAM-X J1627 was recently discovered by Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2022), who reported an extremely long spin period (P = 1091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='17s) together with a possibly large period derivative (| ˙P| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2×10−9 ss−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The magnetic field strength implied, assuming a neutron star undergoing mag- netic dipole braking (though see Loeb & Maoz 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Katz 2022, for a white dwarf interpretation), comfortably exceeds 1016 G when using the best-fit value ˙P = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0 × 10−10 ss−1 (Ruderman & Sutherland 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In this paper, a critical examination of the magnetar interpretation is carried out, though under the proviso that model-based specifics are in- escapable and conclusions cannot be asserted strongly based on the limited data at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Magnetospheric gap models require a minimum mag- netic field strength, for a given period, for the switch-on of the star as a radio pulsar (Goldreich & Julian 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Stur- rock 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Hibschman & Arons 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Medin & Lai 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For canonical stellar parameters, we find in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1 that minimum fields of order ∼ 1016 G appear to be necessary, even when assuming a high degree of multipolarity, long- lived twists in the magnetosphere (Beloborodov 2009), and small curvature radii Rc ∼ R (Medin & Lai 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' If the star has a large radius, R⋆ ≳ 13 km, this requirement may drop to Bp,min ≲ 5 × 1015 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Standard electromagnetic braking theory suggests that the star is between ∼ 10 and 50 kyrs old, depending on the historical braking index and field evo- lution model (though cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Ronchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Gen¸cali, Ertan & Alpar 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Assuming ages much larger than 10 kyr and a present-day ∼ 5 × 1015 G polar field, a Hall-Ohm back- extrapolation implies a birth strength of ≲ 1017 G, and fur- ther that field decay was stalled to some degree, possibly by plastic opposition of the electron fluid motion in the crust (Lander & Gourgouliatos 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Gourgouliatos, De Grandis & Igoshev 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This points towards there having been a large angular momentum reservoir at birth to support in- tense field amplification via some combination of dynamo activity, Kelvin-Helmholtz action, or magneto-rotational in- stabilities (Ciolfi 2020a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' A simple magneto-thermal model is employed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 3 to show that the competition between heating induced by field decay and neutrino cooling implies a particular sur- face luminosity, depending on assumptions on the thermal conductivity, stellar composition, and Urca channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The lack of strong thermal emissions from the source (LX ≲ 1030 erg s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Rea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022) is difficult to reconcile with the radio requirements, unless fast cooling mechanisms are in © ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' RAS, MNRAS 000, 1–12 1035 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='5×1014G 5 4 1034 V 0 B B (erg/s) 3 8 erg/s 1033 8 2 1032 0 1031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='100 1 B16 (G)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='8 1035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='6 1034 L=1033erg/s (erg/s) 1033 G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='4 G G 0 5 8 1032 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='2 V 1031 L=1031erg/s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='0 1030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='50 B16 (G)A magnetar interpretation for GLEAM-X J1627 11 operation (Lattimer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' A heavier star with larger moment of inertia, which is generally easier to cool quickly (Anzuini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022a), could also help to alleviate the ten- sion between the available spin-down power and observed radio luminosity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Another clue about the nature of GLEAM-X J1627 comes from the transient character of its radio pulsations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2022) noted that the source (visibly) pulsated for only 3 months and then abruptly turned off, indicating an overall duty cycle of only ∼ 2% within the ob- servational monitoring window (see also Footnote 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' This could occur, if the source hovers near the death line, with magnetohydrodynamic evolutions triggering its descent into the graveyard around March of 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' For example, a crustal fracture may have injected twist into the magnetosphere prior to the object’s discovery, allowing it to temporarily access line (d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' In the case of the so-called ro- tating radio transients (RRATs), which similarly display high degrees of nulling, it was suggested by Zhang, Gil & Dyks (2007) that concentrated starspots may emerge near the poles, sporadically allowing the host star to rise above the death line (see also Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1, Suvorov, Mastrano & Gep- pert 2016, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Magnetar-like X-ray bursts are known to suppress radio pulsations in many neutron stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' bursts observed in PSR J1119–6127 by XMM-Newton and NuSTAR were coincident with the shut-off of the source as a radio pulsar (Archibald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2017), for example [see also Coti Zelati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' (2018) for a discussion on other sources].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' If GLEAM-X J1627 is regu- larly bursting, as would be expected if Bp ≳ 1016 G and the crust frequently succumbs to Maxwell stresses, this could also explain the high degree of nulling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The absence of any X-ray activity (Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022) casts doubt how- ever on this interpretation, though geometric factors related to beaming and directionality may explain this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Finally, the population study recently conducted by Sheikh & MacDon- ald (2021) indicates that there is a (weak) correlation be- tween the spin period and nulling fraction in radio pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The high nulling fraction and long spin period of GLEAM-X J1627 fits within this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Regardless, further monitor- ing of the source in both the radio and X-ray bands will help to unveil its magnetar nature or otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' If indeed GLEAM-X J1627 boasts a polar field strength greater than 1016 G, as suggested by its place in the P– ˙P dia- gram and the death valley considerations (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='1), it would have been an ample source of gravitational waves when born.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Even in the absence of a toroidal field, the quadrupo- lar ellipticity of the source could easily reach ∼ 10−4 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=', Haskell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' Mastrano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The source, lo- cated ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3 kpc from Earth (Hurley-Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2022), would have been visible to the advanced Laser Interferom- eter Gravitational-Wave Observatory (aLIGO) for P ≪ 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' From the braking analysis given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='3, if the birth pe- riod was at most a few ms (as argued in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content='4), the source would have been sufficiently bright in gravitational waves to enable detection for ∼ years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The existence of GLEAM- X J1627 therefore adds further incentive to perform blind, gravitational-wave searches for magnetar-like sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We extend our thanks to Filippo Anzuini for discussions about direct Urca processes and the heating effects of Landau quantization in magnetars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' AGS thanks Kostas Glampedakis for pointing out the possibility of a Hall attrac- tor state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The research leading to these results has received funding from the European Union’s Horizon 2020 Pro- gramme under the AHEAD2020 project (grant n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' 871158).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFAT4oBgHgl3EQfYx1Q/content/2301.08541v1.pdf'} +page_content=' The constructive criticisms of the anonymous referee, which led to a richer study, are gratefully acknowledged.' metadata={'source': 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b/dtAyT4oBgHgl3EQfwvmK/content/tmp_files/2301.00654v1.pdf.txt @@ -0,0 +1,8751 @@ +arXiv:2301.00654v1 [math.AP] 2 Jan 2023 +ON THE EXISTENCE AND UNIQUENESS OF SOLUTION TO A STOCHASTIC +CHEMOTAXIS-NAVIER-STOKES MODEL +E. HAUSENBLAS˚, B. JIDJOU MOGHOMYE˚ AND P. A. RAZAFIMANDIMBY˚˚ +˚ Department of Mathematics and Information Technology, Montanuniversitaet Leoben, +Leoben Franz Josef Strasse 18, 8700 Leoben, Austria +˚˚ School of Mathematical Science, Dublin City University, Collins Avenue Dublin 9, Ireland +ABSTRACT. In this article, we study a mathematical system which models the dynamic of +the collective behaviour of oxygen-driven swimming bacteria in an aquatic fluid flowing in +a two dimensional bounded domain under stochastic perturbation. This model can be seen +as a stochastic version of Chemotaxis-Navier-Stokes model. +We prove the existence of a +unique (probabilistic) strong solution. In addition, we establish some properties of the strong +solution. More precisely, we prove that the unique solution is non-negative and satisfies the +mass conservation property and an energy inequality. +1. INTRODUCTION +The migration of bacteria cells to a higher concentration of a chemical has been observed +in biological applications concerning aerobic bacteria. +This phenomenon, called chemotaxis, +is presumed to have a deep impact on the time evolution of a bacteria population. +There +are different concepts of chemotaxis depending on the kind of bacteria and the chemical. In +the present article, we focus on the mathematical model describing an oxygen-driven bacteria +suspension swimming in an incompressible fluid like water which was firstly proposed in +[39]. +Mainly, the system consists of three coupled partial differential equations. +The first +equation describes the fluid flow with field velocity u. +The second equation describes the +dynamic of the oxygen concentration c, and the last equation describes the dynamic of the +population density n of the bacteria. Now, the coupled model can be written as +(1.1) +$ +’ +’ +’ +’ +’ +’ +’ +’ +’ +& +’ +’ +’ +’ +’ +’ +’ +’ +’ +% +du ` rpu ¨ ∇qu ` ∇P ´ η∆us dt “ n∇Φdt in r0, Ts ˆ O, +dc ` u ¨ ∇cdt “ rµ∆c ´ nfpcqs dt in r0, Ts ˆ O, +dn ` u ¨ ∇ndt “ rδ∆n ´ ∇ ¨ pnχpcq∇cqs dt in r0, Ts ˆ O, +∇ ¨ u “ 0 in r0, Ts ˆ O, +np0q “ n0, +cp0q “ c0, +up0q “ u0 +in +O. +In addition to the unknows u, c, n, we have the scalar pressure P. The positive number T +is the final observation time, and O Ă R2 is a domain where the cells and the fluid move +and interact. The positive constants η, µ and δ are the corresponding diffusion coefficients +Date: January 3, 2023. +2000 Mathematics Subject Classification. 35R60,35Q35,60H15,76M35,86A05. +Key words and phrases. Navier-Stokes system; +Chemotaxis; +Stochastic; +Probabilistic weak solution; +strong +solution . +1 + +2 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +for the fluid, the oxygen, and the bacteria, respectively. The given functions χ and f denote +the chemotactic sensitivity and the oxygen consumption rate, respectively. +The symbol Φ +denotes a given time-independent potential function representing, e.g., the gravitational force +or centrifugal force. +The mathematical analysis of system (1.1) has been investigated by several authors. The +existence of weak solutions and the existence of a unique classical solution have been proven, +see for instance [9, 10, 15, 16, 18, 25, 36, 37, 42, 43] and references therein. In the case +d “ 2, the existence of a global weak solutions for (1.1) without the nonlinear convective +term pu ¨ ∇qu is obtained in [16, 36, 37] and in [18] with nonlinear diffusion. The existence +of weak global solutions under various assumptions on the data can be found in [15, 25]; the +global existence of smooth solutions has been proven in [10, 42]. Results on the existence +of classical solution are found in [9, 16, 43]. +Fix T ą 0. In this paper, we are interested in the mathematical analysis of a stochastic +version of problem (1.1) in the two-dimensional bounded domain. More precisely, for a given +family of independent, identically distributed standard real-valued Brownian motions tβkuk“1,2, +and a cylindrical Wiener processes W evolving on a fixed separable Hilbert space U defined +on a filtered probability space, pΩ, F, pFtqtPr0,Ts, Pq, we consider the following system +(1.2) +$ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +& +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +’ +% +du ` rpu ¨ ∇qu ` ∇P ´ η∆us dt “ n∇Φdt ` gpu, cqdWt in r0, Ts ˆ O, +dc ` u ¨ ∇cdt “ rµ∆c ´ nfpcqs dt ` γ +2ÿ +k“1 +σk ¨ ∇c ˝ dβk +t in r0, Ts ˆ O, +dn ` u ¨ ∇ndt “ rδ∆n ´ ∇ ¨ pnχpcq∇cqs dt in r0, Ts ˆ O, +∇ ¨ u “ 0 in r0, Ts ˆ O, +Bn +Bν “ Bc +Bν “ 0 +on +r0, Ts ˆ BO, +u “ 0 +on +r0, Ts ˆ BO, +np0q “ n0, +cp0q “ c0, +up0q “ u0 +in +O, +where O Ă R2 is a bounded domain with smooth boundary BO and the positive constant γ is +the intensity of the noise. The symbol ˝ means that the stochastic differential is understood +in the Stratonovich sense. +The main difference between the deterministic model (1.1) and +the stochastic model (1.2) is the presence of the terms gpu, cqdWt and γ ř2 +k“1 σk ¨ ∇c ˝ dβk +t +called noise terms. +The presence of these noise terms weakened the regularity in time of +the velocity field and the concentration of oxygen and so, make the mathematical analysis +more involved. +Our investigation is motivated by the need for a sound mathematical analysis for the +understanding of the effect of small scale perturbations such as random pollution of water or +air which are inherently present in nature (see [11, 29]). The presence of these stochastic +perturbations can lead to new and important phenomena. In fact, in two-dimensional case, many +models such as the Navier-Stokes equation, the Oldroy-B type model, the Landau-Lifshitz-Bloch +equation, and magnetohydrodynamics model with sufficiently degenerate noise for example +have a unique invariant measure and hence exhibit ergodic behavior in the sense that the time +average of a solution is equal to the average over all possible initial data. Despite continuous +efforts in the last 30 years, such property has so far not been found for the deterministic +counterpart of these equations. +This property could lead to profound understanding of the +nature of turbulence. +To the best of our knowledge, +the only papers that consider the + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +3 +mathematical analysis of a stochastic version of chemotaxis-fluid interaction model are [44, 45] +where the authors have proved the existence of both mild and weak solutions for the model +(1.2) with γ “ 0 and gpu, cq “ gpuq in a two and three dimensional bounded domain under +some strong assumptions on the data. +The aim of this article is to study the global resolvability of problem (1.2) with positive +parameters η, µ γ and δ different from zero. We prove the existence and uniqueness of a +probabilistic strong solution in a two dimensional bounded domain. The proof is based on +a Galerkin scheme and the Yamada-Watanabe Theorem. Let us recall that the presence of +the noise on the c-equation makes the mathematical analysis of the model more involved. +In fact, the noise term in c-equation makes impossible the application of the deterministic +maximum principle method for the proof of the non-negativity of solution as is done in the +literature. +Moreover, the stochastic version of maximum principle method where we learn +from [14] need to be adapted in order to conserve the positivity of solutions. +The main +difference between our work and that of [44] is that the model considered in [44] does not +contain any noise on the c-equation and the noise term in the u equation depend only on +the velocity field u. Therefore, the present paper can be seen as a generalization of [44]. +The organisation of this article is as follows. In Section 2, we define various functional +spaces, and introduce assumptions which are used throughout in our paper. +In Section 3, +we state and prove the main result which is the existence of a unique probabilistic strong +solution. In Section 4, we give a detailed proof of important ingredients which have been +useful for the proof of the main result. +In Section 5, we prove the mass conservation +property and the non-negativity of the strong solution. +Besides that, we prove an energy +inequality which may be useful for the study of the invariant measure in future. +2. FUNCTIONAL +SETTING +OF +THE +MODEL +AND +ASSUMPTIONS +Throughout the paper, we assume that O Ă R2 is a bounded domain with boundary BO +of class C8. The symbol LppOq denotes the Lp space with respect to the Lebesgue measure +while W m,ppOq denotes the Sobolev space of functions whose distributional derivatives of +order up to m belong to LppOq. +The spaces of functions φ : O Ñ R2 such that each +component of φ belongs to LppOq or to W m,ppOq are denoted by LppOq or by Wm,ppOq. +We denote by |.|Lp the norm on LppOq or LppOq and by }.}W m,q the norm on W m,ppOq or +Wm,ppOq. For p “ 2 the function space W m,2pOq (resp. Wm,2pOq) is denoted by HmpOq +(resp. HmpOq) and its norm will be denoted by |¨|Hm. By H1 +0pOq we mean the space of +functions in H1 that vanish on the boundary BO. +The inner product on L2pOq will be +denoted by p¨, ¨q. +Following the notations using in [38] for the Navier-Stokes model, we +introduce the following space V “ tv P C8 +c pO; R2q : such that ∇ ¨ v “ 0u, and define the +spaces H and V +as the closure of V in L2pOq and H1 +0pOq, respectively. +We endow H +with the scalar product and norm of L2pOq. +As usual, we equip the space V +with the +gradient-scalar product and the gradient-norm |∇¨|L2, which is equivalent to the H1 +0pOq-norm. +As usual, P denotes the Helmholtz projection from L2pOq onto H. It is also known that +V +is dense in H and that the embedding is continuous and compact. Identifying H with +its dual, we have the Gelfand triple V ãÑ H ãÑ V ˚. +We define the Newmann Laplacian operator on L2pOq by A1φ “ ´∆φ for all φ P DpA1q +where +DpA1q “ tφ P H2pOq : Bφ +Bν “ 0, on BOu. + +4 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +It is known that A1 is a non-negative self-adjoint operator in L2pOq. As we are working on a +bounded domain, A1 has compact resolvent, see e.g. [7]. Hence, there exists an orthonormal +basis tϕiu8 +i“1 Ă C8pOq of L2pOq consisting of the eigenfunctions of the Neumann Laplacian +A1. Also we have the dense and compact embeddings H2pOq ãÑ H1pOq ãÑ L2pOq. +Now we define the Hilbert space H by +H “ H ˆ H1pOq, +endowed with the scalar product whose associated norm is given by +|pu, cq|2 +H “ |u|2 +L2 ` |c|2 +H1 , pu, cq P H. +We introduce the bilinear operators B0, B1 and R2 and their associated trilinear forms +b0, b1 and r2 respectively as follows: +pB0pu, vq, wq “ +ż +O +rpupxq ¨ ∇qvpxqs ¨ wpxqdx “ b0pu, v, wq, @u P V, v P V, w P V, +pB1pu, cq, ψq “ +ż +O +upxq ¨ ∇cpxqψpxqdx “ b1pu, c, ψq, @u P V, c P H1pOq, ψ P H1pOq, +pR2pn, cq, ψq “ +ż +O +∇ ¨ pnpxq∇cpxqqψpxqdx +“ ´ +ż +O +npxq∇cpxq ¨ ∇ψpxqdx “ r2pn, c, ψq, @n P L2pOq, c P H1pOq, ψ P H3pOq. +It is well known in [38, Chapter II, Section 1.2] that the operator B0 is well-defined. The +operator B1 is well-defined for u P V , c P H1pOq and ψ P H1pOq since by the H¨older +inequality and the Sobolev embedding of H1pOq into L4pOq, we have +pB1pu, cq, ψq ď |u|L4 |∇c|L2 |ψ|L4 +ď K |∇u|L2 |c|H1 |ψ|H1 . +In a similar way, we can also check that the operator R2 is well-defined for n P L2pOq, +c P H1pOq and ψ P H1pOq. +In fact, in addition to the H¨older inequality, by using the +Sobolev embedding of H2pOq into L8pOq, we see that +pR2pn, cq, ψq ď |n|L2 |∇c|L2 |∇ψ|L8 +ď |n|L2 |c|H1 |ψ|H3 . +We also introduce the following coupling mappings R0 and R1 +pR0pn, Φq, vq “ +ż +O +npxq∇Φpxq ¨ vpxqdx, @n P L2pOq, v P H, Φ P W 1,8pOq, +pR1pn, cq, ψq “ +ż +O +npxqfpcpxqqψpxqdx, @n P L2pOq, c P L8pOq, ψ P L2pOq, f P L8pRq. +We note that the operators R0 and R1 are well-defined. Indeed, for n P L2pOq, v P H and +Φ P W 1,8pOq we see that +pR0pn, Φq, vq ď |Φ|W 1,8 |n|L2 |v|L2 . +Further, for n P L2pOq, c P L8pOq, ψ P L2pOq and f P L8pRq, we also see that +pR1pn, cq, ψq ď |fpcq|L8 |n|L2 |ψ|L2 . +Hereafter, A :“ pΩ, F, pFtqtPr0,Ts, Pq will be a complete probability space equipped with a +filtration pFtqtPr0,Ts satisfying the usual conditions, i.e. the filtration is right-continuous and + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +5 +all null sets of F are elements of F0. +Let U be a separable Hilbert space with basis +teku8 +k“1 and W +be a cylindrical Wiener process over U. +In particular, according to [12, +Proposition 4.3] the Wiener process t ÞÑ Wt can be expressed as +Wt “ +8 +ÿ +k“1 +W k +t ek, +for all t P r0, Ts, +where tW k : k P Nu is a family of mutually independent standard R-valued Brownian motion +over A. +For any Hilbert space X, we will denote by L2pU; Xq the separable Hilbert space of +Hilbert-Schmidt operators from U into X. For a separable Banach space X, p P r1, 8q and +T ą 0 we denote by Mp +Ap0, T; Xq the space of all processes ψ P LppΩˆp0, Tq, dPbdt; Xq over +A, being tFtutPr0,Ts-progressively measurable. We denote by LppΩ; Cpr0, Ts; Xqq, 1 ď p ă 8, +the +space +of +all +continuous +and +tFtutPr0,Ts-progressively +measurable +X-valued +processes +tψt; 0 ď t ď Tu over A satisfying +E +« +sup +tPr0,Ts +}ψt}p +X +ff +ă `8. +If Y is a Banach space, we will denote by LpX, Y q the space of bounded linear operators. +From the theory of stochastic integration on infinite dimensional Hilbert space (see [12, +Chapter 4]), for any process ρ P M2 +Ap0, T; L2pU; Hqq, the stochastic integral of ρ with respect +to the Wiener process t ÞÑ Wt is denoted by +ż t +0 +ρpsqdWs, +0 ď t ď T, +and is defined as the unique continuous H-valued martingale over A, such that for all h P H, +we have +ˆż t +0 +ρpsqdWs, h +˙ +H +“ +8 +ÿ +k“1 +ż t +0 +pρpsqek, hqHdW k +s , +0 ď t ď T, +where the integral with respect to dW k +s +is understood in the sense of Itˆo. +We introduce now the following conditions on the parameters and functions involved in +the system (1.2). +Assumption 2.1. For the parameter functions χ, f and Φ in (1.2), we assume that χpcq is +a non-negative constant, i.e. χpcq “ χ ą 0 and require that f and Φ satisfy +f P C1pr0, 8qq, +fp0q “ 0, +and +f ą 0, +f 1 ą 0 +in p0, 8q, +Φ is time-independent and Φ P W 1,8pOq. +(2.1) +Throughout this paper, we set +(2.2) +Kf :“ +χ2 +2δ +min +0ďcď|c0|L8 f 1 ` +1 +min +0ďcď|c0|L8 f 1. +Furthermore, we consider a family of vector fields tσ1, σ2u satisfying the following assumptions. +Assumption 2.2. +(A1) For k P t1, 2u, σk :“ pσ1 +k, σ2 +kq P W 1,8pOq ˆ W 1,8pOq and σk “ 0 on BO. +(A2) σk is a divergence free vector fields, that is ∇ ¨ σk “ 0, for k “ 1, 2. + +6 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +(A3) The matrix-valued function q : O ˆ O Ñ R2 b R2 defined by +(2.3) +qi,jpx, yq “ +2ÿ +k“1 +σi +kpxqσj +kpyq, +@i, j “ 1, 2 and @x, y P O, +satisfies qpx, xq “ IdR2 for any x P O. +Before introducing the other standing assumptions used in this paper, we shall make few +important remarks and observations on Assumption 2.2 and the noise +2ÿ +k“1 +σk ¨ ∇c ˝ dβk +t . +Remark 2.1. Setting for k “ 1, 2, +σkpxq “ +$ +& +% +gk +if x P ¯OzBO, +0 +if x P BO, +where tg1, g2u is the canonical basis of R2, the family of vector fields tσ1, σ2u satisfies +(A1), (A2) and (A3). +Hereafter we will use the following notation +(2.4) +|σ|L8 “ +˜ 2ÿ +k“1 +|σk|2 +L8 +¸1{2 +and +|σ|W 1,8 “ +˜ 2ÿ +k“1 +|σk|2 +W 1,8 +¸1{2 +. +Owing to [17, p. +65, Section 4.5.1], the Stratonovich integral γ +şt +0 σk ¨ ∇cpsq ˝ dβk +s +can be +expressed as the Itˆo integral with a correction term as follows: +(2.5) +γ +ż t +0 +σk ¨ ∇cpsq ˝ dβk +s “ 1 +2 +ż t +0 +Dcpγσk ¨ ∇cpsqqpγσk ¨ ∇cpsqqds ` γ +ż t +0 +σk ¨ ∇cpsqdβk +s , +where, Dcpγσk ¨ ∇cq denotes the Fr´echet derivative of γσk ¨ ∇c with respect to c. +Lemma 2.2. If Assumption 2.2 holds, then for all t P r0, Ts, +(2.6) +1 +2 +ż t +0 +2ÿ +k“1 +Dcpγσk ¨ ∇cpsqqpγσk ¨ ∇cpsqqds “ γ2 +2 +ż t +0 +∆cpsqds, c P H2pOq. +Proof. Let c P H2pOq and t P r0, Ts be arbitrary but fixed. Then for all s P r0, ts and k “ 1, 2, +2ÿ +k“1 +Dcpγσk ¨ ∇cqpγσk ¨ ∇cq “ γ +2ÿ +k“1 +σk ¨ ∇pγσk ¨ ∇cq “ γ2 +2ÿ +k“1 +σk ¨ ∇pσk ¨ ∇cq. +Since ∇ ¨ σk “ 0, we remark that σk ¨ ∇c “ ∇ ¨ pcσkq and therefore, +γ2 +2ÿ +k“1 +σk ¨ ∇pσk ¨ ∇cq “ γ2 +2ÿ +k“1 +σk ¨ ∇p∇ ¨ pcσkqq “ γ2 +2ÿ +k“1 +∇ ¨ pσk∇ ¨ pcσkqq . +(2.7) +For the second equality we have used once more the fact that ∇ ¨ σk “ 0 for all k “ 1, 2. +Since σk “ pσ1 +k, σ2 +kq P W 1,8pOq ˆ W 1,8pOq and c P H2pOq ãÑ L8pOq, we can apply the +differentiation of product formula given in [2, Proposition 9.4, P. 269] to obtain, +(2.8) +2ÿ +k“1 +∇ ¨ pσkp∇ ¨ pcσkqq “ +2ÿ +i,j“1 +B2 +BxiBxj +pqijpx, xqcq ´ ∇ ¨ +˜˜ 2ÿ +k“1 +σk ¨ ∇σk +¸ +c +¸ +, + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +7 +where σk ¨ ∇σk is the vector field with components +pσk ¨ ∇σkqi “ +2ÿ +j“1 +σj +k +B +Bxj +σi +k. +Applying the differentiation of product formula once more, for j “ 1, 2, we see that +(2.9) +2ÿ +k“1 +2ÿ +j“1 +p∇σk ¨ σkqi “ +2ÿ +j“1 +B +Bxj +qijpx, xq ´ +2ÿ +k“1 +σi +k∇ ¨ σk “ +2ÿ +j“1 +B +Bxj +δij “ 0. +In (2.9), we have used the fact that ∇ ¨ σk “ 0 and also the fact that qij “ δij (see (A3) of +assumption 2). +From (2.8) and (2.9), we infer that +(2.10) +2ÿ +k“1 +∇ ¨ pσk∇ ¨ pcσkqq “ +2ÿ +i,j“1 +B2 +BxiBxj +pqijpx, xqcq “ +2ÿ +i,j“1 +B2 +BxiBxj +pδijcq “ ∆c. +Combining (2.10) and (2.7), we derive (2.6) which completes the proof of Lemma 2.2. +□ +Define for k P t1, 2u, a map φk : H1pOq Ñ L2pOq by φkpcq “ σk ¨ ∇c. +Then, the map +φ : H1pOq Ñ L2pR2; L2pOqq given by +φpcqphq “ +2ÿ +k“1 +φkpcqhk, +c P H1pOq, h “ ph1, h2q P R2, +is well defined under the condition (A1). +Let tg1, g2u be the orthonormal basis of R2 +then φpcqpgkq “ φkpcq, for all c P H1pOq. Let β “ pβ1, β2q be a standard two dimensional +Brownian motion over A, independent of W. We will repeatedly use the following notation +(2.11) +φpcqdβs “ +2ÿ +k“1 +φkpcqdβk +s . +We recall that throughout this paper, the symbols K, KGN and Ki, i P N will denote positive +constants which may change from one line to another. +Assumption 2.3. Let g : H Ñ L2pU, Hq be a continuous mapping. In particular, there exists +a positive constant Lg such that for any pu, cq P H, +(2.12) +|gpu, cq|L2pU,Hq ď Lgp1 ` |pu, cq|Hq. +Assumption 2.4. Let g : H Ñ L2pU, Hq be a Lipschitz-continuous mapping. +In particular, +there exists a positive constant LLip such that for all pui, ciq P H, i “ 1, 2, +(2.13) +|gpu1, c1q ´ gpu2, c2q|L2pU;Hq ď LLip |pu1 ´ u2, c1 ´ c2q|H . +Using the previous notations, setting ξ “ η ` γ2 +2 , and taking into account Lemma 2.2, the +model (1.2) can formally be written in the following abstract form +uptq ` +ż t +0 +rηA0upsq ` B0pupsq, upsqsds “ u0 ` +ż t +0 +R0pnpsq, Φqds ` +ż t +0 +gpupsq, cpsqqdWs, +cptq ` +ż t +0 +rξA1cpsq ` B1pupsq, cpsqqsds “ c0 ´ +ż t +0 +R1pnpsq, cpsqqds ` γ +ż t +0 +φpcpsqqdβs, +nptq ` +ż t +0 +rδA1npsq ` B1pupsq, npsqqsds “ n0 ´ +ż t +0 +R2pnpsq, cpsqqds. +(2.14) + +8 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +These equations are understood being valid in V ˚, H´2pOq and H´3pOq, respectively. +We end this section by introduce some notations. +Let Y +be a Banach space. +By +Cpr0, Ts : Y q we denote the space of continuous functions v : r0, Ts Ñ Y +with the topology +induced by the norm defined by +|v|Cpr0,Ts;Y q :“ sup +0ďsďT +}vpsq}Y . +With L2p0, T; Y q we denote the space of measurable functions v : r0, Ts Ñ Y +with the +topology generated by the norm +|v|L2p0,T;Y q :“ +ˆż T +0 +}vpsq}2 +Y ds +˙1{2 +, +while by L2 +wp0, T; Y q we denote the space of measurable functions v : r0, Ts Ñ Y with weak +topology. +For a Hilbert space X, we denote by Xw the space X endowed with the weak topology +and by Cpr0, Ts; Xwq we denote the space of functions v : r0, Ts Ñ Xw that are weakly +continuous. +3. THE +MAIN +RESULT: EXISTENCE +OF +PROBABILISTIC +STRONG +SOLUTIONS +This section is devoted to the statement of the main result of this paper. Before proceeding +further, let us state the following definition. +Definition 3.1. A probabilistic strong solution of the problem (1.2) is a HˆH1pOqˆL2pOq-valued +stochastic process pu, c, nq such that +i): We have P-a.e. +u P Cpr0, Ts; Hq X L2p0, T; V q, +c P Cpr0, Ts; H1pOqq X L2p0, T; H2pOqq, +n P Cpr0, Ts; L2 +wpOqq X L2p0, T; H1pOqq X Cpr0, Ts; H´3pOqq. +ii): pu, c, nq : r0, Ts ˆ Ω Ñ H ˆ H1pOq ˆ L2pOq is progessively measurable and for all +p ě 1 +E sup +0ďsďT +|upsq|p +L2 ` E +ˆż T +0 +|∇upsq|2 +L2 ds +˙p +ă 8, +E +ˆż T +0 +|npsq|2 +L2 ds +˙p +ă 8, +(3.1) +and +E sup +0ďsďT +|cpsq|p +H1 ` E +ˆż T +0 +|cpsq|2 +H2 ds +˙p +ă 8. +iii): for all t P r0, Ts the following identity holds P-a.s. +uptq ` +ż t +0 +rηA0upsq ` B0pupsq, upsqqsds “ u0 ` +ż t +0 +R0pnpsq, Φqds ` +ż t +0 +gpupsq, cpsqqdWs, +cptq ` +ż t +0 +rξA1cpsq ` B1pupsq, cpsqqsds “ c0 ´ +ż t +0 +R1pnpsq, cpsqqds ` γ +ż t +0 +φpcpsqqdβs, +nptq ` +ż t +0 +rδA1npsq ` B1pupsq, npsqqsds “ n0 ´ +ż t +0 +R2pnpsq, cpsqqds, +(3.2) + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +9 +in V ˚, H´2pOq and H´3pOq, respectively. +Let us now present the main result of this section. +Theorem 3.2. Let Assumption 2.1, Assumption 2.2, Assumption 2.3, and Assumption 2.4 be +valid. Let us assume that the initial data pu0, c0, n0q belong to +H ˆ L8pOq X H1pOq ˆ L2pOq. +In addition, let us assume that c0pxq ą 0, n0pxq ą 0 for all x P O and +ż +O +n0pxq ln n0pxqdx ă 8, +as well as +4Kf +max +0ďcď|c0|L8 f 2 +min +0ďcď|c0|L8 f 1 +ď δ, γ2 ď +min +´ +ξ, +ξ +2K0 +¯ +6 |σ|2 +L8 +, and γ2p ď +3pξp +22p`1 |σ|2p +L8 8p , +(3.3) +for all p ě 2, where K0 is positive constant such that |ψ|2 +H2 ď K0p|∆ψ|2 +L2 ` |ψ|2 +H1q, for all +ψ P H2pOq (see [35, Proposition 7.2, P. 404] for the existence of such constant). Then, there +exists a unique probabilistic strong solution to the problem (1.2) in the sense of Definition +3.1. +Remark 3.3. We note that in the case where fpcq “ c, then we have Kf “ χ2`2δ +2δ +, and the +first inequality of the condition (3.3) is satisfied if +|c0|L8 ď +δ +? +2 +2 +a +χ2 ` 2δ +. +Furthermore, the condition (3.3) have been introduced in order to control the cell term in the +inequality (??) and the higher regularity of the noise term on the c-equation in the inequalities +(4.36) and (??). +However, it is known in [24, Remark 1.1] that, for the two-dimensional +deterministic chemotaxis system, there exists a critical mass phenomenon. +When the total +initial mass of cells +ş +O n0pxqdx above a critical mass mcrit (i.e. +ş +O n0pxqdx ą mcrit), solutions +blow-up in finite time, otherwise, all solutions remain bounded. While, for the two-dimensional +stochastic chemotaxis system, it is shown in [26] that, if the chemotaxis sensibility χ is +sufficiently large, then blow-up occurs with probability 1. +For the coupled system (1.2), +despite the rapid flow of fluid, we also expect some phenomenons to appear. +Then, it is +important to ask oneself what will happen if the condition (3.3) is violated? The answer to +this question will be given by the study of the blow-up criterion of the system (1.2) in +future. +In order to prove Theorem 3.2, we will first show that problem (1.1) has a probabilistic +weak solution, see Definition 3.2, then prove the non-negativity property and the L8-stability +property of weak solution, which give us the possibility to prove the pathwise uniqueness, +and finally apply the Yamada-Watanabe Theorem. +But before proceeding further, we now +introduce the concept of a probabilistic weak solution. +Definition 3.4. A weak probabilistic solution of the problem (1.2) is a system +p¯Ω, ¯F, ¯F, ¯P, pu, c, nq, p ¯ +W , ¯βqq, +where +i): p¯Ω, ¯F, ¯F, ¯Pq is a filtered probability space, + +10 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +ii): p ¯W, ¯βq is a cylindrical Wiener processes on U ˆ R2 over p¯Ω, ¯F, ¯F, ¯Pq, +iii): and pu, c, nq : r0, Ts ˆ ¯Ω Ñ H ˆ L2pOq is a strong solution to (1.1) with driving +noise p ¯W, ¯βq on the filtered probability space p¯Ω, ¯F, ¯F, ¯Pq. +The existence of weak solution to our problem is given in the following proposition. +Proposition 3.5. Let us assume that Assumption 2.1, Assumption 2.2 and Assumption 2.3 are +satisfied. Let +pu0, c0, n0q P H ˆ L8pOq X H1pOq ˆ L2pOq, +such that c0pxq ą 0, n0pxq ą 0 for all x P O and +ż +O +n0pxq ln n0pxqdx ă 8. +We also assume that (3.3) holds. Then, there exists at least one probabilistic weak solution +to the problem (1.2) in the sense of Definition 3.4. +The proof of Proposition 3.5, which is very technical is postponed to Section 5. +Next, we prove some properties of probabilistic weak solutions to the problem (1.2) such +as the non-negativity and the L8-stability which will be useful for the proof of the pathwise +uniqueness result. In fact, the main ingredient for the pathwise uniqueness is the L8-stability +property but to obtain this property we will need the non-negativity property. +Lemma 3.6. Let Assumption 2.1 and Assumption 2.2 are satisfied. Let p¯Ω, ¯F, ¯F, ¯P, pu, c, nq, p ¯ +W , ¯βqq +be a probabilistic weak solution to the problem (1.2). +If c0 ą 0 and n0 ą 0, then the +following inequality hold ¯P-a.s +(3.4) +nptq ą 0, and cptq ą 0, for all t P r0, Ts. +Proof. We will follow the idea developed in [19, Section 3.1] combined with the idea of +[14, Lemma 14] and [5, Theorem 3.7]. Let t P r0, Ts arbitrary but fixed. We then define +n´ptq :“ maxp´nptq, 0q and remark that n´ptq P W 2,2pOq. +Hence, we multiply equation +p2.14q3 by n´ptq, integrate over O, and use an integration-by-parts to obtain ¯P-a.s. +1 +2 +d +dt |n´ptq|2 +L2 “ ´ +ż +O +upt, xq ¨ ∇n´pt, xqn´pt, xqdx ´ δ |∇n´ptq|2 +L2 +´ χ +ż +O +npt, xq∇cpt, xq∇n´pt, xqdx +“ 1 +2 +ż +O +n2 +´pt, xq∇ ¨ upt, xqdx ´ δ |∇n´ptq|2 +L2 ` χ +ż +O +n´pt, xq∇cpt, xq∇n´pt, xqdx +(3.5) +ď ´δ |∇n´ptq|2 +L2 ` χ |n´ptq|L4 |∇cptq|L4 |∇n´ptq|L2 . +By the Gagliardo-Nirenberg-Sobolev inequality (3.7) and the Young inequality, we note that +χ |n´|L4 |∇c|L4 |∇n´|L2 ď Kp|n´|1{2 +L2 |∇n´|1{2 +L2 ` |n´|L2q |∇c|L4 |∇n´|L2 +ď K |n´|1{2 +L2 |∇c|L4 |∇n´|3{2 +L2 ` K |n´|L2 |∇c|L4 |∇n´|L2 +ď δ +2 |∇n´|2 +L2 ` K |n´|2 +L2 p|∇c|4 +L4 ` |∇c|2 +L4q +(3.6) +ď δ +2 |∇n´|2 +L2 ` K |n´|2 +L2 p|∇c|4 +L4 ` 1q. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +11 +Owing to the fact that +¯P-a.s. +c P Cpr0, Ts; H1pOqq X L2p0, T; H2pOqq, +by the following +Gagliardo-Niremberg inequality +(3.7) +|f|L4 ď KGNp|f|1{2 +L2 |∇f|1{2 +L2 ` |f|L2q, +f P W 1,2pOq, +we note that for all t P r0, Ts and ¯P-a.s. +ż t +0 +p|∇cpsq|4 +L4 ` 1qds ď +ż t +0 +|∇cpsq|4 +L4 ds ` t +ď K +ż T +0 +|∇cpsq|2 +L2 |cpsq|2 +H2 ds ` +ż T +0 +|∇cpsq|4 +L2 ds ` T +ď K sup +0ďsďT +|∇cpsq|2 +L2 +ż T +0 +|cpsq|2 +H2 ds ` sup +0ďsďT +|∇cpsq|2 +L2 +ż T +0 +|∇cpsq|2 +L2 ds ` T +ď K sup +0ďsďT +|cpsq|2 +H1 +ż T +0 +|cpsq|2 +H2 ds ` T ă 8. +Hence, integrating (3.5) over r0, Ts, and using the inequality (3.6), we infer that ¯P-a.s. +|n´ptq|2 +L2 ď |n´p0q|2 +L2 ` K +ż t +0 +p|∇cpsq|4 +L4 ` |∇cpsq|2 +L4q |n´psq|2 +L2 ds. +Thanks to Gronwall’s inequality, we derive that +|n´ptq|2 +L2 ď |pn0q´|2 +L2 exp +ˆ +K +ż t +0 +p|∇cpsq|4 +L4 ` |∇cpsq|2 +L4qds +˙ +, +which implies that ¯P-a.s, n´ptq “ 0 and the non-negativity of nptq follows. +For the proof of the non-negativity property of cptq, the main idea is to apply the Itˆo formula +to the function Ψ : H2pOq Ñ R defined by Ψpzq “ +ş +O z2 +´pxqdx where z´ “ maxp´z; 0q. +Since the function Ψ is not twice Fr´echet differentiable, we will follow the idea of [14, +Lemma 14] (see also [5, Theorem 3.7]) by introducing the following approximation of Ψ. +Let ϕ : R Ñ r´1; 0s be a C8 class increasing function such that +(3.8) +ϕpsq “ +# +´1 if s P p´8, ´2s +0 if s P r´1, `8q. +Let tψhuhPN be a sequence of smooth functions defined by ψhpyq “ y2ϕphyq, for all y P R +and h P N. For any h P N, we consider the following sequence of function Ψh : H2pOq Ñ R +defined by +Ψhpcq “ +ż +O +ψhpcpxqqdx, for c P H2pOq. +We note that the mapping Ψh is twice Fr´echet-differentiable and +Ψ1 +hpcqpkq “ 2 +ż +O +cpxqϕphcpxqqkpxqdx ` h +ż +O +c2pxqϕ1phcpxqqkpxqdx, +@c, k P H2pOq, +as well as +Ψ +2 +hpcqpz, kq “ m2 +ż +O +c2pxqϕ +2phcpxqqzpxqkpxqdx +` 4h +ż +O +cpxqϕ1phcpxqqzpxqkpxqdx ` 2 +ż +O +ϕphcpxqqzpxqkpxqdx, +@c, z, k P H2pOq. + +12 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +By applying the Itˆo formula to t ÞÑ Ψhpcptqq, we obtain ¯P-a.s. +Ψhpcptqq ´ Ψhpcp0qq “ +ż t +0 +Ψ1 +hpcpsqq pupsq ¨ ∇cpsq ` ξ∆cpsq ´ npsqfpcpsqqq ds +` 1 +2 +ż t +0 +2ÿ +k“1 +Ψ +2 +hpcpsqq pγφkpcpsqq, γφkpcpsqqq ds +(3.9) +` γ +2ÿ +k“1 +ż t +0 +Ψ1 +hpcpsqqpφkpcpsqqqd¯βk +s . +Now, we will find a simpler representation of the formula (3.9). +For a fixed k “ 1, 2, we remark that for all h ě 1, +(3.10) +hϕ1phcqσk ¨ ∇c “ σk ¨ phϕ1phcq∇cq “ σk ¨ ∇pϕphcqq, +and also that 2cσk ¨ ∇c “ σk ¨ ∇c2. Hence, for any h ě 1 thanks to an integration-by-parts +and the fact that σk “ 0 on BO, we have that for any h P N, +Ψ1 +hpcqpφkpcqq “ 2 +ż +O +cpxqϕphcpxqqσkpxq ¨ ∇cpxqdx ` h +ż +O +c2pxqϕ1phcpxqqσkpxq ¨ ∇cpxqdx +“ +ż +O +ϕphcpxqqσkpxq ¨ ∇c2pxqdx ` +ż +O +c2pxqσkpxq ¨ ∇pϕphcpxqqqdx +“ ´ +ż +O +c2pxq∇ ¨ pϕphcpxqqσkpxqqdx ` +ż +BO +c2pσqϕphcpσqqσkpσq ¨ νdσ +(3.11) +` +ż +O +c2pxqσkpxq ¨ ∇pϕphcpxqqqdx +“ ´ +ż +O +c2pxq∇ ¨ pϕphcpxqqσkpxqqdx ` +ż +O +c2pxqσkpxq ¨ ∇pϕphcpxqqqdx. +Owing to the fact that ∇ ¨ σk “ 0, we derive that +Ψ1 +hpcqpφkpcqq “ ´ +ż +O +c2pxqϕphcpxqq∇ ¨ σkpxqdx +´ +ż +O +c2pxqσkpxq ¨ ∇pϕphcpxqqqdx ` +ż +O +c2pxqσkpxq ¨ ∇pϕphcpxqqqdx +(3.12) +“ 0. +We note that +2ÿ +k“1 +σk ¨ ∇cσk ¨ ∇c “ +2ÿ +k“1 +2ÿ +i,j“1 +σi +kσj +k +Bc +Bxi +Bc +Bxj +“ +2ÿ +i,j“1 +qijpx, xq Bc +Bxi +Bc +Bxj +(3.13) +“ +2ÿ +i,j“1 +δij +Bc +Bxi +Bc +Bxj +“ +2ÿ +i“1 +Bc +Bxi +Bc +Bxi +“ |∇c|2 . + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +13 +Therefore, +2ÿ +k“1 +Ψ +2 +hpcq pγφkpcq, γφkpcqq “ γ2h2 +ż +O +c2pxqϕ +2phcpxqq |∇cpxq|2 dx +` 4hγ2 +ż +O +cpxqϕ1phcpxqq |∇cpxq|2 dx ` 2γ2 +ż +O +ϕphcpxqq |∇cpxq|2 dx. +On the other hand, by integration-by-parts, we get +γ2Ψ1 +hpcqp∆cq +“ 2γ2 +ż +O +cpxqϕphcpxqq∆cpxqdx ` hγ2 +ż +O +c2pxqϕ1phcpxqq∆cpxqdx +“ ´2γ2 +ż +O +∇cpxq ¨ ∇pcpxqϕphcpxqqqdx ´ hγ2 +ż +O +∇cpxq ¨ ∇pc2pxqϕ1phcpxqqqdx +` 2γ2 +ż +BO +Bcpσq +Bν +ϕphcpσqq∆cpσqdσ ` 2hγ2 +ż +BO +Bcpσq +Bν +cpσqϕ1phcpσqq∆cpσqdσ +“ ´2γ2 +ż +O +ϕphcpxqq |∇cpxq|2 dx ´ 2hγ2 +ż +O +cpxqϕ1phcpxqq |∇cpxq|2 dx +(3.14) +´ 2hγ2 +ż +O +cpxqϕ1phcpxqq |∇cpxq|2 dx ´ γ2h2 +ż +O +c2pxqϕ +2phcpxqq |∇cpxq|2 dx +“ ´ +2ÿ +k“1 +Ψ +2 +hpcq pγφkpcq, γφkpcqq . +In the equality (3.14), we have used the fact that +Bc +Bν vanishes on BO. +Therefore, recalling that ξ “ η ` γ2 +2 +and using (3.12) and (3.14), the equality (3.9) is +equivalent to +ż +O +ψhpcpt, xqqdx ´ +ż +O +ψhpc0pxqqdx “ +ż t +0 +Ψ1 +hpcpsqq pupsq ¨ ∇cpsq ` η∆cpsq ´ npsqfpcpsqqq ds, +from which along with the passage to the limit as h Ñ 8 we infer that +´ +ż +O +c2 +´pt, xqdx ` +ż +O +pc0pxqq2 +´dx +“ ´2 +ż t +0 +ż +O +ppups, xq ¨ ∇cps, xq ` η∆cps, xq ´ nps, xqfpcps, xqqqq cps, xq1tcps,xqă0udxds +“ 2 +ż t +0 +ż +O +´ +η |∇cps, xq|2 ` nps, xqfpcps, xqqcps, xq +¯ +1tcps,xqă0udxds. +We note that, in the last line, we used an integration-by-parts and the fact that ∇ ¨ u “ 0. +By the mean value theorem, the fact that fp0q “ 0, and f 1 ą 0 as well as 1tcă0u ą 0, +c2 ą 0, and n ą 0, we deduce that |c´ptq|2 +L2 ď |pc0q´|2 +L2. This implies that c´ptq “ 0 ¯P-a.s. +and end the proof of Lemma 3.6. +□ +With the non-negativity of probabilistic weak solutions in hand, we are able now to state +and prove the L8-stability. +Corollary 3.7. Under the same assumptions as in Lemma 3.6, if p¯Ω, ¯F, ¯F, ¯P, pu, c, nq, p ¯ +W , ¯βqq +is a probabilistic weak solution to the problem (1.2), then for all t P r0, Ts +(3.15) +|cptq|L8 ď |c0|L8 , +¯P-a.s. + +14 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Proof. The idea of the proof comes from [19, Section 3.2]. We apply the Itˆo formula to +the process t ÞÑ Ψpcptqq :“ +ş +O cppt, xqdx, for any p ě 2 and evaluate the limit as p tends +to 8. Let Ψ : H2pOq Ñ R be the functional defined by Ψpcq “ +ş +O cppxqdx. Note that this +mapping is twice Fr´echet-differentiable and +Ψ1pcqphq “ p +ż +M +cp´1pxqhpxqdx, +@c, h P H2pOq, +Ψ +2pcqph, kq “ ppp ´ 1q +ż +M +cp´2pxqhpxqkpxqdx, +@c, h, k P H2pOq. +Applying the Itˆo formula to the process t ÞÑ Ψpcptqq, yields +Ψpcptqq ´ Ψpcp0qq “ +ż t +0 +Ψ1pcpsqq pupsq ¨ ∇cpsq ` ξ∆cpsq ´ npsqfpcpsqqq ds +` 1 +2 +ż t +0 +2ÿ +k“1 +Ψ +2pcpsqq pγφkpcpsqq, γφkpcpsqqq ds ` γ +2ÿ +k“1 +ż t +0 +Ψ1pcpsqqpφkpcpsqqqd¯βk +s . +(3.16) +By integration-by-parts, the divergence free property of σk and the fact that σk “ 0 on BO, +we remark that for all k ě 1, +Ψ1pcqpφkpcqq “ p +ż +O +cp´1pxqσkpxq ¨ ∇cpxqdx +“ +ż +O +σkpxq ¨ ∇cppxqdx +(3.17) +“ ´ +ż +O +cppxq∇ ¨ σkpxqdx ` +ż +BO +cppσqσkpσq ¨ νdσ “ 0. +This implies that the stochastic term in (3.16) vanishes. +Note that +(3.18) +ż +O +∆cpxqcp´1pxqdx “ ´pp ´ 1q +ż +O +|∇cpxq|2 cpxqp´2dx. +Since ∇ ¨ u “ 0, by integration by part, we infer that +ż +O +upxq ¨ ∇cpxqcp´1pxqdx “ 1 +p +ż +O +upxq ¨ ∇cppxqdx “ 0. +(3.19) +Using the equalities (3.17), (3.18) and (3.19), we deduce from (3.17) that +Ψpcptqq ´ Ψpc0q “ +ż t +0 +ż +O +´ +´ppp ´ 2qξ |∇cps, xq|2 cp´2ps, xq ´ pnps, xqfpcps, xqqcp´1ps, xq +¯ +dxds +` ppp ´ 1q +2 +ż t +0 +ż +O +cp´2psq +2ÿ +k“1 +σkpxq ¨ ∇cps, xqσkpxq ¨ ∇cps, xqdxds. +(3.20) +From the equality (3.13), we get ř2 +k“1 σk ¨ ∇cσk ¨ ∇c “ |∇c|2. +Hence, the equality (3.20) +becomes +Ψpcptqq ´ Ψpc0q “ +ż t +0 +ż +O +´ +´ppp ´ 2q |∇cps, xq|2 cp´2ps, xq ´ pnps, xqfpcps, xqqcp´1ps, xq +¯ +dxds. +Using the non-negative property of n and c proved in Lemma 3.6 combined with the +non-negativity of the function f, we infer from the last equality that for all p ě 2 and +t P r0, Ts, |cptq|Lp ď |c0|Lp, which along with the passage to the limit p Ñ `8 completes +the proof of Theorem 5.1 (see [1, Theorem 2.14] for a detailed proof). +□ + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +15 +We now proceed with the statement and proof of the pathwise uniqueness of the weak +solution. +Proposition 3.8. We assume that the assumptions of Theorem 3.2 hold. If +pΩ, F, tFtutPr0,Ts, P, pu1, c1, n1q, p ¯ +W , ¯βqq and pΩ, F, tFtutPr0,Ts, P, pu2, c2, n2q, p ¯W, ¯βqq +are two weak probabilistic solutions of system (2.14) with the same initial data pu0, c0, n0q, +then +(3.21) +pu1ptq, c1ptq, n1ptqq “ pu2ptq, c2ptq, n2ptqq +P-a.s. +for all t P r0, Ts. +Proof. For t P r0, Ts, let +pwptq, ψptq, ϕptqq “ pu1ptq ´ u2ptq, c1ptq ´ c2ptq, n1ptq ´ n2ptqq. +Then this process satisfies pwp0q, ψp0q, ϕp0qq “ 0 and for all t P r0, Ts, we have +wptq ` +ż t +0 +rηA0wpsq ` B0pwpsq, u1psqq ` B0pu2psq, wpsqqsds +“ +ż t +0 +R0pϕpsq, Φqds ` +ż t +0 +rgpu1psq, c1psqq ´ gpu2psq, c2psqqsdWs, +(3.22) +ψptq ` +ż t +0 +rξA1ψpsq ` B1pwpsq, c1psqq ` B1pu2psq, ψpsqqsds +“ ´ +ż t +0 +rR1pn1psq, c1psqq ´ R1pn2psq, c2psqqsds ` γ +ż t +0 +φpψpsqqdβs, +(3.23) +ϕptq ` +ż t +0 +rδA1ϕpsq ` B1pwpsq, n1psqq ` B1pu2psq, φpsqqsds +“ ´ +ż t +0 +rR2pn1psq, c1psqq ´ R2pn2psq, c2psqqsds. +(3.24) +Using the fact that pB0pu2, wq, wq “ 0, we get by applying the Itˆo formula to t ÞÑ |wptq|2 +L2 +that +|wptq|2 +L2 ` 2η +ż t +0 +|∇wpsq|2 +L2 ds “ ´2 +ż t +0 +pB0pwpsq, u1psqq, wpsqqds ` 2 +ż t +0 +pR0pϕpsq, Φq, wpsqqds +` +ż t +0 +|gpu1psq, c1psqq ´ gpu2psq, c2psqq|2 +L2pU,Hq ds +(3.25) +` 2 +ż t +0 +pgpu1psq, c1psqq ´ gpu2psq, c2psqq, wpsqqdWs. +Using the continuous embeddings V ãÑ H +and H1pOq ãÑ L4pOq as well as the H¨older +inequality and the Young inequality, we derive that +2 |pB0pw, u1q, wq| ď 2 |w|L4 |u1|L4 |w|L2 +ď η +5 |∇w|2 +L2 ` K |∇u1|2 +L2 |w|2 +L2 , +(3.26) + +16 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +and +2 |pR0pϕ, Φq, wq| ď 2 |∇Φ|L8 |ϕ|L2 |w|L2 +ď K |∇Φ|L8 |ϕ|L2 |∇w|L2 +(3.27) +ď η +5 |∇w|2 +L2 ` K |Φ|2 +W 1,8 |ϕ|2 +L2 . +Thanks to (2.13), we have +(3.28) +|gpu1, c1q ´ gpu2, c2q|2 +L2pU,Hq ď L2 +Lipp|w|2 +L2 ` |ψ|2 +H1q. +Since ∇ ¨ σ1 “ ∇ ¨ σ2 “ 0, we obtain pφpψq, ψq “ 0. Futhermore, by the fact that ∇ ¨ u2 “ 0, +we derive that pB1pu2, ψq, ψq “ 0. Next, we recall that (A3) implies +|φpψq|2 +L2pR2;L2q “ +2ÿ +k“1 +ż +O +|σkpxq ¨ ∇ψpxq|2 dx “ |∇ψ|2 +L2 . +Hence, by applying the Itˆo formula to t ÞÑ |ψptq|2 +H1, we see that +|ψptq|2 +H1 ` 2 +ż t +0 +´ +µ |∇ψpsq|2 +L2 ` ξ |A1ψpsq|2 +L2 +¯ +ds +“ ´2 +ż t +0 +pB1pwpsq, c1psqq, ψpsqqds ´ 2 +ż t +0 +pR1pn1psq, c1psqq ´ R1pn2psq, c2psqq, ψpsqqds +` 2 +ż t +0 +pB1pwpsq, c1psqq ` B1pu2psq, ψpsqq, A1ψpsqqds +(3.29) +´ 2 +ż t +0 +pR1pn1psq, c1psqq ´ R1pn2psq, c2psqq, A1ψpsqqds +` γ2 +ż t +0 +|∇φpψpsqq|2 +L2pR2;L2q ds ` 2γ +ż t +0 +p∇φpψpsqq, ∇ψpsqqdβs. +Taking the L2-inner product of the equation (3.24) with ϕ and adding the result to (3.29), +yield +|ϕptq|2 +L2 ` |ψptq|2 +H1 ` 2 +ż t +0 +pµ |∇ψpsq|2 +L2 ` ξ |A1ψpsq|2 +L2 ` δ |∇ϕpsq|2 +L2qds +“ ´2 +ż t +0 +pB1pwpsq, c1psqq, ψpsqqds ´ 2 +ż t +0 +pR1pn1psq, c1psqq ´ R1pn2psq, c2psqq, ψpsqqds +` 2 +ż t +0 +pB1pwpsq, c1psqq ` B1pu2psq, ψpsqq, A1ψpsqqds +´ 2 +ż t +0 +pR1pn1psq, c1psqq ´ R1pn2psq, c2psqq, A1ψpsqqds +(3.30) +´ 2 +ż t +0 +rr2pϕpsq, c1psq, ϕpsqq ` r2pn2psq, ψpsq, ϕpsqqsds ` γ2 +ż t +0 +|∇φpψpsqq|2 +L2pR2;L2q ds +´ 2 +ż t +0 +pB1pwpsq, n1psqq, ϕpsqqds ` 2γ +ż t +0 +p∇φpψpsqq, ∇ψpsqqdβs. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +17 +Now, we give an estimate for the right-hand side of (3.30). Similarly to (3.26), we have +2 |pB1pw, c1q, ψq| ď 2 |w|L4 |∇c1|L2 |ψpsq|L4 +ď K |∇w|L2 |∇c1|L2 |ψ|H1 +(3.31) +ď η +5 |∇w|2 +L2 ` K |∇c1|2 +L2 |ψ|H1 . +Thanks to the continuous embedding H1pOq ãÑ L4pOq and the L8-stability property proved +in Corollary 3.7, we have +2pR1pn1, c1q ´ R1pn2, c2q, ψq ď 2 |R1pn1, c1q ´ R1pn2, c2q|L2 |ψ|L2 +ď 4 |pfpc1q ´ fpc2qqn1|2 +L2 ` 4 |fpc2qψ|2 +L2 ` 2 |ψ|2 +L2 +ď 4 +sup +0ďrď|c0|L8 +pf 1prqq2 |n1ψ|2 +L2 ` 4 +sup +0ďrď|c0|L8 +fprq |ψ|2 +L2 ` 2 |ψ|2 +L2 +ď K |ψ|2 +L4 |n1|2 +L4 ` Kf |ψ|2 +L2 . +Applying the Galiardo-Nirenberg-Sobolev inequality, we arrive at +2pR1pn1, c1q ´ R1pn2, c2q, ψq ď K |ψ|2 +H1 +´ +|∇n1|L2 |n1|L2 ` |n1|2 +L2 +¯ +` Kf |ψ|2 +L2 +ď K +´ +|∇n1|L2 |n1|L2 ` |n1|2 +L2 +¯ +|ψ|2 +H1 ` Kf |ψ|2 +H1 . +(3.32) +Thanks to the Ladyzhenskaya, Galiardo-Nirenberg-Sobolev, and Young inequalities, we find +that +2 |pB1pw, c1q, A1ψq| ď 2 |w|L4 |∇c1|L4 |A1ψ|L2 +ď ξ +6 |A1ψ|2 +L2 ` K |w|L2 |∇w|L2 +´ +|c1|H2 |∇c1|L2 ` |∇c1|2 +L2 +¯ +(3.33) +ď ξ +6 |A1ψ|2 +L2 ` η +5 |∇w|2 +L2 ` K +´ +|c1|2 +H2 |∇c1|2 +L2 ` |∇c1|4 +L2 +¯ +|w|2 +L2 . +We recall that there exist a positive constant K0, such that |ψ|2 +H2 ď K0p|A1ψ|2 ` |ψ|2 +H1q. +Hence, using also the continuous embedding V ãÑ H, we obtain +2 |pB1pu2, ψq, A1ψq| ď 2 |u2|L4 |∇ψ|L4 |A1ψ|L2 +ď ξ +6 |A1ψ|2 +L2 ` K |u2|L2 |∇u2|L2 +´ +|ψ|H2 |∇ψ|L2 ` |∇ψ|2 +L2 +¯ +ď ξ +6 |A1ψ|2 +L2 ` K´1 +0 ξ +6 +|ψ|2 +H2 ` K |u2|2 +L2 |∇u2|2 +L2 |∇ψ|2 +L2 +(3.34) +` K |u2|L2 |∇u2|L2 |∇ψ|2 +L2 +ď ξ +3 |A1ψ|2 +L2 ` ξ +6 |ψ|2 +H1 ` K +´ +|u2|2 +L2 |∇u2|2 +L2 ` |∇u2|2 +L2 +¯ +|ψ|2 +H1 . +Using a similarly argument as in (3.32), we arrive at +2 |pR1pn1, c1q ´ R1pn2, c2q, A1ψ| ď ξ +6 |A1ψ|2 +L2 ` K |R1pn1, c1q ´ R1pn2, c2q|2 +L2 +ď ξ +6 |A1ψ|2 +L2 ` K |ψ|2 +L4 |n1|2 +L4 ` Kf |ψ|2 +L2 +(3.35) +ď ξ +6 |A1ψ|2 +L2 ` Kf |ψ|2 +H1 ` K +´ +|∇n1|L2 |n1|L2 ` |n1|2 +L2 +¯ +|ψ|2 +H1 . + +18 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +By using an integration-by-parts and H¨older, and the Galiardo-Nirenberg-Sobolev inequalities, +we see that +2 |pB1pw, n1q, ϕq| ď 2 +ˇˇˇˇ +ż +O +n1pxqwpxq ¨ ∇ϕpxqdx +ˇˇˇˇ +ď 2 |n1|L4 |w|L4 |∇ϕ|L2 +ď δ +4 |∇ϕ|2 +L2 ` K |w|L2 |∇w|L2 +´ +|∇n1|L2 |n1|L2 ` |n1|2 +L2 +¯ +(3.36) +ď δ +4 |∇ϕ|2 +L2 ` η +5 |∇w|2 +L2 ` K +´ +|∇n1|2 +L2 |n1|2 +L2 ` |n1|4 +L2 +¯ +|w|2 +L2 . +By applying the Young and Galiardo-Nirenberg-Sobolev inequalities we obtain +2 |r2pϕ, c1, ϕq| ď 2 |ϕ|L4 |∇c1|L4 |∇ϕ|L2 +ď δ +4 |∇ϕ|2 +L2 ` K |ϕ|2 +L4 |∇c1|2 +L4 +ď δ +4 |∇ϕ|2 +L2 ` K +´ +|∇ϕ|L2 |ϕ|L2 ` |ϕ|2 +L2 +¯ ´ +|c1|H2 |∇c1|L2 ` |∇c1|2 +L2 +¯ +(3.37) +ď δ +2 |∇ϕ|2 +L2 ` K +´ +|c1|2 +H2 |∇c1|2 +L2 ` |∇c1|4 +L2 ` |c1|H2 |∇c1|L2 ` |∇c1|2 +L2 +¯ +|ϕ|2 +L2 . +In a similarly way we have that +2 |r2pn2, ψ, ϕq| ď 2 |n2|L4 |∇ψ|L4 |∇ϕ|L2 +ď δ +4 |∇ϕ|2 +L2 ` K |n2|2 +L4 +´ +|ψ|H2 |∇ψ|L2 ` |∇ψ|2 +L2 +¯ +ď δ +4 |∇ϕ|2 +L2 ` K´1 +0 ξ +6 +|ψ|2 +H2 ` K |n2|4 +L4 |∇ψ|2 +L2 ` K |n2|2 +L4 |∇ψ|2 +L2 +(3.38) +ď δ +4 |∇ϕ|2 +L2 ` ξ +6 |A1ψ|2 +L2 ` ξ +6 |ψ|2 +H1 ` K |n2|4 +L2 |ψ|2 +H1 +` K +´ +|∇n2|L2 |n2|L2 ` |n2|2 +L2 |∇n2|2 +L2 ` |n2|2 +L2 +¯ +|ψ|2 +H1 . +By using (3.3) we derive that +γ2 |∇φpψq|2 +L2pR2;L2q “ γ2 +2ÿ +k“1 +ż +O +|∇pσkpxq ¨ ∇ψpxqq|2 dx +ď 2γ2 +2ÿ +k“1 +|σk|2 +W 1,8 |∇ψ|2 +L2 ` 2γ2 +2ÿ +k“1 +|σk|2 +L8 |ψ|2 +H2 +(3.39) +ď p1 ` K0q2γ2 |σ|2 +W 1,8 |∇ψ|2 +L2 ` 2γ2K0 |σ|2 +L8 |A1ψ|2 +L2 +ď ξ +6 |A1ψ|2 +L2 ` p1 ` K0q2γ2 |σ|2 +W 1,8 |ψ|2 +H1 . +Now, for t P r0, Ts and s P r0, ts, let us set +Yptq :“ |uptq|2 +L2 ` |cptq|2 +H1 ` |ϕptq|2 +L2 , + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +19 +Zpsq :“ K |∇u1psq|2 +L2 ` K |∇c1psq|2 +L2 ` K +´ +|∇n1psq|L2 |n1psq|L2 ` |n1psq|2 +L2 +¯ +` K +´ +|c1psq|2 +H2 |∇c1psq|2 +L2 ` |∇c1psq|4 +L2 +¯ +` K +´ +|u2psq|2 +L2 |∇u2psq|2 +L2 ` |∇u2psq|2 +L2 +¯ +` K +´ +|∇n1psq|L2 |n1psq|L2 ` |n1psq|2 +L2 +¯ +` K +´ +|∇n1psq|2 +L2 |n1psq|2 +L2 ` |n1psq|4 +L2 +¯ +(3.40) +` K +´ +|c1psq|2 +H2 |∇c1psq|2 +L2 ` |∇c1psq|4 +L2 ` |c1psq|H2 |∇c1psq|L2 ` |∇c1psq|2 +L2 +¯ +` K +´ +|∇n2psq|L2 |n2psq|L2 ` |n2psq|2 +L2 |∇n2psq|2 +L2 ` |n2psq|2 +L2 ` |n2psq|4 +L2 +¯ +, +and +θptq :“ exp +ˆ +´ +ż t +0 +Zpsqds +˙ +. +Applying the Itˆo formula to t ÞÑ θptq |uptq|2 +L2, we derive that +θptq |wptq|2 +L2 ` 2η +ż t +0 +θpsq |∇wpsq|2 +L2 ds ď 2 +ż t +0 +θpsqpB0pwpsq, u1psqq, wpsqqds +` 2 +ż t +0 +θpsqpR0pϕpsqq, wpsqqds ` +ż t +0 +θ1psq |wpsq|2 +L2 ds +` +ż t +0 +θpsq |gpu1psq, c1psqq ´ gpu2psq, c2psqq|2 +L2pU,Hq ds +(3.41) +` 2 +ż t +0 +θpsqpgpu1psq, c1psqq ´ gpu2psq, c2psqq, wpsqqdWs. +Applying the Itˆo formula once more to t ÞÑ θptqp|ϕptq|2 +L2 ` |ψptq|2 +H1q and adding the result +with (3.41) after taking into account the estimates (3.26)-(3.28) and (3.31)-(3.39), we arrive at +θptqYptq ` +ż t +0 +θpsq +´ +η |∇wpsq|2 +L2 ` µ |∇ψpsq|2 +L2 ` ξ |A1ψpsq|2 +L2 +¯ +ds +ď +ˆ +K |Φ|2 +W 1,8 ` L2 +Lip ` 2Kf ` ξ +3 ` p1 ` K0q2γ2 |σ|2 +W 1,8 +˙ ż t +0 +θpsqYpsqds +(3.42) +` 2γ +ż t +0 +θpsqp∇φpψpsqq, ∇ψpsqqdβs +` 2 +ż t +0 +θpsqpgpu1psq, c1psqq ´ gpu2psq, c2psqq, wpsqqdWs. +Next, taking the mathematical expectation yields +EθptqYptq ` E +ż t +0 +θpsq +´ +η |∇wpsq|2 +L2 ` µ |∇ψpsq|2 +L2 ` ξ |A1ψpsq|2 +L2 +¯ +ds +ď +ˆ +K |Φ|2 +W 1,8 ` L2 +Lip ` 2Kf ` ξ +3 ` p1 ` K0q2γ2 |σ|2 +W 1,8 +˙ +E +ż t +0 +θpsqYpsqds. +(3.43) +From which along with the Gronwall inequality we infer that for any t P r0, Ts +EθptqYptq “ 0. +It follows that for all t P r0, Ts, Yptq “ 0 P-a.s. Since the paths of pui, ci, niq, i “ 1, 2 are +continuous P-a.s., then +pu1ptq, c1ptq, n1ptqq “ pu2ptq, c2ptq, n2ptqq, +P-a.s., for all t P r0, Ts. + +20 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +□ +With the existence and pathwise uniqueness results at hand we now prove the existence +of strong solution stated in Theorem 3.2. +Proof of Theorem 3.2. +The existence of a probabilistic weak solution to the problem (1.2) +is shown in Proposition 3.5. The pathwise uniqueness of probabilistic weak solutions is given +by Proposition 3.8. Thus, the existence and uniqueness of a probabilistic strong solution to +the problem (1.2) follows from the Yamada-Watanabe Theorem (see [30, Theorem E.1.8]), +which states that the existence of weak probabilistic solution and the pathwise uniqueness +imply the existence of a unique probabilistic strong solution. +□ +4. PROOF +OF PROPOSITION 3.5 +In this section, we will show Proposition 3.5. We introduce a Galerkin approximation first. +We then discuss the existence of the Galerkin approximation and prove the mass conservation +property, the non-negativity property and the L8-norm satibility in finite dimension. +Using +these properties, we prove priori estimates and by these a priori estimates, we show the +tightness of the family of approximations, and pass in a second step, to the limit in the +deterministic terms and the construction of the noise terms by exploiting the usual martingale +representation theorem proved in [12, Theorem 8.2]. +4.1. Galerkin approximation and a priori uniform estimates. In this subsection, we will +construct +a family of approximations +of the solutions +and prove some crucial +estimates +satisfied uniformly by the approximations. For this propose, let us recall that there exists an +orthonormal basis twiu8 +i“1 of H consisting of the eigenfunctions of the Stokes operator A0 +and an orthonormal basis tϕiu8 +i“1 Ă C8pOq of L2pOq consisting of the eigenfunctions of the +Neumann Laplacian operator A1. For m P N, we will consider the following finite-dimensional +spaces +Hm “ spamtw1, ..., wmu, +Hm “ spamtϕ1, ..., ϕmu, +Hm “ Hm ˆ Hm ˆ Hm, +where we endow Hm with the following norm +|pu, c, nq|2 +Hm “ |u|2 +L2 ` |c|2 +L2 ` |n|2 +L2 , +pu, c, nq P Hm. +Owing to the fact that Hm is a finite dimensional space, the L2pOq, H1pOq and H2pOq-norms +are equivalent on this space. We choose as in [44, P. 335] nm +0 , cm +0 +and um +0 +such that +nm +0 ą 0, nm +0 Ñ n0 in L2pOq, nm +0 ln nm +0 Ñ n0 ln n0 in L1pOq, +cm +0 ą 0, |cm +0 |L8 ď |c0|L8 , cm +0 Ñ c0 in H1pOq, +and um +0 Ñ u0 in H. +(4.1) + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +21 +We then consider on the filtered probability space pΩ, F, tFtutPr0,Ts, Pq the following finite +dimensional problem. For all t P r0, Ts +umptq ` +ż t +0 +rηA0umpsq ` P1 +mB0pumpsq, umpsqqsds +“ um +0 ` +ż t +0 +P1 +mR0pnmpsq, Φqds ` +ż t +0 +P1 +mgpumpsq, cmpsqqdWs, +cmptq ` +ż t +0 +rξA1cmpsq ` P2 +mB1pumpsq, cmpsqqsds +“ cm +0 ´ +ż t +0 +P2 +mR1pnmpsq, cmpsqqds ` γ +ż t +0 +P2 +mφpcmpsqqdβs, +nmptq ` +ż t +0 +rδA1nmpsq ` P2 +mB1pumpsq, nmpsqqsds “ nm +0 ´ +ż t +0 +P2 +mR2pnmpsq, cmpsqqds, +(4.2) +where P1 +m and P2 +m are the projection from H and L2pOq onto Hm and Hm, respectively, +and their operator norms are equal to 1. +For each m, we consider the following mapping Ψm : Hm Ñ Hm defined by +Ψmpu, c, nq “ +¨ +˝ +ηA0u ` P1 +mB0pu, uq ´ P1 +mR0pn, Φq +ξA1c ` P2 +mB1pu, cq ` P2 +mR1pn, cq +δA1n ` P2 +mB1pu, nq ` P2 +mR2pn, cq +˛ +‚. +In the following lemma, we are going to state an important property of the mappings Ψm, +m P N. +Lemma 4.1. Let Assumption 2.1 and Assumption 2.3 be satisfied. +For each m P N, the +mapping Ψm is locally Lipschitz continuous. To be more precise, for each m P N and every +r ą 0, there exists a constant Kr such that +(4.3) +|Ψmpv1q ´ Ψmpv2q|Hm ď Kr |v1 ´ v2|Hm , +for v1 “ pu1, c1, n1q, v2 “ pu2, c2, n2q P Hm with |vi|Hm ď r, i “ 1, 2. +Proof. Let v1 “ pu1, c1, n1q, v2 “ pu2, c2, n2q P Hm and v “ pu, c, nq P Hm. We assume that +|vi|Hm ď r, i “ 1, 2. We have +pΨmpv1q ´ Ψmpv2q, vqHm “ pηA0pu1 ´ u2q ` B0pu1, u1q ´ B0pu2, u2q ´ R0pn1, Φq ` R0pn2, Φq, uq +` pξA1pc1 ´ c2q ` B1pu1, c1q ´ B1pu2, c2q ` R1pn1, c1q ´ R1pn2, c2q, cq +` pδA1pn1 ´ n2q ` B1pu1, n1q ´ B1pu2, n2q ` R2pn1, c1q ´ R2pn2, c2q, nq. +(4.4) +Using the bilinearity of the operator B0, we see that +|pB0pu1, u1q ´ B0pu2, u2q, uq| ď |pB0pu1 ´ u2, u1q, uq| ` |pB0pu2, u1 ´ u2q, uq| +ď 2Kr |u1 ´ u2|L2 |u|L2 . +By the H¨older inequality we also note that +pR0pn1, Φq ´ R0pn2, Φq, uq ď +ż +O +|n1 ´ n2| |∇Φ| |u| dx +ď |∇Φ|L8 |n1 ´ n2|L2 |u|L2 . + +22 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Since the space H1pOq is continuously embedded in the space LqpOq for any q ě 2, we +have +|pB1pu1, c1q ´ B1pu2, c2q, cq| ď |pB1pu1 ´ u2, c1q, cq| ` |pB1pu2, c1 ´ c2q, cq| +ď |u1 ´ u2|L4 |∇c1|L2 |c|L4 ` |u2|L4 |∇pc1 ´ c2q|L2 |c|L4 +ď p|∇pu1 ´ u2q|L2 |∇c1| ` |∇u2|L2 |∇pc1 ´ c2q|L2q |c|H1 +ď Krp|∇pu1 ´ u2q|L2 ` |∇pc1 ´ c2q|L2q |c|H1 . +In a similar way we show that +|pB1pu1, n1q ´ B1pu2, n2q, nq| ď Krp|∇pu1 ´ u2q|L2 ` |∇pn1 ´ n2q|L2q |n|H1 . +Owing to the fact that Hm Ă C8pOq and fp0q “ 0 as well as f P C1pr0, 8qq, we derive that +|pR1pn1, c1q ´ R1pn2, c2q, cq| ď +ż +O +|n1 ´ n2| fpc1q |c| dx ` +ż +O +|n2| |fpc1q ´ fpc2q| |c| dx +ď +max +0ďcď|c1|L8 fpcq +ż +O +|n1 ´ n2| |c| dx +` +max +0ďcďmaxp|c1|L8,|c2|L8q f 1pcq +ż +O +|n2| |c1 ´ c2| |c| dx +ď max +0ďcďr fpcq |n1 ´ n2|L2 |c|L2 ` max +0ďcďr f 1 |n2|L4 |c1 ´ c2|L4 |c|L2 +ď Krp|n1 ´ n2|L2 ` |c1 ´ c2|H1q |c|L2 . +Also, we note that +|pR2pn1, c1q ´ R2pn2, c2q, nq| ď +ż +O +|n1 ´ n2| |∇c1| |∇n| dx ` +ż +O +|n2| |∇pc1 ´ c2q| |∇n| dx +ď |n1 ´ n2|L2 |∇c1|L4 |∇n|L2 ` |n2|L4 |∇pc1 ´ c2q|L4 |∇n|L2 +ď Krp|n1 ´ n2|L2 ` |c1 ´ c2|H2q |n|H1 . +Taking into account the fact that all norms are equivalent in finite dimensional space, and +the fact that the operators A0 and A1 are linear, we infer these previous inequalities and +equality (4.4). +□ +The existence of solutions to the finite dimensional problem (4.2) is classical. +In fact, +due to Lemma 4.1, the mapping Ψm is locally Lipschitz. +Also by the inequality (2.12), +P1 +mgp¨, ¨q is locally Lipschitz. From the linearity of φp.q, we can easily see that P2 +mφp¨q is +Lipschitz. Hence, by well known theory for finite dimensional stochastic differential equations +with locally Lipschitz coefficients (see [31, Theorem 38, P. 303] for full details) there exists +a local solution of system (4.2) with continuous paths in Hm. That is, there exists a stopping +time τm, a process t ÞÑ pumptq, cmptq, nmptqq such that τm ą 0 P-a.s., and the stopped process +t ÞÑ pumpt ^ τmq, cmpt ^ τmq, nmpt ^ τmqq +satisfies the system of Itˆo equation (4.2) and has continuous paths in Hm. Moreover, if a +process +t ÞÑ p¯umptq, ¯cmptq, ¯nmptqq, +and a stopping time σm constitute another local solution, then +pump¨q, cmp¨q, nmp¨qq “ p¯ump¨q, ¯cmp¨q, ¯nmp¨qq, +P-a.s. on r0, τm ^ σms. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +23 +We will show in what follows that the solutions pum, cm, nmq exist almost surely for every +t P r0, Ts. For this goal, it will be enough to show that +(4.5) +τmpωq ą T, for almost all ω P Ω, and all m P N. +To this aim, we will use some idea from [33, P. 132, Proof of Theorem 12.1]. Since for all +m P N, the deterministic integrand Ψm and the stochastic integrand P1 +mg are locally Lipschitz, +for each N P N, we can define the integrands ΨN +m and P1 +mgN, agreeing respectively with +Ψm and P1 +mg on the ball +BN +Hm :“ +␣ +pv, ϕ, ψq P Hm : |pv, ϕ, ψq|Hm ă N +( +, +such that ΨN +m and P1 +mgN +are globally Lipschitz. +As consequence, since P2 +mφ is already +globally Lipschitz, [33, P. 128, Theorem 11.2] guarantees that there is a unique solution +puN +m, cN +m, nN +mq to a system associated to the system (4.2) with ΨN +m and P1 +mgN (instead of +Ψm and P1 +mg) and defined on r0, `8q almost surely. We then define a sequence of stopping +times as follows for all m, N P N +(4.6) +τ m +N :“ inftt ą 0 : +b +|nN +mptq|2 +L2 ` |uN +mptq|2 +L2 ` |cN +mptq|2 +H1 ě Nu ^ N, +where a ^ b :“ minta, bu for any real numbers a and b. +For any fixed m P N, the sequence tτ m +N uNPN is obviously increasing. +Moreover [33, P. +131, Corollary 11.10] implies that for all N P N, +pum, cm, nmq “ puN +m, cN +m, nN +mq on r0, τ m +N s. +From this last equality, we infer that the solution pum, cm, nmq of system (4.2) is defined +on r0, τ m +N s for all N P N and hence, τm ą τ m +N +almost surely for all N P N. Therefore, +τm ě sup +NPN +τ m +N , P-a.s. +In order to prove the inequality (4.5), it is sufficient to prove that +(4.7) +sup +NPN +τ m +N ą T, P-a.s. +Before proving this, in the following lemma, we prove some properties of the local solution +pum, cm, nmq of system (4.2). +Lemma 4.2. Assumption 2.1 and Assumption 2.2. +Then for all m, N P N, the following +equality and inequalities hold P-a.s. +(4.8) +ż +O +nmpt ^ τ m +N , xqdx “ +ż +O +nm +0 pxqdx, for all t P r0, Ts, +(4.9) +nmpt ^ τ m +N q ą 0, and cmpt ^ τ m +N q ą 0, for all t P r0, Ts, +and +(4.10) +|cmpt ^ τ m +N q|L8 ď |c0|L8 , for all t P r0, Ts. +Proof. In order to prove the non-negativity of nmpt^τ m +N q and cmpt^τ m +N q, we will follow the +idea of the proof of Lemma 3.6. But, instead of the Gagliardo-Niremberg-Sobolev inequality, +we will use the equivalence of the norms on finite dimensional space. +Let N, m P N and t P r0, Ts be arbitrary but fixed. For all s P r0, ts define +nm´ps^τ m +N q :“ maxp´nmps ^ τ m +N q, 0q. + +24 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +We remark that nm´ps ^ τ m +N q P W 2,2pOq and +nm´ps ^ τ m +N q “ 0 ¨ 1tnmps^τ m +N qě0u ´ nmps ^ τ m +N q ¨ 1tnmps^τ m +N qă0u, +∇nm´ps ^ τ m +N q “ 0 ¨ 1tnmps^τ m +N qě0u ´ ∇nmps ^ τ m +N q ¨ 1tnmps^τ m +N qă0u, +∆nm´ps ^ τ m +N q “ 0 ¨ 1tnmps^τ m +N qě0u ´ ∆nmps ^ τ m +N q ¨ 1tnmps^τ m +N qă0u. +We can easily see also that for all s P r0, ts, +dnmps ^ τ m +N q +dt +nm´ps ^ τ m +N q “ ´dnm´ps ^ τ m +N q +dt +nm´ps ^ τ m +N q, +nm´ps ^ τ m +N q∇nmps ^ τ m +N q “ ´nm´ps ^ τ m +N q∇nm´ps ^ τ m +N q, +∆nmps ^ τ m +N qnm´ps ^ τ m +N q “ ´∆nm´ps ^ τ m +N qnm´ps ^ τ m +N q. +Hence, we multiply equation p2.14q3 by nm´ps ^ τ m +N q for any s P r0, ts, integrate over O, +and use an integration-by-parts with the fact that ∇ ¨ um “ 0 to obtain +1 +2 +d +dt +ˇˇnm´ps ^ τ m +N q +ˇˇ2 +L2 +“ ´ +ż +O +umps ^ τ m +N , xq ¨ ∇nm´ps ^ τ m +N , xqnm´ps ^ τ m +N , xqdx ´ δ +ˇˇ∇nm´ps ^ τ m +N q +ˇˇ2 +L2 +´ χ +ż +O +nmps ^ τ m +N , xq∇cmps ^ τ m +N , xq∇nm´ps ^ τ m +N , xqdx +“ 1 +2 +ż +O +n2 +m´ps ^ τ m +N , xq∇ ¨ umps ^ τ m +N , xqdx ´ δ +ˇˇ∇nm´ps ^ τ m +N , xq +ˇˇ2 +L2 +` χ +ż +O +nm´ps ^ τ m +N , xq∇cmps ^ τ m +N , xq∇n´ps ^ τ m +N , xqdx +ď ´δ +ˇˇ∇nm´ps ^ τ m +N q +ˇˇ2 +L2 ` χ +ˇˇnm´ps ^ τ m +N q +ˇˇ +L4 |∇cmps ^ τ m +N q|L4 +ˇˇ∇nm´ps ^ τ m +N q +ˇˇ +L2 +ď K +ˇˇnm´ps ^ τ m +N q +ˇˇ2 +H1 |cmps ^ τ m +N q|H2 . +In the last line we have used the continuous embedding of H1pOq into L4pOq. +Since +the L2pOq, H1pOq and H2pOq-norms are equivalent on Hm, we then infer from this last +inequality that for all s P r0, ts, +(4.11) +1 +2 +d +dt +ˇˇnm´ps ^ τ m +N q +ˇˇ2 +L2 ď Kpmq +ˇˇnm´ps ^ τ m +N q +ˇˇ2 +L2 |cmps ^ τ m +N q|L2 , +where Kpmq is a constant depending of m which is the dimension of the space Hm. Owing +to the fact that P-a.s. the paths of cm are continuous, we derive that +sup +0ďsďt +|cmps ^ τ m +N q|L2 ă 8, +P-a.s. +Hence, integrating (4.11) over r0, ts we arrive at +(4.12) +ˇˇnm´pt ^ τ m +N q +ˇˇ2 +L2 ď |pnm +0 q´|2 +L2 ` K +ż t +0 +|cmps ^ τ m +N q|L2 +ˇˇnm´ps ^ τ m +N q +ˇˇ2 +L2 ds. +Thanks to the Gronwall inequality, we derive from the inequality (4.12) that +ˇˇnm´pt ^ τ m +N q +ˇˇ2 +L2 ď |pnm +0 q´|2 +L2 exp +ˆ +K +ż t +0 +|cmps ^ τ m +N q|L2 ds +˙ +, +which implies that P-a.s, nm´pt ^ τ m +N q “ 0 for all t P r0, Ts since by the relation (4.1), +nm +0 ą 0. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +25 +The non-negativity property of cmpt^τ m +N q is quite similar to the proof of Lemma 3.6. We +consider the function Ψ : Hm Ñ R defined by Ψpcq “ +ş +O c2 +´pxqdx where c´ “ maxp´c; 0q. +Let tψhuhPN be a sequence of smooth functions defined by ψhpyq “ y2ϕphyq, for all y P R +and h P N, where the function ϕ is defined by (3.8). +We consider for any h ě 1, the +following sequence of function Ψh : Hm Ñ R defined by Ψh “ +ş +O ψhpcpxqqdx, for c P Hm. +The mapping Ψh is twice (Fr´echet) differentiable and its first and second derivatives are +given by +Ψ1 +hpcqpzq “ 2 +ż +O +cpxqϕphcpxqqzpxqdx ` h +ż +O +c2pxqϕ1phcpxqqzpxqdx, +@c, z P Hm, +and +Ψ +2 +hpcqpz, kq “ h2 +ż +O +c2pxqϕ +2phcpxqqzpxqkpxqdx +` 4h +ż +O +cpxqϕ1phcpxqqzpxqkpxqdx ` 2 +ż +O +ϕphcpxqqzpxqkpxqdx, +@c, z, k P Hm. +Applying the Itˆo formula to t ÞÑ Ψhpcmpt ^ τ m +N qq, we obtain for all t P r0, Ts, +Ψhpcmpt ^ τ m +N qq ´ Ψhpcmp0qq “ +ż t^τ m +N +0 +Ψ1 +hpcmpsqq pumpsq ¨ ∇cmpsq ` ξ∆cmpsq ´ nmpsqfpcmpsqqq ds +` 1 +2 +ż t^τ m +N +0 +2ÿ +k“1 +Ψ +2 +hpcmpsqq pγφkpcmpsqq, γφkpcmpsqqq ds +` γ +2ÿ +k“1 +ż t^τ m +N +0 +Ψ1 +hpcmpsqqpφkpcmpsqqqdβk +s . +Similarly to (3.10), (3.11), (3.12), (3.13) and (3.14), we can infer from this last equality that +ż +O +ψhpcmpt ^ τ m +N , xqqdx ´ +ż +O +ψhpcm +0 pxqqdx +“ +ż t^τ m +N +0 +Ψ1 +hpcmpsqq pumpsq ¨ ∇cmpsq ` η∆cmpsq ´ nmpsqfpcmpsqqq ds. +(4.13) +Now, observe that from the assumptions on the function ϕ, we infer that for all y P R we +have +(4.14) +lim +hÝÑ8 ψhpyq “ ´y2 ¨ 1tyă0u “ ´y2 +´ +and +lim +hÝÑ8 2yϕphyq “ ´2y ¨ 1tyă0u. +We note that for any y P R, we have +(4.15) +lim +hÝÑ8 hϕ1phyq “ 0, +and also that +(4.16) +|ψhpyq| ď Ky2 +and +ˇˇhϕ1phyq +ˇˇ ď K |y| , +for any y P R and for all h ě 1, where K ą 0 is a constant. + +26 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Using (4.14)-(4.16) and applying the Lebesgue Dominated Convergence Theorem, we can +pass to the limit as h tends to infinity in (4.13). In this way, we derive that +´ +ż +O +c2 +m´pt ^ τ m +N , xqdx ` +ż +O +pcm +0 pxqq2 +´dx +“ ´2 +ż t^τ m +N +0 +ż +O +ppumps, xq ¨ ∇cmps, xq ` η∆cmps, xqqqq cmps, xq1tcmps,xqă0udxds +` 2 +ż t^τ m +N +0 +ż +O +nmps, xqfpcmps, xqqcmps, xq1tcmps,xqă0udxds +(4.17) +“ 2 +ż t^τ m +N +0 +ż +O +´ +η |∇cmps, xq|2 ` nmps, xqfpcmps, xqqcmps, xq +¯ +1tcmps,xqă0udxds, +where we have used integration-by-parts and the fact that ∇¨um “ 0. By the mean value theorem +we know that, for all x P O, there exists a number λmpxq P pminp0, cmpxqq, maxp0, cmpxqqq +such that +fpcmpxqq ´ fp0q “ cmpxqf 1pλmpxqq. +By the fact that fp0q “ 0, we infer from (4.17) that +ˇˇcm´pt ^ τ m +N q +ˇˇ2 +L2 ´ |pcm +0 q´|2 +L2 “ ´2 +ż t^τ m +N +0 +ż +O +nmps, xqf 1pλmps, xqqc2 +mps, xq1tcmps,xqă0udxds. +Since f 1 ą 0 and 1tcmă0u ą 0 as well as on r0, t ^ τ m +N s, c2 +m ą 0 and nm ą 0, we deduce that +ˇˇcm´pt ^ τ m +N q +ˇˇ2 +L2 ď |pcm +0 q´|2 +L2. Owing to the fact that by the relation (4.1) we have cm +0 ą 0, +we derive that pcm +0 q´ “ 0 and therefore +ˇˇcm´pt ^ τ m +N q +ˇˇ2 +L2 “ 0. +This gives cm´pt ^ τ m +N q “ 0 +and implies that for all t P r0, Ts, P-a.s, cmpt ^ τ m +N q ą 0. +It remains to prove the inequality (4.10). The proof is similar to the proof of Corollary 3.7. +Let p ě 2 be an integer. Let Ψ : Hm Ñ R be the functional defined by Ψpcq “ +ş +O cppxqdx. +Note that the mapping Ψ is twice (Fr´echet) differentiable and its first and second derivatives +are given by +Ψ1pcqpzq “ p +ż +O +cp´1pxqzpxqdx, +@c, z P Hm, +Ψ +2pcqpz, kq “ ppp ´ 1q +ż +O +cp´2pxqzpxqkpxqdx, +@c, z, k P Hm. +By applying the Itˆo formula to the process t ÞÑ Ψpcmpt ^ τ m +N qq, we derive that for all +t P r0, Ts, +Ψpcmpt ^ τ m +N qq ´ Ψpcmp0qq “ +ż t^τ m +N +0 +Ψ1pcmpsqq pupsq ¨ ∇cmpsq ` ξ∆cmpsq ´ nmpsqGpcmpsqqq ds +` 1 +2 +ż t^τ m +N +0 +2ÿ +k“1 +Ψ +2pcmpsqq pγφkpcmpsqq, γφkpcmpsqqq ds +` γ +2ÿ +k“1 +ż t^τ m +N +0 +Ψ1pcmpsqqpφkpcmpsqqqdβk +s , + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +27 +from which and calculations similar to (3.13), (3.18), (3.19) and (3.20) we derive from the +last equality that +Ψpcmpt ^ τ m +N qq ´ Ψpcm +0 q +“ +ż t^τ m +N +0 +ż +O +´ +´ppp ´ 2q |∇cmps, xq|2 cp´2 +m ps, xq ´ pnmps, xqfpcmps, xqqcp´1 +m ps, xq +¯ +dxds. +Since for all s P r0, ts the quantities nmps ^ τ m +N q, fpcmps ^ τ m +N qq and cmps ^ τ m +N q are positive +P-a.s, we infer from the last equality that for all t P r0, Ts, Ψpcmpt ^ τ m +N qq ď Ψpcm +0 q. This +implies that |cmpt ^ τ m +N q|Lp ď |cm +0 |Lp for all p ě 2. +Using the fact that |.|Lp Ñ |.|L8 as +p Ñ `8 and the inequality (4.1), we obtain the result. +□ +Next, we introduce for any t P r0, Ts and m, N P N, the following Lyapunov functional +Epnm, cm, umqpt ^ τ m +N q “ +ż +O +nmpt ^ τ m +N q ln nmpt ^ τ m +N qdx ` Kf |∇cmpt ^ τ m +N q|2 +L2 +` K4 +η |umpt ^ τ m +N q|2 +L2 ` e´1 |O| , +where K4 is some positive constant to be given later and Kf is defined in (2.2). +Since +x ln x ě ´e´1 for any x ą 0, we can easily see that for all t P r0, Ts, Epnm, cm, umqpt^τ m +N q ě 0. +As in [44] the property (4.1) implies that +(4.18) +Epnm +0 , cm +0 , um +0 q ď Epn0, c0, u0q, +for all m ě 1. +In addition, taking into account the inequality (4.10) and setting K “ minpKf, K4 +η q the following +holds for all t P r0, Ts, +|pumptq, cmpt ^ τ m +N q|2 +H ď K´1Epnm, cm, umqpt ^ τ m +N q ` K´1 |cmpt ^ τ m +N q|2 +L2 +ď K´1Epnm, cm, umqpt ^ τ m +N q ` K´1 |O| |c0|2 +L8 , +P-a.s. +(4.19) +We now proceed to establish some uniform bounds for um, cm, and nm in some suitable +spaces. For this purpose, we recall that hereafter, K will denote a positive constant independent +of m and N, which may change from one term to the next. +Lemma 4.3. Under the same assumptions as in Proposition 3.5, there exists a positive constant +K such that for all m P N and N P N, +(4.20) +sup +0ďsďT +|cmps ^ τ m +N q|2 +L2 ` 2η +ż T^τ m +N +0 +|∇cmpsq|2 +L2 ds ď |O| |c0|2 +L8 , +P-a.s. +E sup +0ďsďT +Epnm, cm, umqps ^ τ m +N q ď K, +E +ż T^τ m +N +0 +ˆˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ` |∆cmpsq|2 +L2 ` |∇umpsq|2 +L2 +˙ +ds ď K. +(4.21) + +28 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Proof. Let t P r0, Ts be arbitrary but fixed. We start by proving the estimate (4.20). To do +this, we take pm, Nq P N2 arbitrary and apply the Itˆo formula to t ÞÑ |cmpt ^ τ m +N q|2 +L2 to get +|cmpt ^ τ m +N q|2 +L2 ` 2ξ +ż t^τ m +N +0 +|∇cmpsq|2 +L2 ds +“ |cm +0 |2 +L2 ´ 2 +ż t^τ m +N +0 +pB1pumpsq, cmpsqq, cmpsqqds ´ 2 +ż t^τ m +N +0 +pR1pnmpsq, cmpsqq, cmpsqqds +(4.22) +` γ2 +ż t^τ m +N +0 +|φpcmpsqq|2 +L2pR2;L2q ` 2γ +ż t^τ m +N +0 +pφpcmpsqq, cmpsqqdβs. +By integration by part, we derive that +pB1pum, cmq, cmq “ 1 +2 +ż +O +umpxq ¨ ∇c2 +mpxqdx “ ´1 +2 +ż +O +c2 +mpxq∇ ¨ umpxqdx “ 0. +By the free divergence property of σk and the fact that σk “ 0 on BO, k “ 1, 2, we get +pφpcmq, cmq “ +2ÿ +k“1 +ż +O +σkpxq ¨ ∇cmpxqcmpxqdx +“ 1 +2 +2ÿ +k“1 +ż +O +σkpxq ¨ ∇c2 +mpxqdx +“ ´1 +2 +2ÿ +k“1 +ż +O +c2 +mpxq∇ ¨ σkpxqdx ` 1 +2 +2ÿ +k“1 +ż +BO +c2 +mpσqσkpσq ¨ νdσ +“ 0. +Taking into account the equality (3.13), we infer that +|φpcmq|2 +L2pR2;L2q “ +2ÿ +k“1 +ż +O +|σkpxq ¨ ∇cmpxq|2 +L2 dx “ |∇cm|2 +L2 . +Using these three last equalities and the fact that |cm +0 |2 +L2 ď |O| |c0|2 +L8 (since by the relation +(4.1), |cm +0 |2 +L8 ď |c0|2 +L8), we infer from the equality (4.22) that for all t P r0, Ts, +(4.23) +|cmpt ^ τ m +N q|2 +L2`2η +ż t^τ m +N +0 +|∇cmpsq|2 +L2 ds`2 +ż t^τ m +N +0 +ż +O +nmps, xqfpcmps, xqqcmps, xqdxds ď |O| |c0|2 +L8 . +Thanks to the non-negativity of nmps ^ τ m +N q, cmps ^ τ m +N q and f over the interval r0, ts given +in Lemma 4.2 and Assumption 2.1, we can deduce from the inequality (4.23) that +(4.24) +sup +0ďtďT +|cmpt ^ τ m +N q|2 +L2 ` 2η +ż T^τ m +N +0 +|∇cmpsq|2 +L2 ds ď |O| |c0|2 +L8 , +P-a.s. +Let us now move to the proof of the estimate (4.21). +Multiplying equation (2.14)3 by 1 ` ln nmps ^ τ m +N q for s P r0, ts and integrate the resulting +equation in O and using an integration-by-parts as well as the divergence free property of +um, we have +d +dt +ż +O +nmps ^ τ m +N , xq ln nps ^ τ m +N , xqdx ` δ +ż +O +|∇nmps ^ τ m +N , xq|2 +nmps ^ τ m +N , xq +dx +“ χ +ż +O +∇nmps ^ τ m +N , xq ¨ ∇cmps ^ τ m +N , xqdx. +(4.25) + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +29 +In the equality (4.25), we have used the fact that um “ Bnm +Bν “ 0 on BO and the fact that +´ +ż +O +∆nmpxq lnpnmpxqqdx “ +ż +O +∇nmpxq ¨ ∇ lnpnmpxqqdx ´ +ż +BO +Bnmpσq +Bν +lnpnmpσqqdσ +“ +ż +O +∇nmpxq ¨ ∇nmpxq +nmpxq +dx, +as well as +ż +O +umpxq ¨ ∇nmpxq lnpnmpxqqdx “ ´ +ż +O +nmpxq∇ ¨ pumpxq lnpnmpxqqqdx +` +ż +BO +nmpσq lnpnmpσqqumpσq ¨ νdσ +“ ´ +ż +O +nmpxqumpxq ¨ ∇ lnpnmpxqqdx +´ +ż +O +nmpxq lnpnmpxqq∇ ¨ umpxqdx +“ ´ +ż +O +umpxq ¨ ∇nmpxqdx. +It follows from the Young inequality and the Cauchy-Schwarz inequality that +χ +ż +O +∇nmpxq ¨ ∇cmpxqdx ď δ +2 +ż +O +|∇nmpxq|2 +nmpxq +dx ` χ2 +2δ +ż +O +nmpxq |∇cmpxq|2 dx. +Since +ż +O +|∇nmpxq|2 +nmpxq +dx “ 4 +ż +O +ˇˇˇ∇ +a +nmpxq +ˇˇˇ +2 +dx, +we may combine the last inequality with equality (4.25) to obtain +ż +O +nmpt ^ τ m +N , xq ln nmpt ^ τ m +N , xqdx ` 2δ +ż t^τ m +N +0 +ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ds +ď +ż +O +nm +0 pxq ln nm +0 pxqdx ` χ2 +2δ +ż t^τ m +N +0 +ˇˇˇ +a +nmpsq∇cmpsq +ˇˇˇ +2 +L2 ds. +(4.26) +Applying the Itˆo formula once more to t ÞÑ |∇cmpt ^ τ m +N q|2 +L2, yields +|∇cmpt ^ τ m +N q|2 +L2 ` 2ξ +ż t^τ m +N +0 +|∆cmpsq|2 +L2 ds +“ |∇cm +0 |2 +L2 ´ 2 +ż t^τ m +N +0 +p∇B1pumpsq, cmpsqq, ∇cmpsqqds +´ 2 +ż t^τ m +N +0 +p∇R1pnmpsq, cmpsqq, ∇cmpsqqds +(4.27) +` γ2 +ż t^τ m +N +0 +|∇φpcmpsqq|2 +L2pR2;L2q ` 2γ +ż t^τ m +N +0 +p∇φpcmpsqq, ∇cmpsqqdβs. + +30 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Since um is solenoidal and vanishes on BO, we derive that +p∇B1pum, cmq, ∇cmq “ +ż +O +∇pumpxq ¨ ∇cmpxqq ¨ ∇cmpxqdx +“ +ż +O +∇umpxq∇cmpxq ¨ ∇cmpxqdx +` +ż +O +∇cmpxq ¨ D2cmpxqumpxqdx +ď +ż +O +|∇umpxq| |∇cmpxq|2 dx ` 1 +2 +ż +O +umpxq ¨ ∇ |∇cmpxq|2 dx +(4.28) +ď |∇um|L2 |∇cm|2 +L4 . +We use the Gagliardo-Niremberg inequality to obtain +|∇cm|4 +L4 ď KGN |cm|2 +L8 +ˇˇD2cm +ˇˇ2 +L2 ` KGN |cm|4 +L8 , +To cancel +ˇˇD2cm +ˇˇ +L2, we invoke the pointwise identity +|∆cm|2 “ ∇ ¨ p∆cm∇cmq ´ ∇cm ¨ ∇∆cm, +and ∆ |∇cm|2 “ 2∇cm ¨ ∇∆cm ` 2 +ˇˇD2cm +ˇˇ2, as well as the integration-by-parts to rewrite +|∆cm|2 +L2 as +|∆cm|2 +L2 “ ´ +ż +O +∇cmpxq ¨ ∇∆cmpxqdx +“ +ˇˇD2cm +ˇˇ2 +L2 ´ 1 +2 +ż +O +∆ |∇cmpxq|2 dx +(4.29) +“ +ˇˇD2cm +ˇˇ2 +L2 ´ 1 +2 +ż +BO +B |∇cmpσq|2 +Bν +dσ. +Invoking [27, Lemma 4.2] we obtain +(4.30) +1 +2 +ż +BO +B |∇cmpσq|2 +Bν +dσ ď κpOq +ż +BO +|∇cmpσq|2 dσ, +where κpOq is an upper bound for the curvatures of BO. +Thanks to the trace theorem (see [21, (ii) of Proposition 4.22 with (i) of Theorem 4.24]), +it holds that +ż +BO +|∇cmpσq|2 dσ ď KpO, ςq |cm|2 +H +3`ς +2 +for any ς P p0, 1q, +where KpO, ςq ą 0 depends only on O and ς, which can be fixed for instance ς “ 1{2. On +the other hand, the interpolation inequality, the Young inequality and the inequality (4.10) of +Lemma 4.2 imply the existence of K1 and K2 depending on O such that +κpOqKpO, ςq |cm|2 +H +7 +4 ď K1p +ˇˇD2cm +ˇˇ7{4 +L2 |cm|1{4 +L2 ` |cm|2 +L2q +ď 1 +4 +ˇˇD2cm +ˇˇ2 +L2 ` K2 |c0|2 +L8 . +Using this previous inequality and (4.30), we infer from the equality (4.29) that +(4.31) +ˇˇD2cm +ˇˇ2 +L2 ď 4 +3 |∆cm|2 +L2 ` 4K2 +3 +|c0|2 +L8 , + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +31 +and therefore +|∇cm|4 +L4 ď 4KGN |c0|2 +L8 +3 +|∆cm|2 +L2 ` +ˆ4K2 +3 +` 1 +˙ +KGN |c0|4 +L8 . +By the inequality (4.28) and the Young inequality, we infer that +p∇B1pum, cmq, ∇cmq ď |∇um|L2 |∇cm|2 +L4 +ď +3ξ +16KGN |c0|2 +L8 +|∇cm|4 +L4 ` 4KGN |c0|2 +L8 +3ξ +|∇um|2 +L2 +ď ξ +4 |∆cm|2 +L2 ` 4KGN |c0|2 +L8 +3ξ +|∇um|2 +L2 ` ξp4K2 ` 3q +16 +|c0|2 +L8 . +Due to the Assumption 1 and the inequality (4.10) of Lemma 4.2, we note that +´p∇R1pnm, cmq, ∇cmq “ ´ +ż +O +∇pnmpxqfpcmpxqqq ¨ ∇cmpxqdx +“ ´ +ż +O +f 1pcmpxqq |∇cmpxq|2 nmpxqdx ´ +ż +O +fpcmpxqq∇cmpxq ¨ ∇nmpxqdx +ď ´ +min +0ďcď|c0|L8 f 1pcq +2 +ż +O +nmpxq |∇cmpxq|2 dx +` +1 +2 +min +0ďcď|c0|L8 f 1pcq +ż +O +f 2pcmpxqq|∇nmpxq|2 +nmpxq +dx +ď ´ +min +0ďcď|c0|L8 f 1pcq +2 +|?nm∇cm|2 +L2 ` +2 +max +0ďcď|c0|L8 f 2pcq +min +0ďcď|c0|L8 f 1pcq |∇?nm|2 +L2 . +Combining these two last inequalities, we derive from equality (4.27) that +|∇cmpt ^ τ m +N q|2 +L2 ` 3ξ +2 +ż t^τ m +N +0 +|∆cmpsq|2 +L2 ds ` +min +0ďcď|c0|L8 f 1pcq +ż t^τ m +N +0 +ˇˇˇ +a +nmpsq∇cmpsq +ˇˇˇ +2 +L2 ds +ď |∇cm +0 |2 +L2 ` ξp4K2 ` 3q +8 +|c0|2 +L8 t ` 8KGN |c0|2 +L8 +3ξ +ż t^τ m +N +0 +|∇umpsq|2 +L2 ds +` +4 +max +0ďcď|c0|L8 f 2pcq +min +0ďcď|c0|L8 f 1pcq +ż t^τ m +N +0 +ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ds +` γ2 +ż t^τ m +N +0 +|∇φpcmpsqq|2 +L2pR2;L2q ds ` 2γ +ż t^τ m +N +0 +p∇φpcmpsqq, ∇cmpsqqdβs. + +32 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Multiplying this last inequality by Kf and adding the result with inequality (4.26) we obtain +ż +O +nmpt ^ τ m +N , xq ln nmpt ^ τ m +N , xqdx ` Kf |∇cmpt ^ τ m +N q|2 +L2 ` ξKf +4 +ż t^τ m +N +0 +|∆cmpsq|2 +L2 ds +` 2δ +ż t^τ m +N +0 +ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ds ` +ż t +0 +ˇˇˇ +a +nmpsq∇cmpsq +ˇˇˇ +2 +L2 ds +ď Kf |∇cm +0 |2 +L2 ` +ż +O +nm +0 pxq ln nm +0 pxqdx ` Kfξp4KfK2 ` 3q +8 +|c0|2 +L8 t +` 8KfKGN |c0|2 +L8 +3ξ +ż t^τ m +N +0 +|∇umpsq|2 +L2 ds ` +4Kf +max +0ďcď|c0|L8 f 2pcq +min +0ďcď|c0|L8 f 1pcq +ż t^τ m +N +0 +ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ds +` γ2Kf +ż t^τ m +N +0 +|∇φpcmpsqq|2 +L2pR2;L2q ds ` 2γKf +ż t^τ m +N +0 +p∇φpcmpsqq, ∇cmpsqqdβs. +By using the first inequality of (3.3), we see that the previous inequality reduces to +ż +O +nmpt ^ τ m +N , xq ln nmpt ^ τ m +N , xqdx ` Kf |∇cmpt ^ τ m +N q|2 +L2 ` 3ξKf +2 +ż t^τ m +N +0 +|∆cmpsq|2 +L2 ds +` 2δ +ż t^τ m +N +0 +ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ds ` +ż t^τ m +N +0 +ˇˇˇ +a +nmpsq∇cmpsq +ˇˇˇ +2 +L2 ds +ď Kf |∇cm +0 |2 +L2 ` +ż +O +nm +0 pxq ln nm +0 pxqdx ` Kfξp4KfK2 ` 3q +8 +|c0|2 +L8 t +` 8KfKGN |c0|2 +L8 +3ξ +ż t^τ m +N +0 +|∇umpsq|2 +L2 ds ` γ2Kf +ż t^τ m +N +0 +|∇φpcmpsqq|2 +L2pR2;L2q ds +(4.32) +` 2γKf +ż t^τ m +N +0 +p∇φpcmpsqq, ∇cmpsqqdβs. +Now, we use the equality (4.8) of Lemma 4.2 and the inequality (3.7) to obtain that +|nm|L2 ď KGN +´ +|?nm|L2 |∇?nm|L2 ` |?nm|2 +L2 +¯ +ď KGN +ˆ +|nm +0 | +1 +2 +L1 |∇?nm|L2 ` |nm +0 |L1 +˙ +, +(4.33) +By the relation (4.1), we have nm +0 Ñ n0 in L2pOq. Thanks to the continuous embedding of +L2pOq into L1pOq, we derive that nm +0 Ñ n0 in L1pOq and therefore the sequence tnm +0 umě1 +is bounded in L1pOq. This implies that the inequality (4.33) can be controlled as follows +|nm|L2 ď KGNK1{2 |∇?nm|L2 ` K, +(4.34) +where K is a constant independent of m and N. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +33 +Next, applying the Itˆo formula to t ÞÑ |umpt ^ τ m +N q|2 +L2 and using the estimation (4.34), we +infer the existence of K3 ą 0 such that +|umpt ^ τ m +N q|2 +L2 ` 2η +ż t^τ m +N +0 +|∇umpsq|2 +L2 ds +ď 2 +ż t^τ m +N +0 +|∇Φ|L8 |nmpsq|L2 |umpsq|L2 ds +` +ż t^τ m +N +0 +|gpumpsq, cmpsqq|2 +L2pU;Hq ds ` 2 +ż t^τ m +N +0 +pgpumpsq, cmpsqq, umpsqqdWs +ď |um +0 |2 +L2 ` δη +K4 +ż t^τ m +N +0 +ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ds ` K3 |∇Φ|2 +L8 +ż t^τ m +N +0 +|umpsq|2 +L2 ds +` 1 +2t ` 1 +2 |∇Φ|2 +L8 K2 +ż t^τ m +N +0 +|umpsq|2 +L2 ds +` +ż t^τ m +N +0 +|gpumpsq, cmpsqq|2 +L2pU;Hq ds ` 2 +ż t^τ m +N +0 +pgpumpsq, cmpsqq, umpsqqdWs, +with K4 “ 8Kf KGN|c0|2 +L8 +3ξ +. +Multiplying this inequality by +K4 +η , and adding the result with +inequality (4.32) after using the inequality (4.18), we see that there exists positive constants +K5 and K6 such that for all t P r0, Ts, P-a.s. +Epnm, cm, umqpt ^ τ m +N q ` δ +ż t^τ m +N +0 +ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ds +` +ż t^τ m +N +0 +„3ξKf +2 +|∆cmpsq|2 +L2 ` K4 |∇umpsq|2 +L2 ` +ˇˇˇ +a +nmpsq∇cmpsq +ˇˇˇ +2 +L2 + +ds +ď Epn0, c0, u0q ` K5T ` K6 +ż t^τ m +N +0 +|umpsq|2 +L2 ds ` γ2Kf +ż t^τ m +N +0 +|∇φpcmpsqq|2 +L2pR2;L2q ds +(4.35) +` 2K4 +η +ż t^τ m +N +0 +pgpumpsq, cmpsqq, umpsqqdWs +` K4 +η +ż t^τ m +N +0 +|gpumpsq, cmpsqq|2 +L2pU,Hq ds ` 2γKf +ż t^τ m +N +0 +p∇φpcmpsqq, ∇cmpsqqdβs. +Now, since γ satisfies the relation (3.3), taking into account the inequality (4.31), we note +that +γ2Kf |∇φpcmq|2 +L2pR2;L2q ď 2γ2Kf +2ÿ +k“1 +ż +O +|∇σkpxq∇cpxq|2 dx ` 2γ2Kf +2ÿ +k“1 +ż +O +ˇˇD2cpxqσkpxq +ˇˇ2 dx +ď 2γ2Kf |∇c|2 +L2 +2ÿ +k“1 +|σk|2 +W 1,8 ` |∆c|2 +L2 8γ2Kf +3 +2ÿ +k“1 +|σk|2 +L8 +(4.36) +` 8γ2KfK2 +3 +|c0|L8 +2ÿ +k“1 +|σk|2 +L8 +ď K |∇c|2 +L2 ` ξKf +2 +|∆c|2 +L2 ` K. + +34 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +By the inequalities (2.12) and (4.19), we also note that +|gpum, cmq|2 +L2pU,Hq ď 2L2 +g |pum, cmq|2 +H ` 2L2 +g +ď KEpnm, cm, umq ` K |c0|2 +L8 ` 2L2 +g. +(4.37) +From the estimates (4.35) until (4.37), we derive that +E sup +0ďsďT +Epnm, cm, umqps ^ τ m +N q ` δE +ż T^τ m +N +0 +ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ds +` E +ż T^τ m +N +0 +„ +ξKf |∆cmpsq|2 +L2 ` K4 |∇umpsq|2 +L2 ` +ˇˇˇ +a +nmpsq∇cmpsq +ˇˇˇ +2 +L2 + +ds +ď Epn0, c0, u0q ` KT ` KE +ż T^τ m +N +0 +Epnmpsq, cmpsq, umpsqqds ` 2L2 +gT +` 2γKfE sup +0ďsďT +ˇˇˇˇ +ż s^τ m +N +0 +p∇φpcmpsqq, ∇cmpsqqdβs +ˇˇˇˇ +(4.38) +` 2K4 +η E sup +0ďsďT +ˇˇˇˇˇ +8 +ÿ +k“1 +ż s^τ m +N +0 +pgpumpsq, cmpsqqek, umpsqqdW k +s +ˇˇˇˇˇ . +Now, by making use of the Burholder-Davis-Gundy, Cauchy-Schwarz, Young inequalities and +the fact that γ satisfies the relation (3.3), we infer that +2γKfE sup +0ďsďT +ˇˇˇˇ +ż s^τ m +N +0 +p∇φpcmpsqq, ∇cmpsqqdβs +ˇˇˇˇ +ď KE +ˆż T^τ m +N +0 +|p∇φpcmpsqq, ∇cmpsqq|2 ds +˙1{2 +ď KE +ˆż T^τ m +N +0 +|∇φpcmpsqq|2 +L2pR2;L2q |∇cmpsq|2 +L2 ds +˙1{2 +ď Kf +4 E sup +0ďsďT +|∇cmps ^ τ m +N q|2 +L2 ` KE +ż T^τ m +N +0 +|∇φpcmpsqq|2 +L2pR2;L2q ds +ď 1 +4E sup +0ďsďT +Epnmpsq, cmpsq, umpsqqps ^ τ m +N q +` ξKf +2 E +ż T^τ m +N +0 +|∆cmpsq|2 +L2 ds ` KE +ż T^τ m +N +0 +|∇cmpsq|2 +L2 ds ` KT. +Similarly, +2K4 +η E sup +0ďsďT +ˇˇˇˇˇ +8 +ÿ +k“1 +ż s^τ m +N +0 +pgpumpsq, cmpsqqek, umpsqqdW k +s +ˇˇˇˇˇ +ď 1 +4E sup +0ďsďT +Epnm, cm, umqps ^ τ m +N q ` KE +ż T^τ m +N +0 +|gpumpsq, cmpsqq|2 +L2pU;Hq ds +ď 1 +4E sup +0ďsďT +Epnm, cm, umqps ^ τ m +N q ` KE +ż T^τ m +N +0 +|pumpsq, cmpsqq|2 +H ds ` KTL2 +g. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +35 +It follows from the estimates (4.38) that +E sup +0ďsďT +Epnm, cm, umqps ^ τ m +N q +` E +ż T^τ m +N +0 +„ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ` Kf |∆cmpsq|2 +L2 ` |∇umpsq|2 +L2 ` +ˇˇˇ +a +nmpsq∇cmpsq +ˇˇˇ +2 +L2 + +ds +(4.39) +ď KEpn0, c0, u0q ` KT ` KE +ż T^τ m +N +0 +Epnm, cm, umqpsqds ` K, +where K is a constant depending on the initial data and T but independent of m and N. +Now, the Gronwall lemma yields +E sup +0ďsďT +Epnm, cm, umqps ^ τ m +N q +` E +ż T^τ m +N +0 +„ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ` |∆cmpsq|2 +L2 ` |∇umpsq|2 +L2 ` +ˇˇˇ +a +nmpsq∇cmpsq +ˇˇˇ +2 +L2 + +ds ď K, +from which we deduce the estimates (4.21) and hence completing the proof of Lemma 4.3. +□ +Lemma 4.4. Under the same assumptions as in Lemma 4.3, for all p ě 1, there exists a +positive constant K such that we have for all m P N and N P N, +(4.40) +sup +0ďsďT +|cmps ^ τ m +N q|2p +L2 ` +ˆż T^τ m +N +0 +|∇cmpsq|2 +L2 ds +˙p +ď |O|p |c0|2p +L8 , +P-a.s., +E sup +0ďsďT +Eppnm, cm, umqps ^ τ m +N q ` E +ˆż T^τ m +N +0 +ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ds +˙p +ď K, +and E +ˆż T^τ m +N +0 +|∆cmpsq|2 +L2 ds +˙p +` E +ˆż T^τ m +N +0 +|∇umpsq|2 +L2 ds +˙p +ď K. +(4.41) +Proof. The inequality (4.40) follows directly from the estimates (4.20) of Lemma 4.3. Next, +we are going to derive estimate (4.41). We start with the inequality (4.38) and invoke the +Jensen inequality to derive that for all p ě 2, +E sup +0ďsďT +Eppnm, cm, umqps ^ τ m +N q ` E +ˆż T^τ m +N +0 +ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ds +˙p +` E +ˆż T^τ m +N +0 +ξKf |∆cmpsq|2 +L2 ds +˙p +` E +ˆż T^τ m +N +0 +K4 |∇umpsq|2 +L2 ds +˙p +ď Eppn0, c0, u0q ` KT p ` KE +ˆż T^τ m +N +0 +Epnm, cm, umqpsqds +˙p +(4.42) +` Kp ` 2pγpKp +fE sup +0ďsďT +ˇˇˇˇ +ż s^τ m +N +0 +p∇φpcmpsqq, ∇cmpsqqdβs +ˇˇˇˇ +p +` KE sup +0ďsďT +ˇˇˇˇˇ +8 +ÿ +k“1 +ż s^τ m +N +0 +pgpumpsq, cmpsqqek, umpsqqdW k +s +ˇˇˇˇˇ +p +. +Invoking the H¨older inequality, we see that +KE +ˆż T^τ m +N +0 +Epnm, cm, umqpsqds +˙p +ď KT +p +p´1E +ż T^τ m +N +0 +Eppnm, cm, umqpsqds. + +36 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Thanks to the Burkholder-Davis-Gundy inequality, we see that +KE sup +0ďsďT +ˇˇˇˇˇ +8 +ÿ +k“1 +ż s^τ m +N +0 +pgpumpsq, cmpsqqek, umpsqqdW k +s +ˇˇˇˇˇ +p +ď KE +8 +ÿ +k“1 +ˆż T^τ m +N +0 +|pgpumpsq, cmpsqqek, umpsqq|2 ds +˙p{2 +ď KE sup +0ďsďT +|umps ^ τ m +N q|p +L2 +ˆż T^τ m +N +0 +|gpumpsq, cmpsqq|2 +L2pU;Hq ds +˙p{2 +ď 1 +4E sup +0ďsďT +Eppnm, cm, umqps ^ τ m +N q ` KE +ˆż T^τ m +N +0 +|gpumpsq, cmpsqq|2 +L2pU;Hq ds +˙p +ď 1 +4E sup +0ďsďT +Eppnm, cm, umqps ^ τ m +N q ` KE +ż T^τ m +N +0 +|pumpsq, cmpsqq|2p +H ds ` KT +p +p´1L2p +g . +Taking into account the fact that γ is sufficiently small such that the relation (3.3) is satisfied, +we also arrive at +2pγpKp +fKE sup +0ďsďT +ˇˇˇˇ +ż s^τ m +N +0 +p∇φpcmpsqq, ∇cmpsqqdβs +ˇˇˇˇ +p +ď 2pγpKp +fE +ˆż T^τ m +N +0 +|∇φpcmpsqq|2 +L2pR2;L2q |∇cmpsq|2 +L2 ds +˙p{2 +ď 1 +4E sup +0ďsďT +|∇cmps ^ τ m +N q|2p +L2 ` 22pγ2pK2p +f E +ˆż T^τ m +N +0 +|∇φpcmpsqq|2 +L2pR2;L2q ds +˙p +ď 1 +4E sup +0ďsďT +Eppnm, cm, umqps ^ τ m +N q +` 1 +2E +ˆż T^τ m +N +0 +ξKf |∆cmpsq|2 +L2 ds +˙p +` KT +p +p´1E +ż T^τ m +N +0 +|∇cmpsq|2p +L2 ds ` KT p. +It follows from the estimates (4.42) that +E sup +0ďsďT +Eppnm, cm, umqps ^ τ m +N q ` E +ˆż T^τ m +N +0 +ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ds +˙p +` E +ˆż T^τ m +N +0 +|∆cmpsq|2 +L2 ds +˙p +` E +ˆż T^τ m +N +0 +|∇umpsq|2 +L2 ds +˙p +ď KEppn0, c0, u0q ` KT p ` KE +ż T^τ m +N +0 +Eppnm, cm, umqpsqds ` K. +Now, the Gronwall lemma yields +E sup +0ďsďT +Eppnm, cm, umqps ^ τ m +N q ` E +ˆż T^τ m +N +0 +ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ds +˙p +` E +ˆż T^τ m +N +0 +|∆cmpsq|2 +L2 ds +˙p +` E +ˆż T^τ m +N +0 +|∇umpsq|2 +L2 ds +˙p +ď K, +and the estimate (4.41) follows directly from this last inequality. This completes the proof +of Lemma 4.4. +□ +In order to control the process t ÞÑ nmpt ^ τ m +N q, we prove the following lemma. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +37 +Lemma 4.5. Under the same assumptions as in Lemma 4.3, there exists a positive constant +η0 ą 1 such that for all m P N, N P N and P-a.s., +sup +0ďsďT +|nmps ^ τ m +N q|2 +L2 ` +ż T^τ m +N +0 +|nmpsq|2 +H1 ds ď η0 exp +ˆ +K +ż T^τ m +N +0 +|∇cmpsq|4 +L4 ds +˙ +. +(4.43) +Proof. Let t P r0, Ts be arbitrary but fixed. Multiplying the last equation of (4.2) by nmps^τ m +N q +for 0 ď s ď t, and using the fact that ∇ ¨ um “ 0 and the inequality (3.7) as well as the +H¨older inequality and the Young inequality, we obtain +1 +2 +d +dt |nmps ^ τ m +N q|2 +L2 ` δ |∇nmps ^ τ m +N q|2 +L2 +“ ξ +ż +O +nmps ^ τ m +N , xq∇cmps ^ τ m +N , xq ¨ ∇nmps ^ τ m +N , xqdx +ď ξ |nmps ^ τ m +N q|L4 |∇cmps ^ τ m +N q|L4 |∇nmps ^ τ m +N q|L2 +ď Kp|nmps ^ τ m +N q|1{2 +L2 |∇nmps ^ τ m +N q|1{2 +L2 ` |nmps ^ τ m +N q|L2q |∇cmps ^ τ m +N q|L4 |∇nmps ^ τ m +N q|L2 +ď K |nmps ^ τ m +N q|1{2 +L2 |∇cmps ^ τ m +N q|L4 |∇nmps ^ τ m +N q|3{2 +L2 +` K |nmps ^ τ m +N q|L2 |∇cmps ^ τ m +N q|L4 |∇nmps ^ τ m +N q|L2 +ď δ +2 |∇nmps ^ τ m +N q|2 +L2 ` K |nmps ^ τ m +N q|2 +L2 p|∇cmps ^ τ m +N q|4 +L4 ` |∇cmps ^ τ m +N q|2 +L4q +ď δ +2 |∇nmps ^ τ m +N q|2 +L2 ` K |nmps ^ τ m +N q|2 +L2 +´ +|∇cmps ^ τ m +N q|4 +L4 ` 1 +¯ +. +This implies that for all t P r0, Ts, +sup +0ďsďt +|nmps ^ τ m +N q|2 +L2 ` δ +ż t^τ m +N +0 +|∇nmpsq|2 +L2 ds ď |nm +0 |2 +L2 ` K +ż t^τ m +N +0 +|nmpsq|2 +L2 +´ +|∇cmpsq|4 +L4 ` 1 +¯ +ds. +Since n0 +m Ñ n0 in L2pOq, |nm +0 |2 +L2 is uniformly bounded. Thus, applying the Gronwall lemma, +we obtain that +sup +0ďsďt +|nmps ^ τ m +N q|2 +L2 ` +ż t^τ m +N +0 +|nmpsq|2 +H1 ds ď Kδ exp +ˆ +K +ż t^τ m +N +0 +´ +|∇cmpsq|4 +L4 ` 1 +¯ +ds +˙ +ď pKδ ` 1qeKT exp +ˆ +K +ż t^τ m +N +0 +|∇cmpsq|4 +L4 ds +˙ +, +and complete the proof of Lemma 4.5. +□ +Corollary 4.6. Under the same assumptions as in Lemma 4.3, for any p ě 1, there exists a +positive constant K such that for all m P N and N P N, +E sup +0ďsďT +|umps ^ τ m +N q|2p +L2 ` E +ˆż T^τ m +N +0 +|∇umpsq|2 +L2 ds +˙p +ď K, +(4.44) +E +ˆż T^τ m +N +0 +|nmpsq|2 +L2 ds +˙p +ď K, +(4.45) +E sup +0ďsďT +|cmps ^ τ m +N q|2p +H1 ` E +ˆż T^τ m +N +0 +|cmpsq|2 +H2 ds +˙p +ď K, +(4.46) +E +ż T^τ m +N +0 +|∇cmpsq|4 +L2 ds ď K. +(4.47) + +38 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Proof. The estimate (4.44) is a consequence of the estimates (4.21) and (4.41). +From the +inequalities (4.34), (4.21) and (4.41), we infer that +E +ˆż T^τ m +N +0 +|nmpsq|2 +L2 ds +˙p +ď E +ˆż T^τ m +N +0 +ˆ +KGNK1{2 ˇˇˇ∇ +a +nmpsq +ˇˇˇ +2 +L2 ` K +˙ +ds +˙p +ď K, +which proves the second estimate of inequality (4.45). +According to [35, Proposition 7.2, P. 404], we have +|cm|2 +H2 ď Kp|∆cm|2 +L2 ` |cm|2 +H1q, +from which along with (4.21) and (4.41) we deduce (4.46). +By applying the inequality (3.7), we obtain that +|∇cm|4 +L4 ď Kp|cm|2 +H2 |∇cm|2 +L2 ` |∇cm|4 +L2q. +Therefore, +E +ż T^τ m +N +0 +|∇cmpsq|4 +L4 ds ď KE +ż T^τ m +N +0 +|cmpsq|2 +H2 |∇cmpsq|2 +L2 ds ` KE +ż T^τ m +N +0 +|∇cmpsq|4 +L2 ds +ď KE sup +0ďsďT +|cmps ^ τ m +N q|2 +H1 +ż T +0 +|cmpsq|2 +H2 ds ` KTE sup +0ďsďT +|cmps ^ τ m +N q|4 +H1 +ď KE sup +0ďsďT +|cmps ^ τ m +N q|4 +H1 ` KE +ˆż T^τ m +N +0 +|cmpsq|2 +H2 ds +˙2 +, +from which along with (4.46) we deduce (4.47). +This completes the proof of Corollary +4.6. +□ +In the following lemma, we state and prove a result concerning the stopping time τ m +N . +More precisely, we prove that sup +NPN +τ N +m ě 2T with probability 1 such that the inequality (4.7) +holds. +Lemma 4.7. Let τ m +N , m, N P N be the stopping times defined in (4.6). Then, under the same +assumptions as in Lemma 4.3, it holds that +(4.48) +P +" +ω P Ω : sup +NPN +τ N +m pωq ě 2T +* +“ 1. +Consequently, the solutions pum, cm, nmq of system (4.2) exist almost surely for every t P r0, Ts. +Proof. We notice that the inequalities of Corollary 4.6 hold for every T ą 0. Hence, for a +fixed T ą 0, we set +˜T “ 2T and note that for all J P N, +" +ω P Ω : sup +NPN +τ N +m pωq ă ˜T +* +Ă +! +ω P Ω : τ J +mpωq ă ˜T +) +, +which implies that +(4.49) +P +" +ω P Ω : sup +NPN +τ N +m pωq ă 2T +* +ď +lim +NÝÑ8 P +! +ω P Ω : τ N +m pωq ă ˜T +) +, +and therefore, it is enough to show that the second term of the right hand side of this last +equality converges to zero as N Ñ 8. To this end, let +AN “ +! +ω P Ω : τ N +m ă ˜T +) + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +39 +and +BN “ +" +ω P Ω : +ˇˇˇnmp ˜T ^ τ N +m q +ˇˇˇ +2 +L2 ` +ˇˇˇump ˜T ^ τ N +m q +ˇˇˇ +2 +L2 ` +ˇˇˇcmp ˜T ^ τ N +m q +ˇˇˇ +2 +H1 ě N 2 +* +. +Then, we have AN Ă BN for N ą ˜T. Indeed, let ω P AN, then ˜T ^ τ N +m pωq “ τ N +m pωq. Thus, +by the definition of the stopping time τ N +m , we see that for N ą ˜T, +ˇˇˇnmp ˜T ^ τ N +m q +ˇˇˇ +2 +L2 ` +ˇˇˇump ˜T ^ τ N +m q +ˇˇˇ +2 +L2 ` +ˇˇˇcmp ˜T ^ τ N +m q +ˇˇˇ +2 +H1 “ +ˇˇnmpτ N +m q +ˇˇ2 +L2 ` +ˇˇumpτ N +m q +ˇˇ2 +L2 ` +ˇˇcmpτ N +m q +ˇˇ2 +H1 +ě N 2. +We then conclude that ω P BN. +Now, for N ą ˜T, using the inclusion AN Ă BN we derive that +P +! +ω P Ω : τ N +m ă ˜T +) +ď P +" +ω P Ω : +ˇˇˇnmp ˜T ^ τ N +m q +ˇˇˇ +2 +L2 ` +ˇˇˇump ˜T ^ τ N +m q +ˇˇˇ +2 +L2 ` +ˇˇˇcmp ˜T ^ τ N +m q +ˇˇˇ +2 +H1 ě N 2 +* +ď P +" +ω P Ω : +ˇˇˇnmp ˜T ^ τ N +m q +ˇˇˇ +2 +L2 ě N 2 +3 +* +` P +" +ω P Ω : +ˇˇˇump ˜T ^ τ N +m q +ˇˇˇ +2 +L2 ě N 2 +3 +* +(4.50) +` P +" +ω P Ω : +ˇˇˇcmp ˜T ^ τ N +m q +ˇˇˇ +2 +H1 ě N 2 +3 +* +. +According to the estimates (4.44) and (4.46) of Corollary 4.6 as well as the Markov inequality, +we derive that for N ą ˜T +P +" +ω P Ω : +ˇˇˇcmp ˜T ^ τ N +m q +ˇˇˇ +2 +H1 ě N 2 +3 +* +ď P +# +ω P Ω : sup +0ďsď ˜T +|cmps ^ τ m +N q|2 +H1 ě N 2 +3 ++ +ď +3 +N 2 E sup +0ďsď ˜T +|cmps ^ τ m +N q|2 +H1 +ď K +N 2 , +and +P +" +ω P Ω : +ˇˇˇump ˜T ^ τ N +m q +ˇˇˇ +2 +L2 ě N 2 +3 +* +ď P +# +ω P Ω : sup +0ďsď ˜T +|umps ^ τ m +N q|2 +L2 ě N 2 +3 ++ +ď +3 +N 2 E sup +0ďsď ˜T +|umps ^ τ m +N q|2 +L2 +ď K +N 2 . +Also for N ą maxp?3η0, ˜Tq (where η0 is a constant obtained in Lemma 4.5), we use the +inequality (4.43) of Lemma 4.5 to infer that +P +" +ω P Ω : +ˇˇˇnmp ˜T ^ τ N +m q +ˇˇˇ +2 +L2 ě N 2 +3 +* +ď P +# +ω P Ω : sup +0ďsď ˜T +|nmps ^ τ m +N q|2 +L2 ě N 2 +3 ++ +ď P +# +ω P Ω : η0 exp +˜ +K +ż ˜T^τ m +N +0 +|∇cmpsq|4 +L4 ds +¸ +ě N 2 +3 ++ +ď P +# +ω P Ω : +ż ˜T^τ m +N +0 +|∇cmpsq|4 +L4 ds ě +lnp N2 +3η0 q +K ++ +. + +40 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Invoking the Markov inequality and using the estimate (4.47) of Corollary 4.6, we see that +P +" +ω P Ω : +ˇˇˇnmp ˜T ^ τ N +m q +ˇˇˇ +2 +L2 ě N 2 +3 +* +ď +K +lnp N2 +3η0 q +E +ż ˜T^τ m +N +0 +|∇cmpsq|4 +L4 ds +ď +K +2 lnpNq ´ lnp3η0q. +Plugging these inequalities into the inequality (4.50), we arrive at +P +! +ω P Ω : τ N +m ă ˜T +) +ď K +N 2 ` +K +2 lnpNq ´ lnpη0q ´ lnp3q, +for all for N ą maxp?3η0, ˜Tq. Letting N to infinity in this last inequality we get +lim +NÝÑ8 P +! +ω P Ω : τ N +m ă ˜T +) +“ 0, +which along with (4.49) imply (4.48). +By the equality (4.48) we infer the inequality (4.7) and therefore, the relation (4.5) hold +and the lemma is then proved. +□ +Since pT ^ τ N +m qNPN is increasing, we have T ^ τ N +m Ñ T a.s., as N Ñ 8. With this almost +surely convergence in hand, we are going to give some consequences of Lemma 4.5 and +Corollary 4.6. +Corollary 4.8. Under the same assumptions as in Lemma 4.3, for any p ě 1, there exists a +positive constant K such that for all m P N, +sup +0ďsďT +|nmpsq|2 +L2 ` +ż T +0 +|nmpsq|2 +H1 ds ď η0 exp +ˆ +K +ż T +0 +|∇cmpsq|4 +L4 ds +˙ +, P-a.s. +(4.51) +E sup +0ďsďT +|umpsq|2p +L2 ` E +ˆż T +0 +|∇umpsq|2 +L2 ds +˙p +ď K, +(4.52) +E +ˆż T +0 +|nmpsq|2 +L2 ds +˙p +ď K, +(4.53) +E sup +0ďsďT +|cmpsq|2p +H1 ` E +ˆż T +0 +|cmpsq|2 +H2 ds +˙p +ď K, +(4.54) +E +ż T +0 +|∇cmpsq|4 +L2 ds ď K, +(4.55) +where η0 ą 1 is a constant obtained in Lemma 4.5. +Proof. Since +T ^ τ N +m Ñ T +a.s., +as +N Ñ 8, +by +the +path +continuity +of +the +process +t ÞÑ pumptq, cmptq, nmptqq, we can let N Ñ 8 in the inequality (4.43) of Lemma 4.5 and +derive the inequality (4.51). In addition to the almost surely convergence of T ^ τ N +m to T +and the path continuity of the process t ÞÑ pumptq, cmptq, nmptqq, we invoke the Fatou lemma +and pass to the limit as N Ñ 8 in the inequalities (4.44), (4.45), (4.46) and (4.47) and +derive the estimate (4.52), (4.53), (4.54) and (4.55). +□ +Corollary 4.9. Under the same assumptions as in Lemma 4.3, there exists a positive constant +K such that for all m P N, +(4.56) +E |nm|2 +C1{2pr0,Ts;H´3q ď K. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +41 +Proof. Let v P H3pOq. +We recall that |∇v|L8 ď K |v|H3. +So, using an integration by part +and the H¨older inequality, we derive that +|pA1nm, vq| “ |pnm, ∆vq| ď |nm|L2 |∆v|L2 ď |nm|L2 |v|H3 , +ˇˇpP1 +mB1pum, nmq, vq +ˇˇ “ +ˇˇpB1pum, nmq, P1 +mvq +ˇˇ +“ +ˇˇpnmum, ∇P1 +mvq +ˇˇ +ď K |nm|L2 |um|L2 +ˇˇ∇P1 +mv +ˇˇ +L8 +ď K |nm|L2 |um|L2 |v|H3 , +and +ˇˇpP1 +mR2pnm, cmq, vq +ˇˇ “ ξ +ˇˇpnm∇cm, ∇P1 +mvq +ˇˇ +ď K |nm|L2 |∇cm|L2 +ˇˇ∇P1 +mv +ˇˇ +L8 +ď K |nm|L2 |∇cm|L2 |v|H3 . +Due to the continuous Sobolev embeddings W 1,2p0, T; H´3pOqq ãÑ C1{2p0, T; H´3pOqq, and +L2pOq ãÑ H´3pOq, we have +E |nm|2 +C1{2p0,T;H´3q ď E |nm|2 +W 1,2p0,T;H´3q +“ E +ż T +0 +|nmpsq|2 +H´3 ds ` E +ż T +0 +ˇˇˇˇ +d +dtnmpsq +ˇˇˇˇ +2 +H´3 ds +ď KE +ż T +0 +|nmpsq|2 +L2 ds ` E +ż T +0 +ˇˇˇˇ +d +dtnmpsq +ˇˇˇˇ +2 +H´3 ds. +Using the estimates (4.52), (4.53) and (4.54), we arrive at +E |nm|2 +C1{2p0,T;H´3qq ď K ` KE +ż T +0 +|A1nmpsq|2 +H´3 ds +` KE +ż T +0 +”ˇˇP1 +mB1pumpsq, nmpsqq +ˇˇ2 +H´3 ` +ˇˇP1 +mR2pnmpsq, cmpsqq +ˇˇ2 +H´3 +ı +ds +ď K ` KE +ż T +0 +” +|nmpsq|2 +L2 ` |umpsq|2 +L2 |nmpsq|2 +L2 ` |nmpsq|2 +L2 |∇cmpsq|2 +L2 +ı +ds +ď K ` KE sup +0ďsďT +|umpsq|2 +L2 +ż T +0 +|nmpsq|2 +L2 ds ` KE sup +0ďsďT +|∇cmpsq|2 +L2 +ż T +0 +|nmpsq|2 +L2 ds +ď K ` KE sup +0ďsďT +|umpsq|4 +L2 ` KE sup +0ďsďT +|∇cmpsq|4 +L2 ` KE +ˆż T +0 +|nmpsq|2 +L2 ds +˙2 +ď K. +□ +Lemma 4.10. Under the same assumptions as in Lemma 4.3, there exists a positive constant +K such that for all m P N, +E +ż T +0 +” +|A1cmpsq|2 +L2 ` +ˇˇP2 +mB1pumpsq, cmpsqq +ˇˇ2 +L2 ` +ˇˇP2 +mR1pnmpsq, cmpsqq +ˇˇ2 +L2 +ı +ds ď K, +E +ż T +0 +” +|A0umpsq|2 +V ˚ ` +ˇˇP2 +mB0pumpsq, umpsqq +ˇˇ2 +V ˚ ` +ˇˇP2 +mR0pnmpsq, Φq +ˇˇ2 +V ˚ +ı +ds ď K. +(4.57) + +42 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Proof. Thanks to the inequalities (4.52), (4.53) and (4.54) once more, we note that +E +ż T +0 +|A1cmpsq|2 +L2 ds “ E +ż T +0 +|∆cmpsq|2 +L2 ds ď KE +ż T +0 +|cmpsq|2 +H2 ds ď K, +and +E +ż T +0 +ˇˇP2 +mB1pumpsq, cmpsqq +ˇˇ2 +L2 ds ď KE +ż T +0 +|umpsq ¨ ∇cmpsq|2 +L2 ds +ď KE sup +0ďsďT +|umpsq|2 +L2 +ż T +0 +|∇cmpsq|2 +L2 +ď KE sup +0ďsďT +|umpsq|4 +L2 ` KE +ˆż T +0 +|∇cmpsq|2 +L2 ds +˙2 +ď K, +as well as +E +ż T +0 +ˇˇP2 +mR1pnmpsq, cmpsqq +ˇˇ2 +L2 ds ď KE +ż T +0 +|nmpsqfpcmpsqq|2 +L2 ds +ď K +sup +0ďsď|c0|L8 +f 2psqE +ż T +0 +|nmpsq|2 +L2 ď K, +and +E +ż T +0 +|A0umpsq|2 +V ˚ ds ď E +ż T +0 +|∇umpsq|2 +L2 ds ď K. +In the same way, +E +ż T +0 +ˇˇP2 +mB0pumpsq, umpsqq +ˇˇ2 +V ˚ ds ď KE +ż T +0 +|um|2 +L2 |∇umpsq|2 +L2 ds +ď KE sup +0ďsďT +|umpsq|2 +L2 +ż T +0 +|∇umpsq|2 +L2 ds +ď KE sup +0ďsďT +|umpsq|4 +L2 ` KE +ˆż T +0 +|∇umpsq|2 +L2 ds +˙2 +ď K, +and +E +ż T +0 +|R0pnmpsq, Φq|2 +V ˚ ds ď |Φ|2 +W 1,8 E +ż T +0 +|nmpsq|2 +L2 ds ď K. +Combining all these inequalities, we obtain the relation (4.57). +□ +4.2. Tightness result and passage to the limit. This subsection is devoted to the study of +the tightness of the approximations solutions and the proof of several convergences which +will enable us to pass to the limit and construct a weak probabilistic solution to our problem +via the martingale representation theorem given in [12, Theorem 8.2]. For this purpose, we +consider the following spaces: +Zn “ L2 +wp0, T; H1pOqq X L2p0, T; L2pOqq X Cpr0, Ts; H´3pOqq X Cpr0, Ts; L2 +wpOqq, +Zu “ L2 +wp0, T; V q X L2p0, T; Hq X Cpr0, Ts; V ˚q X Cpr0, Ts; Hwq, +Zc “ L2 +wp0, T; H2pOqq X L2p0, T; H1pOqq X Cpr0, Ts; L2pOqq X Cpr0, Ts; H1 +wpOqq, +Z “ Zn ˆ Zu ˆ Zc. +(4.58) +By making appropriate use of Lemma A.3, Corollary A.8, and Corollary A.9, we will now +show that the sequence of probability law Lm “ Lpnmq ˆ Lpumq ˆ Lpcmq is tight in Z. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +43 +Lemma 4.11. We suppose that the hypotheses of Proposition 4.3 hold. Then the family of +probability laws pLmqmPN is tight on the space Z. +Proof. We firstly prove that pLpnmqqm is tight on Zn. For any ε ą 0 we set Kε “ η0eK{ε ą η0 +where η0 ą 1 is given by Lemma 4.5. From the inequality (4.51), we deduce that +sup +m P +! +ω P Ω : +|nm|2 +L8p0,T;L2q ą Kε +) +ď sup +m P +" +ω P Ω : +η0 exp +ˆ +K +ż T +0 +|∇cmpsq|4 +L4 ds +˙ +ą Kε +* +ď sup +m +P +" +ω P Ω : +K +ż T +0 +|∇cmpsq|4 +L4 ds ą ln +ˆKε +η0 +˙* +. +Using the Markov inequality and inequality (4.55), we infer that +sup +m P +! +ω P Ω : +|nm|2 +L8p0,T;L2q ą Kε +) +ď +1 +ln +´ +Kε +η0 +¯E +ˆ +K +ż T +0 +|∇cmpsq|4 +L4 ds +˙ +ď ε +KE +ˆ +K +ż T +0 +|∇cmpsq|4 +L4 ds +˙ +ď ε. +Similarly, we can also prove that +sup +m P +! +ω P Ω : +|nm|2 +L2p0,T;H1q ą Kε +) +ď sup +m P +" +ω P Ω : +η0 exp +ˆ +K +ż T +0 +|∇cmpsq|4 +L4 ds +˙ +ą Kε +* +ď ε. +Thanks to inequality (4.56) we derive that +sup +m P +" +ω P Ω : +|nm|2 +C1{2pr0,Ts;H´3q ą K +ε +* +ď ε +KE |nm|2 +C1{2pr0,Ts;H´3q ď ε. +Since these three last inequalities hold, we can apply Lemma A.3 and conclude that the law +of nm form a family of probability measures which is tight on Zn. +Secondly, we will prove that the laws of um and cm are tight on Zu ˆ Zc. +From +inequalities (4.52) and (4.54), we obtain the first two conditions of Corollaries A.8 and A.9 +for um and cm respectively. Hence, it is sufficient to prove that the sequences pumqm and +pcmq satisfy the Aldous condition in the spaces V ˚ and L2pOq respectively. +Let θ ą 0 +pτℓqℓě1 be a sequence of stopping times such that 0 ď τℓ ď T. From the second equation of +system (4.2) we have +cmpτℓ ` θq ´ cmpτℓq “ ξ +ż τℓ`θ +τℓ +A1cmpsqds ´ +ż τℓ`θ +τℓ +P2 +mB1pumpsq, cmpsqqds +` +ż τℓ`θ +τℓ +P2 +mR1pnmpsq, cmpsqqds ` γ +ż τℓ`θ +τℓ +P2 +mφpcmpsqqdβs. +(4.59) +By the Fubini theorem, the H¨older inequality and inequality (4.57), we have the following +estimates +E +ˇˇˇˇξ +ż τℓ`θ +τℓ +A1cmpsqds +ˇˇˇˇ +2 +L2 +ď ξ2θ1{2E +ż τℓ`θ +τℓ +|A1cmpsq|2 +L2 ds +ď ξ2θ1{2E +ż T +0 +|A1cmpsq|2 +L2 ds ď Kθ1{2, + +44 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +E +ˇˇˇˇ +ż τℓ`θ +τℓ +P2 +mB1pumpsq, cmpsqqds +ˇˇˇˇ +2 +L2 +ds ď θ1{2E +ż τℓ`θ +τℓ +ˇˇP2 +mB1pumpsq, cmpsqq +ˇˇ2 +L2 ds +ď θ1{2E +ż T +0 +ˇˇP2 +mB1pumpsq, cmpsqq +ˇˇ2 +L2 ds ď Kθ1{2, +and +E +ˇˇˇˇ +ż τℓ`θ +τℓ +P2 +mR1pnmpsq, cmpsqqds +ˇˇˇˇ +2 +L2 +ds ď θ1{2E +ż τℓ`θ +τℓ +ˇˇP2 +mR1pnmpsq, cmpsqq +ˇˇ2 +L2 ds +ď θ1{2E +ż T +0 +ˇˇP2 +mR1pnmpsq, cmpsqq +ˇˇ2 +L2 ds ď Kθ1{2. +By the Itˆo isometry, we note that +E +ˇˇˇˇγ +ż τℓ`θ +τℓ +P2 +mφpcmpsqqdβs +ˇˇˇˇ +2 +L2 +ď γ2E +ż τℓ`θ +τℓ +|φpcmpsqq|2 +L2pR2,L2q +ď γ2 +2ÿ +k“1 +|σk|2 +L2 E +ż τℓ`θ +τℓ +|∇cmpsq|2 +L2 ds +ď KθE sup +0ďsďT +|∇cmpsq|2 +L2 ď Kθ. +Combining these inequalities, we infer from equality (4.59) that the condition (A.5) is satisfies +for pcmqmě1 in L2pOq. Hence by Lemma A.7 the sequence pcmqmě1 satisfies the Aldous +condition in the space L2pOq. +Now we will consider the sequence pumqmě1. We first observe that from the first equation +of system (4.2) we infer that +umpτℓ ` θq ´ umpτℓq “ ´η +ż τℓ`θ +τℓ +A0umpsqds ´ +ż τℓ`θ +τℓ +P1 +mB0pumpsq, umpsqqds +` +ż τℓ`θ +τℓ +P1 +mR0pnmpsq, Φqds ` +ż τℓ`θ +τℓ +P1 +mgpumpsq, cmpsqqdWs. +(4.60) +Thanks to the H¨older inequality and (4.57), we have the following estimates +E +ˇˇˇˇη +ż τℓ`θ +τℓ +A0umpsqds +ˇˇˇˇ +2 +V ˚ +ď η2θ1{2E +ż τℓ`θ +τℓ +|A0umpsq|2 +V ˚ ds +ď η2θ1{2E +ż T +0 +|A0umpsq|2 +V ˚ ds ď Kθ1{2, +and +E +ˇˇˇˇ +ż τℓ`θ +τℓ +P2 +mB0pumpsq, umpsqqds +ˇˇˇˇ +2 +V ˚ +ds ď θ1{2E +ż τℓ`θ +τℓ +ˇˇP2 +mB1pumpsq, umpsqq +ˇˇ2 +V ˚ ds +ď θ1{2E +ż T +0 +ˇˇP2 +mB1pumpsq, umpsqq +ˇˇ2 +V ˚ ds ď Kθ1{2, + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +45 +as well as +E +ˇˇˇˇ +ż τℓ`θ +τℓ +P2 +mR0pnmpsq, Φqds +ˇˇˇˇ +2 +V ˚ +ds ď θ1{2E +ż τℓ`θ +τℓ +ˇˇP2 +mR0pnmpsq, Φq +ˇˇ2 +V ˚ ds +ď θ1{2E +ż T +0 +ˇˇP2 +mR0pnmpsq, Φq +ˇˇ2 +V ˚ ds ď Kθ1{2. +Thanks to the Itˆo isometry and the assumption on g we obtain +E +ˇˇˇˇ +ż τℓ`θ +τℓ +P1 +mgpumpsq, cmpsqqdWs +ˇˇˇˇ +2 +V ˚ +ď KE +ˇˇˇˇ +ż τℓ`θ +τℓ +P1 +mgpumpsq, cmpsqqdWs +ˇˇˇˇ +2 +L2 +ď KE +ż τℓ`θ +τℓ +ˇˇP1 +mgpumpsq, cmpsqq +ˇˇ2 +L2pU,Hq ds +ď KE +ż τℓ`θ +τℓ +p1 ` |pumpsq, cmpsqq|2 +Hqds +ď K +ˆ +1 ` E sup +0ďsďT +|pumpsq, cmpsqq|2 +H +˙ +θ ď Kθ. +From these inequalities and equality (4.60), we can conclude by Lemma A.7 that the sequence +pumqmě1 satisfies the Aldous condition in the space V ˚. +Hence, by applying Corollary +A.8 and Corollary A.9, we see that the laws of cm and um are tight on Zc and Zu, +respectively. +□ +Since pLmqm is tight on Z, invoking [28, Corollary 2, Appendix B] (see also [7, Theorem +4.13]) there exists a probability space +pΩ1, F1, P1q, +and a subsequence of random vectors p¯umk, ¯cmk, ¯nmkq with values in Z such that +i): p¯umk, ¯cmk, ¯nmkq have the same probability distributions as pumk, cmk, nmkq, +ii): p¯umk, ¯cmk, ¯nmkq converges in the topology of Z to a random element pu, c, nq P Z +with probability 1 on pΩ1, F1, P1q as k Ñ 8. +To simplify the notation, we will simply denote these sequences by pum, cm, nmqmě1 and +p¯um, ¯cm, ¯nmqmě1, respectively. +Next, from the definition of the space Z, we deduce that P1-a.s., +¯um Ñ u in L2 +wp0, T; V q X L2p0, T; Hq X Cpr0, Ts; V ˚q X Cpr0, Ts; Hwq, +¯cm Ñ c in L2 +wp0, T; H2pOqq X L2p0, T; H1pOqq X Cpr0, Ts; L2pOqq X Cpr0, Ts; H1 +wpOqq, +¯nm Ñ n in L2 +wp0, T; H1pOqq X L2p0, T; L2pOqq X Cpr0, Ts; H´3pOqq X Cpr0, Ts; L2 +wpOqq. +(4.61) +According to [40, Theorem 1.10.4 and Addendum 1.10.5], +a family of measurable map +Ψm : Ω1 Ñ Ω can be constructed such that together with the new probability space pΩ1, F1, P1q +satisfy the property +¯umpω1q “ um ˝ Ψmpω1q, +¯nmpω1q “ nm ˝ Ψmpω1q, +¯cmpω1q “ cm ˝ Ψmpω1q, and P “ P1 ˝ Ψ´1 +m , +(4.62) + +46 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +for all ω1 P Ω1. Taking into account the fact that inequality (4.10) holds, we can derive that +for almost every pt, ω1q P r0, Ts ˆ Ω1, +(4.63) +ˇˇ¯cmpt, ω1q +ˇˇ +L8 “ +ˇˇcmpt, Ψmpω1qq +ˇˇ +L8 ď |c0|L8 , +for all m ě 1. +Since the laws of pum, cm, nmq and p¯um, ¯cm, ¯nmq are equal in the space Zu ˆ Zc ˆ Zn, we +have the estimates (4.52), (4.54) and +(4.64) +E1 +ż T +0 +|¯cmpsq|2 +H2 ds ď K, +E1 +ż T +0 +|∇¯umpsq|2 +L2 ds ď K, +as well as +(4.65) +E1 +ż T +0 +|¯nmpsq|2 +L2 ds ď K. +From (4.64) and (4.65) and the Banach-Alaoglu Theorem, we conclude that, there exists a sub- +sequence of p¯umqmě1, p¯cmqmě1, and p¯nmqmě1 weakly convergent in L2pΩ1, F1, P1; L2p0, T; V qq, +L2pΩ1, F1, P1; L2p0, T; H2pOqqq, and L2pΩ1, F1, P1; L2p0, T; L2pOqqq respectively. i.e. +u P L2pΩ1, F1, P1; L2p0, T; V qq, +c P L2pΩ1, F1, P1; L2p0, T; H2pOqqq, +n P L2pΩ1, F1, P1; L2p0, T; L2pOqqq. +(4.66) +On the other hand, from estimates (4.52), (4.53) and (4.54) of Corollary 4.8, and the equalities +given by (4.62), we get for any p ě 1, +E1 sup +0ďsďT +|¯umpsq|2p +L2 ` E1 +ˆż T +0 +|∇¯umpsq|2 +L2 ds +˙p +ď K, +(4.67) +E1 +ˆż T +0 +|¯nmpsq|2 +L2 ds +˙p +ď K, +(4.68) +E1 sup +0ďsďT +|¯cmpsq|p +H1 ` E1 +ˆż T +0 +|¯cmpsq|2 +H2 ds +˙p +ď K. +(4.69) +Then, invoking the Fatou lemma, we infer that for p ě 2, we have +(4.70) +E1 sup +0ďsďT +|upsq|p +L2 ă 8, +E1 sup +0ďsďT +|cpsq|p +H1 ă 8. +and +E1 +ˆż T +0 +|∇upsq|2 +L2 ds +˙p +ă 8, E1 +ˆż T +0 +|npsq|2 +L2 ds +˙p +ă 8, E1 +ˆż T +0 +|cpsq|2 +H2 ds +˙p +ă 8. +(4.71) +Now, we prove three lemmata which show how convergence in Z given by (4.61) will be +used for the convergence of the deterministic terms appearing in the Galerkin approximation. +We start by noting that since nm +0 , cm +0 +and um +0 +have been chosen such that (4.1) holds, we +can derive that for all ψ P H3pOq and pψ, vq P L2pOq ˆ V , +(4.72) +lim +mÝÑ8pnm +0 , ψq “ pn0, ψq, +lim +mÝÑ8pcm +0 , ψq “ pc0, ψq, and +lim +mÝÑ8pum +0 , vq “ pu0, vq. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +47 +Lemma 4.12. For any r, t P r0, Ts with r ď t and ψ P H3pOq, the following convergences +hold P1-a.s. +lim +mÝÑ8p¯nmptq, ψq “ pnptq, ψq, +lim +mÝÑ8 +ż t +r +pA1¯nmpsq, ψqds “ +ż t +r +pA1npsq, ψqds +lim +mÝÑ8 +ż t +r +pP2 +mB1p¯umpsq, ¯nmpsqq, ψqds “ +ż t +r +pB1pupsq, npsqq, ψqds, +lim +mÝÑ8 +ż t +r +pP2 +mR2p¯nmpsq, ¯cmpsqq, ψqds “ +ż t +r +pR2pnpsq, cpsqq, ψqds. +(4.73) +Proof. Let ψ P H3pOq and t P r0, Ts be arbitrary but fixed. By the H¨older inequality we have +|p¯nmptq, ψq ´ pnptq, ψq| ď |¯nmptq ´ nptq|H´3 |ψ|H3 +ď |¯nm ´ n|Cpr0,Ts;H´3q |ψ|H3 , +(4.74) +which along with (4.61) implies the first convergence in (4.73). +Now, we also fix r P r0, Ts such that r ď t. By an integration-by-parts and the H¨older +inequality we note that +ˇˇˇˇ +ż t +r +pA1¯nmpsq, ψqds ´ +ż t +r +pA1npsq, ψqds +ˇˇˇˇ dt ď +ż T +0 +|pA1¯nmpsq ´ A1npsq, ψq| ds +ď +ż T +0 +|p¯nmpsq ´ npsq, A1ψq| ds +(4.75) +ď T +sup +0ďsďT +|p¯nmpsq ´ npsq, A1ψq| . +From the convergence (4.61) we infer that ¯nm Ñ n in Cpr0, Ts; L2 +wpOq, P1-a.s. This means +that sup0ďsďT |p¯nmpsq ´ npsq, ϕq| tends to zero for all ϕ P L2pOq as m goes to infinity with +probability one. We plug ϕ “ A1ψ and pass to the limit in (4.75) and derive the second +convergence of (4.73). We have for all ω P Ω, +ˇˇˇˇ +ż t +r +pP2 +mB1p¯umpsq, ¯nmpsqq, ψqds ´ +ż t +r +pB1pupsq, npsqq, ψqds +ˇˇˇˇ +ď +ż T +0 +ˇˇpB1p¯umpsq, ¯nmpsqq, P2 +mψ ´ ψq +ˇˇ ` +ż T +0 +|pB1p¯umpsq, ¯nmpsqq ´ B1pupsq, npsqq, ψq| ds +Since ¯um Ñ u in L2p0, T; Hq, and ¯nm Ñ n in L2p0, T; L2pOqq P1-a.s., by integration-by-parts, +we derive that +ż T +0 +ˇˇpB1p¯umpsq, ¯nmpsqq, P2 +mψ ´ ψq +ˇˇ ds +ď +ż T +0 +ˇˇp¯nmpsq¯umpsq, ∇pP2 +mψ ´ ψqq +ˇˇ +ď +ˇˇ∇pP2 +mψ ´ ψq +ˇˇ +L8 +ż T +0 +|¯nmpsq|L2 |¯umpsq|L2 ds +ď +ˇˇP2 +mψ ´ ψ +ˇˇ +H3 +ˆż T +0 +|¯umpsq|2 +L2 ds +˙1{2 ˆż T +0 +|¯nmpsq|2 +L2 ds +˙1{2 +ď K +ˇˇP2 +mψ ´ ψ +ˇˇ +H3 . + +48 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +By using an integration-by-parts and the fact that ∇ ¨ u “ 0 we get +ż T +0 +|pB1p¯umpsq, ¯nmpsqq ´ B1pupsq, npsqq, ψq, ψq| ds +ď +ż T +0 +|pp¯umpsq ´ upsqq∇¯nmpsq, ψq| ds ` +ż T +0 +|pupsq∇p¯nmpsq ´ npsqq, ψq| ds +ď +ż T +0 +|p¯nmpsq, p¯umpsq ´ upsqq ¨ ∇ψq| ds ` +ż T +0 +|pp¯nmpsq ´ npsqq, upsq ¨ ∇ψq| ds +ď |ψ|L8 +ż T +0 +|¯umpsq ´ upsq|L2 |¯nmpsq|L2 ds ` |∇ψ|L8 +ż T +0 +|¯nmpsq ´ npsq|L2 |upsq|L2 ds. +Using the fact that |∇ψ|L8 ď |ψ|H3, we infer from the two last inequalities that +ˇˇˇˇ +ż t +0 +pP2 +mB1p¯umpsq, ¯nmpsqq, ψqds ´ +ż t +0 +pB1pupsq, npsqq, ψqds +ˇˇˇˇ +ď T |ψ|H3 +ˆż T +0 +|¯umpsq ´ upsq|2 +L2 ds +˙1{2 ˆż T +0 +|¯nmpsq|2 +L2 ds +˙1{2 +` T |ψ|H3 +ˆż T +0 +|¯nmpsq ´ npsq|2 +L2 ds +˙1{2 ˆż T +0 +|upsq|2 +L2 ds +˙1{2 +` K +ˇˇP2 +mψ ´ ψ +ˇˇ +H3 +(4.76) +ď K +ˆż T +0 +|¯umpsq ´ upsq|2 +L2 ds +˙1{2 +` K +ˆż T +0 +|¯nmpsq ´ npsq|2 +L2 ds +˙1{2 ˆż T +0 +|upsq|2 +L2 ds +˙1{2 +` K +ˇˇP2 +mψ ´ ψ +ˇˇ +H3 , +which upon letting n Ñ 8, implies the third convergence in (4.73). +Similarly, we have +ˇˇˇˇ +ż t +r +pP2 +mR2p¯nmpsq, ¯cmpsqq, ψqds ´ +ż t +r +pR2pnpsq, cpsqq, ψqds +ˇˇˇˇ +ď +ż T +0 +|pR2p¯nmpsq, ¯cmpsqq ´ R2pnpsq, cpsqq, ψq| ds +(4.77) +` +ż T +0 +ˇˇpR2p¯nmpsq, ¯cmpsqq, P2 +mψ ´ ψq +ˇˇ ds. +Since p¯cm, ¯nmq Ñ pc, nq in Zc ˆ Zn, we see that P1-a.s, +ż T +0 +ˇˇpR2p¯nmpsq, ¯cmpsqq, P2 +mψ ´ ψq +ˇˇ ds +ď +ˇˇ∇pP2 +mψ ´ ψq +ˇˇ +L8 +ż T +0 +|¯nmpsq|L2 |∇¯cmpsq|L2 ds +ď K +ˇˇP2 +mψ ´ ψ +ˇˇ +H3 . + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +49 +On the other hand, we obtain +ż T +0 +|pR2p¯nmpsq, ¯cmpsqq ´ R2pnpsq, cpsqq, ψq, ψq| ds +ď +ż T +0 +|pp¯nmpsq ´ npsqq∇¯cmpsq, ∇ψq| ds ` +ż T +0 +|pnpsq∇p¯cmpsq ´ cpsqq, ∇ψq| ds +ď |∇ψ|L8 +ż T +0 +|¯nmpsq ´ npsq|L2 |∇¯cmpsq|L2 ds +` |∇ψ|L8 +ż T +0 +|∇p¯cmpsq ´ cpsqq|L2 |npsq|L2 ds +ď K +ˆż T +0 +|¯nmpsq ´ npsq|2 +L2 ds +˙1{2 +` K +ˆż T +0 +|∇p¯cmpsq ´ cpsqq|2 +L2 ds +˙1{2 ˆż T +0 +|npsq|2 +L2 ds +˙1{2 +, +which along with (4.61) implies the fourth convergence in (4.73). +□ +Lemma 4.13. For any r, t P r0, Ts with r ď t and ψ P H2pOq, the following convergences +hold P1-a.s. +lim +mÝÑ8p¯cmptq, ψq “ pcptq, ψq, +lim +mÝÑ8 +ż t +r +pA1¯cmpsq, ψqds “ +ż t +r +pA1cpsq, ψqds, +lim +mÝÑ8 +ż t +r +pP2 +mB1p¯umpsq, ¯cmpsqq, ψqds “ +ż t +r +pB1pupsq, cpsqq, ψqds, +lim +mÝÑ8 +ż t +r +pP2 +mR1p¯nmpsq, ¯cmpsqq, ψqds “ +ż t +r +pR1pnpsq, cpsqq, ψqds. +(4.78) +Proof. Since ¯cm Ñ c in Cpr0, Ts; L2pOqq, P1-a.s., the first convergence is done exactly using +a similarly inequality as (4.74). By an integration by part and the H¨older inequality we note +that +ˇˇˇˇ +ż t +r +pA1¯cmpsq, ψqds ´ +ż t +r +pA1cpsq, ψqds +ˇˇˇˇ ď +ż T +0 +|pA1¯cmpsq ´ A1cpsq, ψq| ds +ď +ż T +0 +|p∇p¯cmpsq ´ cpsqq, ∇ψq| ds +ď T 1{2 |ψ|H1 +ˆż T +0 +|¯cmpsq ´ cpsq|2 +H1 ds +˙1{2 +, +which altogether with (4.61) implies the second convergence in (4.78). + +50 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Next, using the Sobolev embedding H1pOq ãÑ L4pOq, we get +ˇˇˇˇ +ż t +r +pP2 +mB1p¯umpsq, ¯cmpsqq, ψqds ´ +ż t +r +pB1pupsq, cpsqq, ψqds +ˇˇˇˇ +ď +ż T +0 +|pB1p¯umpsq, ¯cmpsqq ´ B1pupsq, cpsqq, ψq, ψq| ds ` +ż T +0 +ˇˇpB1p¯umpsq, ¯cmpsqq, P2 +mψ ´ ψq +ˇˇ ds +ď +ż T +0 +|pp¯umpsq ´ upsqq∇¯cmpsq, ψq| ds ` +ż T +0 +|pupsq∇p¯cmpsq ´ cpsqq, ψq| ds +` T 1{2 ˇˇP2 +mψ ´ ψ +ˇˇ +L2 +ˆż T +0 +|B1p¯umpsq, ¯cmpsqq|2 +L2 ds +˙1{2 +ď |ψ|L4 +ż T +0 +|¯umpsq ´ upsq|L2 |∇¯cmpsq|L4 ds ` |ψ|L4 +ż T +0 +|∇p¯cmpsq ´ cpsqq|L2 |upsq|L4 ds +` K +ˇˇP2 +mψ ´ ψ +ˇˇ +L2 +ď T |ψ|H1 +ż T +0 +|¯umpsq ´ upsq|L2 |¯cmpsq|H2 ds +` T |ψ|H1 +ż T +0 +|∇p¯cmpsq ´ cpsqq|L2 |∇upsq|L2 ds ` K +ˇˇP2 +mψ ´ ψ +ˇˇ +L2 . +Since the convergence (4.61) holds, we arrive at +ˇˇˇˇ +ż t +r +pP2 +mB1p¯umpsq, ¯cmpsqq, ψqds ´ +ż t +r +pB1pupsq, cpsqq, ψqds +ˇˇˇˇ +ď T |ψ|H1 +ˆż T +0 +|¯umpsq ´ upsq|2 +L2 ds +˙1{2 ˆż T +0 +|¯cmpsq|2 +H2 ds +˙ 1 +2 +` T |ψ|H1 +ˆż T +0 +|¯cmpsq ´ cpsq|2 +H1 ds +˙ 1 +2 ˆż T +0 +|∇upsq|2 +L2 ds +˙ 1 +2 +` K +ˇˇP2 +mψ ´ ψ +ˇˇ +L2 , +which along with (4.61) implies the third convergence in (4.78). +Now we prove the last convergence. To this purpose, we note that +ˇˇˇˇ +ż t +r +pP2 +mR1p¯nmpsq, ¯cmpsqq, ψqds ´ +ż t +r +pR1pnpsq, cpsqq, ψqds +ˇˇˇˇ +ď +ż T +0 +|pR1p¯nmpsq, ¯cmpsqq ´ R1pnpsq, cpsqq, ψq| ds +` +ż T +0 +ˇˇpR1p¯nmpsq, ¯cmpsqq, P2 +mψ ´ ψq +ˇˇ ds +(4.79) +ď +ż T +0 +|pp¯nmpsq ´ npsqqfp¯cmpsqq, ψq| ds +` +ż T +0 +|npsqpfp¯cmpsqq ´ fpcpsqqq, ψq| ds ` K +ˇˇP2 +mψ ´ ψ +ˇˇ +L2 . + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +51 +Using (4.63), we derive that +ż T +0 +|pp¯nmpsq ´ npsqqfp¯cmpsqq, ψq| ds +ď |ψ|L8 +ż T +0 +ż +O +|¯nmpsq ´ npsq| |fp¯cmpsqq| dxds +ď T 1{2 |O|1{2 |ψ|H2 +sup +0ďsď|c0|L8 +fpsq +ˆż T +0 +|¯nmpsq ´ npsq|2 +L2 ds +˙1{2 +. +In a similar way, we see that +ż T +0 +|nps, xqpfp¯cmps, xqq ´ fpcps, xqqq, ψq| dsdx +ď |ψ|H2 +ż T +0 +ż +O +|nps, xqfp¯cmps, xqq ´ nps, xqfpcps, xqq| dxds. +(4.80) +Since the strong convergence ¯cm Ñ c in L2p0, T; H1pOqq, P1-a.s., holds, we derive that up +to a subsequence +¯cm Ñ c +dt b dx-a.e +Owing to the fact that f is continuous, we infer that P1-a.s., +nfp¯cmq Ñ nfpcq +a.e in +ˆ p0, Tq ˆ O. +We also note that P-a.s., tnfp¯cmqumě1 is uniformly integrable over p0, Tq ˆ O. Indeed, we +have +ż +p0,TqˆO +|nps, xqfp¯cmps, xqq|2 dxdsdP ď +sup +0ďsď|c0|L8 +f 2psq +ż T +0 +ż +O +|nps, xq|2 dxds +ď K +ż T +0 +|npsq|2 +L2 ds. +Therefore, by the Vitali Convergence Theorem, we derive that P1-a.s., the right answer of +the inequality (4.80) tends to zero as m tends to 8. Owing to this result, we can pass to +the limit in the inequality (4.79) and obtain the last convergence of (4.78). +□ +Next we prove the following convergences. +Lemma 4.14. For any r, t P r0, Ts with r ď t and v P V , the following convergences hold +P1-a.s. +lim +mÝÑ8p¯umptq, vq “ puptq, vq, +lim +mÝÑ8 +ż t +r +pA0¯umpsq, vqds “ +ż t +r +pA0upsq, vqds, +lim +mÝÑ8 +ż t +r +pP1 +mB0p¯umpsq, ¯umpsqq, vqds “ +ż t +r +pB0pupsq, upsqq, vqds, +lim +mÝÑ8 +ż t +r +pP1 +mR0p¯nmpsq, Φq, vqds “ +ż t +r +pR0pnpsq, Φq, vqds. +(4.81) +Proof. The proof is similar to the proof of Lemma 4.12 and Lemma 4.13. +□ + +52 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +In what follows, we will combine the convergence result from Lemma 4.12, Lemma 4.13 +and Lemma 4.14 as well as martingale representation theorem to construct a probabilistic +weak solution to the problem (1.2). +In order to simplify the notation, we define on the +probability space pΩ1, F1, P1q the processes N 1 +m, N 2 +m, and N 3 +m, respectively, by for t P r0, Ts, +N 1 +mptq :“ ´¯umptq ´ +ż t +0 +rηA0¯umpsq ` P1 +mB0p¯umpsq, ¯umpsqqsds ` um +0 ` +ż t +0 +P1 +mR0p¯nmpsq, Φqds, +N 2 +mptq :“ ´¯cmptq ´ +ż t +0 +rξA1¯cmpsq ` P2 +mB1p¯umpsq, ¯cmpsqqsds ` cm +0 ´ +ż t +0 +P2 +mR1p¯nmpsq, ¯cmpsqqds, +and +N 3 +mptq :“ ´¯nmptq ´ +ż t +0 +rδA1¯nmpsq ` P2 +mB1p¯umpsq, ¯nmpsqqsds ` nm +0 ´ +ż t +0 +P2 +mR2p¯nmpsq, ¯cmpsqqds. +Lemma 4.15. For all m P N and for any t P r0, Ts, we have +(4.82) +N 3 +mptq “ 0, +P1-a.s. +Proof. Let m P N and t P r0, Ts be arbitrary but fixed. On the probability space pΩ, F, Pq, +we define the processes M3 +mptq by +M3 +mptq :“ ´nmptq ´ +ż t +0 +rδA1nmpsq ` P2 +mB1pumpsq, nmpsqqsds ` nm +0 ´ +ż t +0 +P2 +mR2pnmpsq, cmpsqqds. +We also define the following subsets of Ω and Ω1 +AN +mptq :“ +␣ +ω1 P Ω1 : N 3 +mptq “ 0 +( +and AM +m ptq :“ +␣ +ω P Ω : M3 +mptq “ 0 +( +. +We note that, since the last equation of (4.2) holds, PpAM +m ptqq “ 1. Furthermore, by (4.62), +we derive that for all ω1 P Ω, N 3 +mpt, ω1q “ M3 +mpt, Ψmpω1qq and therefore we observe that +AN +mptq “ Ψ´1 +m pAM +m ptqq. Invoking (4.62) once more, we deduce that +P1pAN +mptqq “ P1pΨ´1 +m pAM +m ptqqq “ PpAM +m ptqq “ 1, +which completes the proof of Lemma 4.15. +□ +Using the convergences (4.72) and (4.73) as well as Lemma 4.15 we see that for all +t P r0, Ts, P1-a.s. +(4.83) +nptq ` +ż t +0 +rδA1npsq ` B1pupsq, npsqqsds “ n0 ´ +ż t +0 +R2pnpsq, cpsqqds, +in H´3pOq. +Now, on the probability space pΩ1, F1, P1q we define a the Hm ˆ Hm-valued processes Nm +by Nmptq “ pN 1 +mptq, N 2 +mptqq for all m ě 1 and t P r0, Ts. Since +(4.84) +Hm ˆ Hm Ă H ˆ L2pOq ãÑ V ˚ ˆ H´2pOq, +the process Nm can be seen as a V ˚ ˆ H´2pOq-valued process. +Next, we collect the necessary ingredients for the application of the martingale representation +theorem from [12, Theorem 8.2]. +To this aim, we consider the following Gelfand triple +V ãÑ H ãÑ V ˚ and H2pOq ãÑ L2pOq ãÑ H´2pOq. Let i1 : V ãÑ H be the usual embedding +and i1˚ its Hilbert-space-adjoint such that pix, yq “ px, i1˚yqV +for all x P V +and y P H. In +a very similar way, we denote the usual embedding H2pOq ãÑ L2pOq by i2 and by i˚2 its + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +53 +Hilbert-space-adjoint. We define the embedding i : V ˆ H2pOq ãÑ H ˆ L2pOq and its adjoint +i˚ : H ˆ L2pOq ÝÑ V ˆ H2pOq respectively by +i “ +ˆ +i1 +0 +0 +i2 +˙ +, +i˚ “ +ˆ +i1˚ +0 +0 +i2˚ +˙ +. +Further, +we set +L1 “ pi1˚q1 : V ˚ ÝÑ H +as the dual +operator +of i1˚ +such +that +for +all +x P H +and +y P V ˚, +pLy, xq “ xy, xy. +Similarly, +the +dual +operator +of +i2˚ +will +be +denoted +by +L2 : H´2pOq ÝÑ L2pOq. +We +then +define +the +following +dual +operator +L :“ pi˚q1 : V ˚ ˆ H´2pOq ÝÑ H ˆ L2pOq by +L “ +ˆ +L1 +0 +0 +L2 +˙ +. +On the space Hm ˆ Hm, we define a mapping Gm by +Gmpv, ψq “ +ˆ +L1P1 +mgpv, ψq +0 +0 +L2P2 +mφpψq +˙ +, +pv, ψq P Hm ˆ Hm. +Here pP1 +mgpv, ψq, P2 +mφpψqq “ pP1 +mgpv, ψq, P2 +mφpψqq is seen as an element of V ˚ ˆ H´2pOq +owing to the inclusion (4.84). +In the following lemma, we prove the martingale property of the process LNm. +Lemma 4.16. For each m ě 1, the process LNm is an H ˆ L2pOq-valued continuous square +integrable martingale with respect to the filtration +F +1m “ +␣ +σ +` +σ pp¯umpsq, ¯cmpsq, ¯nmpsqq; s ď tq Y N 1˘( +tPr0,Ts , +where N 1 is the set of null sets of F1. The quadratic variation of LNm is given by +(4.85) +xxLNmyyt “ +ż t +0 +Gmp¯umpsq, ¯cmpsqqGmp¯umpsq, ¯cmpsqq˚ds, +where Gmp¯um, ¯cmq˚ : H ˆ L2pOq Ñ U ˆ R2 is the adjoint of the operator Gmp¯um, ¯cmq and +is given by +Gmp¯um, ¯cmq˚v “ +˜ 8 +ÿ +k“1 +pP1 +mgp¯um, ¯cmqek, i1˚wqek, +2ÿ +k“1 +pP2 +mφp¯cmqgk, i2˚ψqgk +¸ +, +for all v “ pw, ψq P H ˆ L2pOq. +Proof. For any m ě 1 we define the V ˚ ˆ H´2pOq-valued processes Mm by +Mmptq “ pM1 +mptq, M2 +mptqq, +t P r0, Ts, +where +M1 +mptq :“ ´umptq ´ +ż t +0 +rηA0umpsq ` P1 +mB0pumpsq, umpsqqsds ` um +0 ` +ż t +0 +P1 +mR0pnmpsq, Φqds, +M2 +mptq :“ ´cmptq ´ +ż t +0 +rξA1cmpsq ` P2 +mB1pumpsq, cmpsqqsds ` cm +0 ´ +ż t +0 +P2 +mR1pnmpsq, cmpsqqds. +Let us set Ws :“ pWs, βsq. Then, since pum, cm, nmq is a solution of the finite dimensional +system (4.2), we deduce that LMm can be represented as +LMmptq “ +ż t +0 +Gmpumpsq, cmpsqqdWs, +P-a.s. +for all t P r0.Ts. + +54 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Using the continuity property of the operators L1 and L2 as well as Corollary 4.8, the +estimate +E +ż T +0 +|Gmpumpsq, cmpsqq|2 +L2pUˆR2,HˆL2q ds +ď KE +ż T +0 +ˇˇP1 +mgpumpsq, cmpsqq +ˇˇ2 +L2pU,Hq ds ` KE +ż T +0 +ˇˇP2 +mφpcmpsqq +ˇˇ2 +L2pR2,L2q ds +ď KE +ż T +0 +p1 ` |pumpsq, cmpsqq|2 +Hqds ` γ2 +2ÿ +k“1 +|σk|2 +L2 E +ż T +0 +|∇cmpsq|2 +L2 ds +ď K +ˆ +1 ` E sup +0ďsďT +|pumpsq, cmpsqq|2 +H +˙ +` KE sup +0ďsďT +|∇cmpsq|2 +L2 ă 8, +yields that +Mm +is a square integrable +continuous +martingale +over the probability +space +pΩ, F, pFtqtPr0,Ts, Pq. Moreover, from the definition of Mm we derive that for each t P r0.Ts, +Mmptq is measurable with respect to the σ-field +Fm “ tσ pσ ppumpsq, cmpsq, nmpsqq; s ď tq Y NqutPr0,Ts , +where N is the set of null sets of F. Hence, invoking [12, Theorem 4.27] we infer that +Mm is a Fm-martingale with quadratic variation +xxMmyyt “ +ż t +0 +Gmpumpsq, cmpsqqGmpumpsq, cmpsqq˚ds. +This means that for all s, t P r0, Ts, s ď t, all vi “ pwi, ψiq P H ˆ L2pOq, i “ 1, 2, and all +bounded and continuous real-valued functions h “ ph1, h2, h3q on Cpr0, Ts; H ˆL2pOqˆL2pOqq, +we have +E +” +pLMmptq ´ LMmpsq, v1qHˆL2pOq h1pum|r0,ssqh2pcm|r0,ssqh3pnm|r0,ssq +ı +“ 0, +and +E +”´ +pLMmptq, v1qHˆL2pOq pLMmptq, v2qHˆL2pOq ´ pLMmpsq, v1qHˆL2pOq pLMmpsq, v2qHˆL2pOq +´ +ż t +0 +pGmpumpsq, cmpsqq˚v1, Gmpumpsq, cmpsqq˚v2qUˆR2 ds +˙ +ˆ +ˆh1pum|r0,ssqh2pcm|r0,ssqh3pnm|r0,ssq +‰ +“ 0. +Since pum, cm, nmq and p¯um, ¯cm, ¯nmq have the same laws on Cpr0, Ts; Hmq, we deduce from +these two last equalities that +(4.86) +E1 ” +pLNmptq ´ LNmpsq, v1qHˆL2pOq h1p¯um|r0,ssqh2p¯cm|r0,ssqh3p¯nm|r0,ssq +ı +“ 0, +and +E1 ”´ +pLNmptq, v1qHˆL2pOq pLNmptq, v2qHˆL2pOq +´ pLNmpsq, v1qHˆL2pOq pLNmpsq, v2qHˆL2pOq +´ +ż t +0 +pGmp¯umpsq, ¯cmpsqq˚v1, Gmp¯umpsq, ¯cmpsqq˚v2qUˆR2 ds +˙ +ˆ +(4.87) +ˆh1p¯um|r0,ssqh2p¯cm|r0,ssqh3p¯nm|r0,ssq +‰ +“ 0, +for all s, t P r0, Ts, s ď t, all vi “ pwi, ψiq P H ˆL2pOq, i “ 1, 2, and all (real-valued) function +hi, i “ 1, 2, 3 bounded and continuous on Cpr0, Ts; Hmq, Cpr0, Ts; Hmq, and Cpr0, Ts; Hmq + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +55 +respectively, This implies that LNm is a continuous square integrable martingale with respect +to F +1m and the quadratic variation is given as claimed by equality (4.85). +□ +On the new probability space pΩ1, F1, P1q, we consider the V ˚ ˆH´2pOq-valued continuous +process N defined by Nptq “ pN 1ptq, N 2ptqq for all t P r0, Ts, where +N 1ptq :“ ´uptq ´ +ż t +0 +rηA0upsq ` B0pupsq, upsqqsds ` u0 ` +ż t +0 +R0pnpsq, Φqds, +N 2ptq :“ ´cptq ´ +ż t +0 +rξA1cpsq ` B1pupsq, cpsqqsds ` c0 ´ +ż t +0 +R1pnpsq, cpsqqds. +In the next lemma, we state that LN “ pL1N 1, L2N 2q is also an H ˆL2pOq-valued martingale. +Lemma 4.17. The process LN is an H ˆL2pOq-valued continuous square integrable martingale +with respect to the filtration F1 “ tσ ppupsq, cpsq, npsqq; s ď tqutPr0,Ts. The quadratic variation +is given by +xxLNyyt “ +ż t +0 +Gpupsq, cpsqqGpupsq, cpsqq˚ds, +where +Gpu, cq “ +ˆ +L1gpu, cq +0 +0 +L2φpcq +˙ +, +and Gpu, cq˚ : H ˆ L2pOq Ñ U ˆ R2 is the adjoint of the operator Gpu, cq given by +Gpu, cq˚v “ +˜ 8 +ÿ +k“1 +pL1gpupsq, cpsqqek, wqek, +2ÿ +k“1 +pL2φpcpsqqgk, ψqgk +¸ +, +for all v “ pw, ψq P H ˆ L2pOq. +Proof. Let t P r0, Ts. We first prove that LN is an H ˆL2pOq-valued square integrable random +variable. Thanks to the continuity of L, it will be sufficient to prove that E |N|2 +V ˚ˆH´2 ă 8. +Using Lemma 4.13 and Lemma 4.14, we conclude that +lim +mÝÑ8 Nmptq “ Nptq +P1-a.s. in +V ˚ ˆ H´2pOq. +By the continuity of the injection H ˆ L2pOq ãÑ V ˚ ˆ H´2pOq, the Burkholder-Gundy-Davis +inequality for continuous martingales and equality (4.85) as well as inequalities (4.67) and +(4.69), we have +E1 sup +0ďsďT +|Nmpsq|4 +V ˚ˆH´2 ď KE1 sup +0ďsďT +|Nmpsq|4 +L2ˆL2 +ď KE1 +ˆż T +0 +|Gmp¯umpsq, ¯cmpsqq|2 +L2pUˆR2,HˆL2q ds +˙2 +“ 2KE1 +ˆż T +0 +ˇˇP1 +mgp¯umpsq, ¯cmpsqq +ˇˇ2 +L2pU,Hq ds +˙2 +` 2KE1 +ˆż T +0 +ˇˇP2 +mφp¯cmpsqq +ˇˇ2 +L2pR2,L2q ds +˙2 +(4.88) +ď K +ˆ +1 ` E1 sup +0ďsďT +|p¯umpsq, ¯cmpsqq|4 +H +˙ +` KE1 sup +0ďsďT +|∇¯cmpsq|4 +L2 ă K. + +56 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Hence, by the Vitali Theorem, we infer that Nptq P L2pΩ1; V ˚ ˆ H´2pOqq and +lim +mÝÑ8 Nmptq “ Nptq +in +L2pΩ1; V ˚ ˆ H´2pOqq. +Next, let v “ pw, ψq P V ˚ ˆH´2pOq, and hi, i “ 1, 2, 3 be a bounded and continuous function +on Cpr0, Ts; V ˚q, Cpr0, Ts; H´2pOqq, and Cpr0, Ts; H´3pOqq respectively. Let s, t P r0, Ts such +that s ď t. Let +Fmpt, sq :“ pLNmptq ´ LNmpsq, vqHˆL2pOq h1p¯um|r0,ssqh2p¯cm|r0,ssqh3p¯nm|r0,ssq, +Fpt, sq :“ pLNptq ´ LNpsq, vqHˆL2pOq h1pu|r0,ssqh2pc|r0,ssqh3pn|r0,ssq. +We will prove that +(4.89) +lim +mÝÑ8 E1Fmpt, sq “ E1Fpt, sq. +To this aim, we start by noting that by the P1-a.s.-convergence p¯um, ¯cm, ¯nmq Ñ pu, c, nq in +Z and Lemma 4.13 as well as Lemma 4.14, we infer that +lim +mÝÑ8 Fmpt, sq “ Fpt, sq, +P1-a.s. +We will now show that the function tFmpt, squmě1 are uniformly integrable. +We use the +estimate (4.88) to derive that +E1 |Fmpt, sq|4 ď K |h1|4 +L8 |h2|4 +L8 |h3|4 +L8 |v|4 +HˆL2 E1 ” +|Nmptq|4 +L2ˆL2 ` |Nmpsq|4 +L2ˆL2 +ı +ď K |h1|4 +L8 |h2|4 +L8 |h3|4 +L8 |v|4 +HˆL2 . +Invoking the Vitali Theorem, we get the convergence (4.89). +Let 0 ď s ď t ď T and vi “ pwi, ψiq P H ˆ L2pOq, i “ 1, 2. Let +Qmpt, sq : “ +´ +pLNmptq, v1qHˆL2pOq pLNmptq, v2qHˆL2pOq +´ pLNmpsq, v1qHˆL2pOq pLNmpsq, v2qHˆL2pOq +¯ +h1p¯um|r0,ssqh2p¯cm|r0,ssqh3p¯nm|r0,ssq, +Qpt, sq : “ +´ +pLNptq, v1qHˆL2pOq pLNptq, v2qHˆL2pOq +´ pLNpsq, v1qHˆL2pOq pLNpsq, v2qHˆL2pOq +¯ +h1pu|r0,ssqh2pc|r0,ssqh3pn|r0,ssq. +Our purpose now is to prove that +(4.90) +E1Qpt, sq “ +lim +mÝÑ8 E1Qmpt, sq, +imitating the proof before. Indeed, by P1-a.s.-convergence p¯um, ¯cm, ¯nmq Ñ pu, c, nq in Z and +Lemma 4.13 as well as Lemma 4.14 once more, we obtain +lim +mÝÑ8 Qmpt, sq “ Qpt, sq, +P1-a.s. +We now prove the uniform integrability of Qmpt, sq. For this purpose, by (4.88) we find that +E1 |Qmpt, sq|2 ď K |h1|2 +L8 |h2|2 +L8 |h3|2 +L8 E1 +„ˇˇˇpNmptq, v1qHˆL2pOq pNmptq, v2qHˆL2pOq +ˇˇˇ +2 +` +ˇˇˇpNmpsq, v1qHˆL2pOq pNmpsq, v2qHˆL2pOq +ˇˇˇ +2 +ď K |h1|2 +L8 |h2|2 +L8 |h3|2 +L8 |v1|2 +HˆL2 |v2|2 +HˆL2 E1 ” +|Nmptq|4 +L2ˆL2 ` |Nmpsq|4 +L2ˆL2 +ı +ď K |h1|2 +L8 |h2|2 +L8 |h3|2 +L8 |v|2 +HˆL2 . + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +57 +As before, the Vitali Theorem yields equality (4.90). +Finally, we also define +Rmpt, sq :“ +ˆż t +s +pGmp¯umprq, ¯cmprqq˚v1, Gmp¯umprq, ¯cmprqq˚v2qUˆR2 dr +˙ +ˆ +ˆ h1p¯um|r0,ssqh2p¯cm|r0,ssqh3p¯nm|r0,ssq, +and +Rpt, sq :“ +ˆż t +s +pGpuprq, cprqq˚v1, Gpuprq, cprqq˚v2qUˆR2 dr +˙ +h1pu|r0,ssqh2pc|r0,ssqh3pn|r0,ssq, +We claim that +(4.91) +lim +mÝÑ8 E1Rmpt, sq “ E1Rpt, sq. +In order to establish this claim we first show that +(4.92) +lim +mÝÑ8 Rmpt, sq “ Rpt, sq, +P1-a.s. +Since h1p¯um|r0,ssqh2p¯cm|r0,ssqh3p¯nm|r0,ssq Ñ h1pu|r0,ssqh2pc|r0,ssqh3pn|r0,ssq P-a.s., in order to +prove (4.92), it is sufficient to prove that +lim +mÝÑ8 +ż t +s +pGmp¯umprq, ¯cmprqq˚v1, Gmp¯umprq, ¯cmprqq˚v2qUˆR2 dr +“ +ż t +s +pGpuprq, cprqq˚v1, Gpuprq, cprqq˚v2qUˆR2 dr, +P1-a.s. +(4.93) +For all r P rs, ts, we set +Jprq :“ pGmp¯umprq, ¯cmprqq˚v1, Gmp¯umprq, ¯cmprqq˚v2qUˆR2 +´ pLGpuprq, cprqq˚v1, LGpuprq, cprqq˚v2qUˆR2 . +Then, we note that +ż t +s +|Jprq| dz ď +ż T +0 +ˇˇpGmp¯umprq, ¯cmprqq˚v1 ´ LGpuprq, cprqq˚v1, Gmp¯umprq, ¯cmprqq˚v2qUˆR2 +ˇˇ dr +` +ż T +0 +ˇˇpGpuprq, cprqq˚v1, Gmp¯umprq, ¯cmprqq˚v2 ´ Gpuprq, cprqq˚v2qUˆR2 +ˇˇ dr +(4.94) +“ I1pmq ` I2pmq. +Using the Cauchy-Schwarz inequality and the H¨older inequality, we derive that +I1pmq ď +ˆż T +0 +|Gmp¯umprq, ¯cmprqq˚v1 ´ Gpuprq, cprqq˚v1q|2 +UˆR2 dr +˙ 1 +2 +ˆ +ˆ +ˆż T +0 +|Gmp¯umprq, ¯cmprqq˚v2|2 +UˆR2 dr +˙ 1 +2 +. +Owing to the fact that P1 +mgp¯um, ¯cmqek P H and P2 +mφp¯cmqgk P L2pOq, we infer that +pL1P1 +mgp¯um, ¯cmqek, w1q :“ xP1 +mgp¯um, ¯cmqek, i1˚w1y “ pgpu, cqek, i1˚w1q. +and +pL2P2 +mφp¯cmqgk, ψ2q :“ xP2 +mφp¯cmqgk, i2˚ψ2y “ pP2 +mφp¯cmqgk, i2˚ψ2q. + +58 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Thus, using the inequality (2.12) and the fact that tekukě1 and tgkuk“1,2 are orthonormal +basis of U and R2 respectively, we derive that +ż T +0 +|Gmp¯umprq, ¯cmprqq˚v2|2 +UˆR2 dr +“ +ż T +0 +¨ +˝ +ˇˇˇˇˇ +8 +ÿ +k“1 +pL1P1 +mgp¯umprq, ¯cmprqqek, w2qek +ˇˇˇˇˇ +2 +U +` +ˇˇˇˇˇ +2ÿ +k“1 +pL2P2 +mφp¯cmprqqgk, ψ2qgk +ˇˇˇˇˇ +2 +R2 +˛ +‚dr +ď +ż T +0 +8 +ÿ +k“1 +ˇˇpP1 +mgp¯umprq, ¯cmprqqek, i1˚w2q +ˇˇ2 dr ` +ż T +0 +2ÿ +k“1 +ˇˇpP2 +mφp¯cmprqqgk, i2˚ψ2q +ˇˇ2 dr +(4.95) +ď +ˇˇi1˚w2 +ˇˇ2 +L2 +ż T +0 +|gp¯umprq, ¯cmprqq|2 +L2pU,Hq dr ` +ˇˇi2˚ψ2 +ˇˇ2 +L2 +ż T +0 +|φp¯cmprqq|2 +L2pR2,L2q dr +ď K +ż T +0 +p1 ` |p¯umprq, ¯cmprqq|2 +Hqdr ` K +ż T +0 +|∇¯cmprqq|2 +L2 dr +ď K, +P1-a.s. +In the last line we used the fact that ¯cm Ñ c in L2p0, T; H1pOqq and ¯um Ñ u in L2p0, T; Hq +P1-a.s. +On the other hand, we note that +ż T +0 +|Gmp¯umprq, ¯cmprqq˚v1 ´ Gpuprq, cprqq˚v1q|2 +UˆR2 dr +ď +ż T +0 +ˇˇˇˇˇ +« 8 +ÿ +k“1 +pL1P1 +mgp¯umprq, ¯cmprqqek, w1q ´ +8 +ÿ +k“1 +pL1gpuprq, cprqqek, w1q +ff +ek +ˇˇˇˇˇ +2 +U +dr +` +ż T +0 +ˇˇˇˇˇ +« 2ÿ +k“1 +pL2P2 +mφp¯cmprqqgk, ψ1q ´ +2ÿ +k“1 +pL2φpcprqqgk, ψ1q +ff +gk +ˇˇˇˇˇ +2 +R2 +dr. +Then by this last inequality and the inequality (4.95), we infer that +I2 +1pmq ď K +ż T +0 +ˇˇˇˇˇ +8 +ÿ +k“1 +pgp¯umprq, ¯cmprqqek, P1 +mi1˚w1q ´ +8 +ÿ +k“1 +pgpuprq, cprqqek, i1˚w1q +ˇˇˇˇˇ +2 +dr +` K +ż T +0 +ˇˇˇˇˇ +2ÿ +k“1 +pφp¯cmprqqgk, P2 +mi2˚ψ1q ´ +2ÿ +k“1 +pφpcprqqgk, i2˚ψ1q +ˇˇˇˇˇ +2 +dr +ď K +ˇˇi1˚w1 +ˇˇ2 +L2 +ż T +0 +|gp¯umprq, ¯cmprqq ´ gpuprq, cprqq|2 +L2pU,Hq dr +(4.96) +` K +ˇˇP1 +mi1˚w1 ´ i1˚w1 +ˇˇ2 +L2 +ż T +0 +|gp¯umprq, ¯cmprqq|2 +L2pU,Hq dr +` +ˇˇi2˚ψ1 +ˇˇ2 +L2 +ż T +0 +|φp¯cmprqq ´ φpcprqq|2 +L2pR2,L2q dr +` +ˇˇP2 +mi2˚ψ1 ´ i2˚ψ1 +ˇˇ2 +L2 +ż T +0 +|φp¯cmprqq|2 +L2pR2,L2q dr +:“ II1pmq ` II2pmq ` II3pmq ` II4pmq. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +59 +By means of the continuity of g, the P1-a.s.-convergence p¯um, ¯cm, ¯nmq Ñ pu, c, nq in Z, +the inequality (2.12) and the Vitali Theorem, +we can derive that limmÝÑ8 II1pmq “ 0. +Furthermore, since +ż T +0 +|gp¯umprq, ¯cmprqq|2 +L2pU,Hq dr ` +ż T +0 +|φp¯cmprqq|2 +L2pR2,L2q dr +ď K +ż T +0 +p1 ` |p¯umprq, ¯cmprqq|2 +Hqdr ` K +ż T +0 +|∇¯cmprq|2 +L2 dr +ď K +P1-a.s., +we deduce that +lim +mÝÑ8 II2pmq “ +lim +mÝÑ8 II4pmq “ 0. +Now, we study the II3pmq. We see that +II3pmq ď |ψ1|2 +L2 γ2 |σ|2 +L8 +ż T +0 +|∇¯cmprq ´ ∇cprq|2 +L2 dr +ď |ψ1|2 +L2 γ2 |σ|2 +L8 +ż T +0 +|¯cmprq ´ cprq|2 +H1 dr. +By using the fact that ¯cm Ñ c in L2p0, T; H1pOqq, P1-a.s., we can pass to the limit in this +last inequality and infer that limmÝÑ8 II3pmq “ 0. Hence passing to the limit in (4.96) we +get limmÝÑ8 I1pmq “ 0. In a similar fashion, we can also prove that limmÝÑ8 I2pmq “ 0. +Therefore, passing to the limit in (4.94), we obtain the convergence (4.93) and completes the +proof of the almost surely convergence (4.92). +To finish the proof of equality (4.91), it remains to prove the uniform integrability of +Rmpt, sq. For this purpose, using the Young inequality, a similar calculations as in inequality +(4.95) and the estimates (4.67) and (4.69), we arrive at +E1 |Rmpt, sq|2 ď +3 +ź +i“1 +|hi|2 +L8 E1 +ˆż t +s +pGmp¯umprq, ¯cmprqq˚v1, Gmp¯umprq, ¯cmprqq˚v2qUˆR2 dr +˙2 +ď Kpt ´ sqE1 +ż t +s +|Gmp¯umprq, ¯cmprqq˚v1|2 +UˆR2 |Gmp¯umprq, ¯cmprqq˚v2|2 +UˆR2 dr +ď KE1 +ż T +0 +|Gmp¯umprq, ¯cmprqq˚v1|4 +UˆR2 dr ` KE1 +ż T +0 +|Gmp¯umprq, ¯cmprqq˚v2|4 +UˆR2 dr +ď KE1 +ż T +0 +|gp¯umprq, ¯cmprqq|4 +L2pU,Hq dr ` KE1 +ż T +0 +|φp¯cmprqq|4 +L2pR2,L2q dr +ď KE1 sup +0ďrďT +p1 ` |p¯umprq, ¯cmprqq|4 +Hq ` KE1 sup +0ďrďT +|∇¯cmprqq|4 +L2 +ď K, +which prove the uniform integrability of Rmpt, sq. +Thus, invoking the Vitali Theorem, we +obtain the convergence (4.91). +Taking into account the convergences (4.89), (4.90) and (4.91), we can pass to the limit +in the equalities (4.86) and (4.87) to get +E +” +pLNptq ´ LNpsq, v1qHˆL2pOq h1pu|r0,ssqh2pc|r0,ssqh3pn|r0,ssq +ı +“ 0, + +60 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +and +E +”´ +pLNptq, v1qHˆL2pOq pLNptq, v2qHˆL2pOq ´ pLNpsq, v1qHˆL2pOq pLNpsq, v2qHˆL2pOq +´ +ż t +0 +pGpupsq, cpsqq˚v1, Gpupsq, cpsqq˚v2qUˆR2 ds +˙ +h1pu|r0,ssqh2pc|r0,ssqh3pn|r0,ssq + +“ 0, +which complete the proof of Lemma 4.17. +□ +Thanks to Lemma 4.17, we apply the usual martingale representation theorem proved in +[12, Theorem 8.2] to the process LN and conclude that there exists a probability space +p˜Ω, ˜F, ˜Pq, a filtration ˜F and a U ˆ R2-cylindrical Wiener process +¯ +Ws :“ p ¯Ws, ¯βsq defined on +the probability space p¯Ω, ¯F, ¯Pq “ pΩ1 ˆ ˜Ω, F1 b ˜F, P1 b ˜Pq adapted to the filtration ¯F “ F1 b ˜F +such that +LNpt, ω1, ˜ωq “ +ż t +0 +Gpups, ω1, ˜ωq, cps, ω1, ˜ωqqd ¯ +Wspω1, ˜ωq, +t P r0, Ts, +pω1, ˜ωq P ¯Ω, +where +LNpt, ω1, ˜ωq “ LNpt, ω1q, pups, ω1, ˜ωq, cps, ω1, ˜ωqq “ pups, ω1q, cps, ω1qq, t P r0, Ts, pω1, ˜ωq P ¯Ω. +This implies that in the probability space p¯Ω, ¯F, ¯Pq, for t P r0, Ts and ¯P-a.s. +(4.97) +$ +’ +’ +’ +& +’ +’ +’ +% +L1N 1ptq “ +ż t +0 +L1gpupsq, cpsqqd ¯ +Ws, in H, +L2N 2ptq “ +ż t +0 +L2φpcpsqqd¯βs, in L2pOq. +Thanks to (2.12) and (4.70) the estimate +¯E +ż T +0 +|gpupsq, cpsqq|2 +L2pU,V ˚q ds ď K¯E +ż T +0 +|gpupsq, cpsqq|2 +L2pU,Hq ds +ď K +ˆ +1 ` E1 sup +0ďsďT +|pupsq, cpsqq|2 +H +˙ +ă 8, +and +¯E +ż T +0 +|φpcpsqq|2 +L2pR2,H´2q ds ď K¯E +ż T +0 +|φpcpsqq|2 +L2pR2,L2q ds +ď K +ˆ +1 ` E1 sup +0ďsďT +|cpsq|2 +H1 +˙ +ă 8, +yield that L1N 1 and L2N 2 in (4.97) are continuous martingale in H and L2pOq respectively. +In a similar fashion as in [6, Proof of Theorem 1.1], using the continuity of the operators +L1 and L2, we get +ż t +0 +L1gpupsq, cpsqqd ¯ +Ws “ L1 +ˆż t +0 +gpupsq, cpsqqd ¯ +Ws +˙ +and +ż t +0 +L2φpcpsqqd¯βs “ L2 +ˆż t +0 +φpcpsqqd¯βs +˙ +, +for all t P r0, Ts. Combining these two last inequalities with the injectivity of the operators +L1 and L2, we infer from the system (4.97) that for t P r0, Ts, +(4.98) +$ +’ +’ +’ +& +’ +’ +’ +% +N 1ptq “ +ż t +0 +gpupsq, cpsqqd ¯ +Ws, in V ˚, +N 2ptq “ +ż t +0 +φpcpsqqd¯βs, in H´2pOq. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +61 +On the new probability space p¯Ω, ¯F, ¯Pq, we also extend the random variable nptq by +npt, ω1, ˜ωq “ npt, ω1q, t P r0, Ts, pω1, ˜ωq P ¯Ω, +and infer that the equality (4.83) also hods in p¯Ω, ¯F, ¯Pq. Using this, the definition of N 1 and +N 2, and the system (4.98), we derive that p¯Ω, ¯F, ¯F, ¯P, pu, c, nq, p ¯ +W , ¯βqq satisfies the system +(3.2). In particular, we have for all t P r0, Ts and ¯P-a.s. +$ +’ +’ +’ +& +’ +’ +’ +% +uptq “ u0 ´ +ż t +0 +rηA0upsq ` B0pupsq, upsqq ` R0pnpsq, Φqsds ` +ż t +0 +gpupsq, cpsqqd ¯ +Ws, in V ˚, +cptq “ c0 ´ +ż t +0 +rξA1cpsq ` B1pupsq, cpsqq ´ R1pnpsq, cpsqqsds ` γ +ż t +0 +φpcpsqqd¯βs, in H´2pOq, +which can be written as +$ +’ +’ +’ +& +’ +’ +’ +% +uptq “ u0 ´ +ż t +0 +G0psqds ` +ż t +0 +S0psqd ¯Ws, in V ˚, +cptq “ c0 ´ +ż t +0 +G1psqds ` +ż t +0 +S1psqd¯βs, in H´2pOq, +where for all t P r0, Ts, +G0ptq :“ ηA0uptq ` B0puptq, uptqq ` R0pnptq, Φq, +G1ptq :“ ξA1cptq ` B1puptq, cptqq ´ R1pnptq, cptqq, +S0ptq :“ gpuptq, cptqq, +and +S1ptq :“ γφpcptqq. +Since the identities +(4.66), +(4.70) and (4.71) hold, +following +the idea of the proof +of +estimate +(4.57), +we +can +see +that +G0 P L2pr0, Ts ˆ ¯Ω; V ˚q, +G1 P L2pr0, Ts ˆ ¯Ω; L2pOqq, +S0 P L2pr0, Ts ˆ ¯Ω; Hq and S1 P L2pr0, Ts ˆ ¯Ω; H1pOqq. +Therefore, +it follows from [23, +Theorem 3.2] that there exists ¯Ω0 P ¯F such that ¯Pp¯Ω0q “ 1 and for all ω P ¯Ω0, the function +u and c take values in H and in H1pOq respectively and are continuous in H and H1pOq +with respect to t. Owing to the fact that pu, c, nq is Zu ˆ Zc ˆ Zn-valued random variable +and progressively measurable over the filtration ¯F, we derive that p¯Ω, ¯F, ¯F, ¯P, pu, c, nq, p ¯ +W , ¯βqq +is a probabilistic weak solution of system (1.2). We recall that the inequalities (3.1) directly +follows from relations (4.66), (4.70), and (4.71). +5. PROPERTIES +OF +SOLUTION +AND +ENERGY +INEQUALITY +In this section we prove the mass conservation property, the non-negativity property and the +L8-norm stability for the prrobabilistic strong solution of system (1.2). By these properties, +we also prove an energy inequality which may be useful for the study of the invariant +measure of system (1.2) which is still an opened problem according to our knowledge. +5.1. Non-negativity and mass conservation. The following theorem gives the conservation of +the total mass property and the non-negativity of the strong solutions of system (1.2). +Theorem 5.1. Let A “ pΩ, F, tFtutPr0,Ts, Pq be a filtered probability space, U be a separable +Hilbert space, W be cylindrical Wiener process on U over A, and β “ pβ1, β2q be a two +dimensional standard Brownian motion over A independent of W. If pu, c, nq is a probabilistic +strong solution of system (1.2), then the following equality holds for all t P r0, Ts +(5.1) +ż +O +npt, xqdx “ +ż +O +n0pxqdx, P-a.s. + +62 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +Furthermore, if c0 ą 0 and n0 ą 0, then the following inequality hold P-a.s +(5.2) +nptq ą 0, and cptq ą 0, for all t P r0, Ts. +Proof. We Note that, the conservation of the total mass (5.1) follows straightforwardly from +the fact that ∇ ¨ u “ 0 and the proof of (5.2) is very similar to the proof of Lemma 3.6. +□ +The following theorem gives the L8-stability of the probabilistic strong solution of system +(1.2). +Theorem 5.2. Let A “ pΩ, F, tFtutPr0,Ts, Pq be a filtered probability space, U be a separable +Hilbert space, W be cylindrical Wiener process on U over A, and β “ pβ1, β2q be a two +dimensional standard Brownian motion over A independent of W. If pu, c, nq is a probabilistic +strong solution of system (1.2) in the filtered probability space A, then for all t P r0, Ts +(5.3) +|cptq|L8 ď |c0|L8 , +P-a.s. +Proof. The proof is similar to the proof of Corollary 3.7. +□ +5.2. Energy inequality. In this subsection, we will derive an energy inequality. The probabilistic +strong solution pu, n, cq involving the following Lyapunov functional +Epn, c, uqptq “ +ż +O +nptq ln nptqdx`Kf |∇cptq|2 +L2 ` 8KfKGN |c0|2 +L8 +3ξη +|uptq|2 +L2 `e´1 |O| , +t P r0, Ts, +where KGN is a constant given by the Gagliardo-Niremberg inequality (3.7) and Kf is defined +in (2.2). +Proposition 5.3. Suppose that Assumption 1, Assumption 2 and the following inequality +(5.4) +4Kf +max +0ďcď|c0|L8 f 2 +min +0ďcď|c0|L8 f 1 +ď δ, +are satisfied. Let A “ pΩ, F, tFtutPr0,Ts, Pq be a filtered probability space, U be a separable +Hilbert space, W be cylindrical Wiener process on U over A, and β “ pβ1, β2q be a two +dimensional standard Brownian motion over A independent of W. +Then, any probabilistic +strong solution pu, c, nq of system (1.2) in the filtered probability space A satisfies the following +entropy functional relations for almost all t P r0, Ts, +|cptq|2 +L2 ` 2η +ż t +0 +|∇cpsq|2 +L2 ds ` 2 +ż t +0 +pnpsqfpcpsqq, cpsqqds “ |c0|2 +L2 , +(5.5) +Epn, c, uqptq ` +ż t +0 +« +δ +ˇˇˇ∇ +a +npsq +ˇˇˇ +2 +L2 ` 3ξKf +2 +|∆cpsq|2 +L2 ` 8KfKGN |c0|2 +L8 +3ξ +|∇upsq|2 +L2 ` +ˇˇˇ +a +npsq∇cpsq +ˇˇˇ +2 +L2 +ff +ds +ď Epn0, c0, u0q ` K5t ` K6 +ż t +0 +|upsq|2 +L2 ds ` γ2Kf +ż t +0 +|∇φpcpsqq|2 +L2pR2;L2q ds +` 8KfKGN |c0|2 +L8 +3ξη +ż t +0 +|gpupsq, cpsqq|2 +L2pU;Hq ds ` 2γKf +ż t +0 +p∇φpcpsqq, ∇cpsqqdβs +(5.6) +` 16KfKGN |c0|2 +L8 +3ξη +ż t +0 +pgpupsq, cpsqq, upsqqdWs, +P-a.s., where K5 and K6 are some positive constant to be given later. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +63 +Proof. The equality (5.5) follows directly from the application of the Itˆo formula to t ÞÑ |cptq|2 +L2 +and the fact that +pB1pu, cq, cq “ 1 +2 +ż +O +upxq ¨ ∇c2pxqdx “ ´1 +2 +ż +O +c2pxq∇ ¨ upxqdx “ 0, +as well as +pφpcq, cq “ +2ÿ +k“1 +ż +O +σkpxq ¨ ∇cpxqcpxqdx “ 1 +2 +2ÿ +k“1 +ż +O +σkpxq ¨ ∇c2pxqdx “ 0 +and +|φpcq|2 +L2pR2;L2q “ |∇c|2 +L2 . +Next, we multiply equation (2.14)3 by 1 ` ln npsq for s P r0, ts and integrate the resulting +equation in O to obtain +(5.7) +d +dt +ż +O +nps, xq ln nps, xqdx ` δ +ż +O +|∇nps, xq|2 +nps, xq +dx “ χ +ż +O +∇nps, xq ¨ ∇cps, xqdx. +Thanks to the Young inequality and the Cauchy-Schwartz inequality we note that +χ +ż +O +∇npxq ¨ ∇cpxqdx ď 2δ +ż +O +ˇˇˇ∇ +a +npxq +ˇˇˇ +2 +dx ` χ2 +2δ +ż +O +npxq |∇cpxq|2 dx. +Combining the last inequality with equality (5.7) we arrive at +ż +O +npt, xq ln npt, xqdx ` 2δ +ż t +0 +ˇˇˇ∇ +a +npsq +ˇˇˇ +2 +L2 ds ď +ż +O +n0pxq ln n0pxqdx +` χ2 +2δ +ż t +0 +ˇˇˇ +a +npsq∇cpsq +ˇˇˇ +2 +L2 ds. +(5.8) +By applying the Itˆo formula to t ÞÑ |∇cptq|2 +L2, we find that +|∇cptq|2 +L2 ` 2ξ +ż t +0 +|∆cpsq|2 +L2 ds “ |∇c0|2 +L2 ´ 2 +ż t +0 +p∇B1pupsq, cpsqq, ∇cpsqqds +´ 2 +ż t +0 +p∇R2pnpsq, cpsqq, ∇cpsqqds +` γ2 +ż t +0 +|∇φpcpsqq|2 +L2pR2;L2q ` 2γ +ż t +0 +p∇φpcpsqq, ∇cpsqqdβs. +(5.9) +Due to the Assumption 1 and the L8-norm stability obtained in Theorem 5.2, we obtain +p∇B1pu, cq, ∇cq ď |∇u|L2 |∇c|2 +L4 +ď +3ξ +16KGN |c0|2 +L8 +|∇c|4 +L4 ` 4KGN |c0|2 +L8 +3ξ +|∇u|2 +L2 +ď ξ +4 |∆c|2 +L2 ` 4KGN |c0|2 +L8 +3ξ +|∇u|2 +L2 ` ξp4K2 ` 3q +16 +|c0|2 +L8 . + +64 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +and +´p∇R2pn, cq, ∇cpsqqds ď ´ +min +0ďcď|c0|L8 f 1pcq +2 +ż +O +npxq |∇cpxq|2 dx +` +1 +2 +min +0ďcď|c0|L8 f 1 +ż +O +f 2pcpxqq|∇npxq|2 +npxq +dx +ď ´ +min +0ďcď|c0|L8 f 1pcq +2 +ˇˇ?n∇c +ˇˇ2 +L2 ` +2 +max +0ďcď|c0|L8 f 2 +min +0ďcď|c0|L8 f 1pcq +ˇˇ∇?n +ˇˇ2 +L2 . +Thus, we see from (5.9) that +|∇cptq|2 +L2 ` 3ξ +2 +ż t +0 +|∆cpsq|2 +L2 ds ` +min +0ďcď|c0|L8 f 1 +ż t +0 +ˇˇˇ +a +psq∇cpsq +ˇˇˇ +2 +L2 ds +ď |∇c0|2 +L2 ` ξp4K2 ` 3q +8 +|c0|2 +L8 t ` 8KGN |c0|2 +L8 +3ξ +ż t +0 +|∇upsq|2 +L2 ds +` +4 +max +0ďcď|c0|L8 f 2 +min +0ďcď|c0|L8 f 1 +ż t +0 +ˇˇˇ∇ +a +npsq +ˇˇˇ +2 +L2 ds +` γ2 +ż t +0 +|∇φpcpsqq|2 +L2pR2;L2q ds ` 2γ +ż t +0 +p∇φpcpsqq, ∇cpsqqdβs. +Now, we multiply this last inequality by Kf, add the result with inequality (5.8), and use +the inequality (5.4) to obtain +ż +O +npt, xq ln npt, xqdx ` Kf |∇cptq|2 +L2 ` 3ξKf +2 +ż t +0 +|∆cpsq|2 +L2 ds +` 2δ +ż t +0 +ˇˇˇ∇ +a +npsq +ˇˇˇ +2 +L2 ds ` +ż t +0 +ˇˇˇ +a +npsq∇cpsq +ˇˇˇ +2 +L2 ds +ď Kf |∇c0|2 +L2 ` +ż +O +n0pxq ln n0pxqdx ` Kfξp4KfK2 ` 3q +8 +|c0|2 +L8 t +` 8KfKGN |c0|2 +L8 +3ξ +ż t +0 +|∇upsq|2 +L2 ds ` γ2Kf +ż t +0 +|∇φpcpsqq|2 +L2pR2;L2q ds +(5.10) +` 2γKf +ż t +0 +p∇φpcpsqq, ∇cpsqqdβs. +Using the equality (5.1) and the inequality (3.7) we note that +|n|L2 ď KGN +´ˇˇ?n +ˇˇ +L2 +ˇˇ∇?n +ˇˇ +L2 ` +ˇˇ?n +ˇˇ2 +L2 +¯ +ď KGN +ˆ +|n0| +1 +2 +L1 +ˇˇ∇?n +ˇˇ +L2 ` |n0|L1 +˙ +, +(5.11) + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +65 +which altogether with the Itˆo formula to t ÞÑ |uptq|2 +L2 implies the existence of K3 ą 0 such +that +|uptq|2 +L2 ` 2η +ż t +0 +|∇upsq|2 +L2 ds ď 2 +ż t +0 +|∇Φ|L8 |npsq|L2 |upsq|L2 ds +` +ż t +0 +|gpupsq, cpsqq|2 +L2pU;Hq ds ` 2 +ż t +0 +pgpupsq, cpsqq, upsqqdWs +ď |u0|2 +L2 ` δη +K4 +ż t +0 +ˇˇˇ∇ +a +npsq +ˇˇˇ +2 +L2 ds ` K3 |∇Φ|2 +L8 |n0|L1 +ż t +0 +|upsq|2 +L2 ds +(5.12) +` 1 +2t ` 1 +2 |∇Φ|2 +L8 |n0|2 +L1 +ż t +0 +|upsq|2 +L2 ds +` +ż t +0 +|gpupsq, cpsqq|2 +L2pU;Hq ds ` 2 +ż t +0 +pgpupsq, cpsqq, upsqqdWs, +with K4 “ 8Kf KGN|c0|2 +L8 +3ξ +. Multiplying the inequality (5.12) by +K4 +η , and adding the result with +inequality (5.10), we obtain some positive constants K5 and K6 such that the inequality (5.6) +holds. +□ +APPENDIX A. COMPACTNESS +AND +TIGHTNESS +CRITERIA +In this appendix we recall several compactness and tightness criteria that are frequently +used in this paper. +We start with the following lemma based on the Dubinsky Theorem. +Lemma A.1. Let us consider the space +(A.1) +˜Z0 “ L2 +wp0, T; H1pOqq X L2p0, T; L2pOqq X Cpr0, Ts; H´3pOqq +and ˜T0 be the supremum of the corresponding topologies. Then a set ¯¯K0 Ă ˜Z0 is ˜T0-relatively +compact if the following three conditions hold +(a) sup +ϕP ¯¯ +K0 +Tż +0 +|ϕpsq|2 +H1ds ă 8, i.e., +¯¯K0 is bounded in L2p0, T; H1pOqq, +(b) Dγ ą 0: +sup +ϕP ¯¯ +K0 +|ϕ|Cγpr0,Ts;H´3q ă 8. +Proof. We note that the following embedding is continuous H1pOq ãÑ L2pOq ãÑ H´3pOq with +H1pOq ãÑ L2pOq compact. By the Banach-Alaoglu Theorem condition (a) yields that +¯¯K0 is +compact in L2 +wp0, T; H1pOqq. Moreover (b) implies that the functions ϕ P ¯¯K0 are equicontinuous, +i.e. for all ε ą 0, there exists δ ą 0 such that if |t ´ s| ă δ then |ϕptq ´ ϕpsq|H´3 ă ε for +all ϕ P ¯¯K0. +We can then apply Dubinsky’s Theorem (see [41, Theorem IV.4.1]) since by +condition (a), +¯¯K0 is bounded in L2p0, T; H1pOqq. +□ +Following the same method as in [8, Lemma 3.3 ], we obtain the following compactness +result. +Lemma A.2. Let us consider the space +(A.2) +˜Zn “ L2 +wp0, T; H1pOqq X L2p0, T; L2pOqq X Cpr0, Ts; H´3pOqq X Cpr0, Ts; L2 +wpOqq, +and ˜T0 be the supremum of the corresponding topologies. Then a set ¯¯K0 Ă ˜Zn is ˜T0-relatively +compact if the following three conditions hold + +66 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +(a) sup +ϕP ¯¯ +K0 +|ϕ|L8p0,T;L2q ă 8, +(b) sup +ϕP ¯¯ +K0 +Tż +0 +|ϕpsq|2 +H1ds ă 8, i.e., +¯¯K0 is bounded in L2p0, T; H1pOqq, +(c) Dγ ą 0: +sup +ϕP ¯¯ +K0 +|ϕ|Cγpr0,Ts;H´3q ă 8. +From this lemma we also get the following tightness criteria for stochastic processes with +paths in +˜Zn where the proof is the same as the proof of [3, Lemma 5.5]. +Lemma A.3 (Tightness criterion for n). Let γ ą 0 be a given parameters and pϕnqn be a +sequence of continuous tFtutPr0,Ts-adapted H´3pOq-valued processes. Let Lm be the law of +ϕn on +˜Zn. If for any ε ą 0 there exists a constant Ki, i “ 1, ..., 3 such that +sup +m P +´ +|ϕm|L8p0,T;L2q ą K1 +¯ +ď ε, +sup +m P +´ +|ϕm|L2p0,T;H1q ą K2 +¯ +ď ε, +sup +m P +´ +|ϕm|Cγp0,T;H´3q ą K3 +¯ +ď ε, +then the sequence pLmqm is tight on +˜Zn. +The following compactness results are due to [7, Theorem 4.4 and Theorem 4.5] (see also +[28]), where we can see the details of the proof. +Lemma A.4. Let us consider the space +(A.3) +˜Zu “ L2 +wp0, T; V q X L2p0, T; Hq X Cpr0, Ts; V ˚q X Cpr0, Ts; Hwq, +and ˜T1 be the supremum of the corresponding topologies. Then a set ¯¯K1 Ă ˜Zu is ˜T1-relatively +compact if the following three conditions hold +(a) sup +vP ¯¯ +K1 +sup +tPr0,Ts +|vptq|L2 ă 8, +(b) sup +vP ¯¯ +K1 +Tż +0 +|∇vpsq|2 +L2ds ă 8, i.e., +¯¯K2 is bounded in L2p0, T; V q, +(c) lim +δÑ0 sup +vP ¯¯ +K1 +sup +s,tPr0,Ts,|t´s|ďδ +|vptq ´ vpsq|V ˚ “ 0. +Lemma A.5. Let us consider the space +(A.4) +˜Zc “ L2 +wp0, T; H2pOqq X L2p0, T; H1 +wpOqq X Cpr0, Ts; L2pOqq X Cpr0, Ts; H1 +wpOqq, +and ˜T2 be the supremum of the corresponding topologies. Then a set ¯¯K2 Ă ˜Zc is ˜T2-relatively +compact if the following three conditions hold +(a) sup +ϕP ¯¯ +K2 +sup +tPr0,Ts +|ϕptq|H1 ă 8, +(b) sup +ϕP ¯¯ +K2 +Tż +0 +|ϕpsq|2 +H2ds ă 8, i.e., +¯¯K2 is bounded in L2p0, T; H2pOqq, +(c) lim +δÑ0 sup +ϕP ¯¯ +K2 +sup +s,tPr0,Ts,|t´s|ďδ +|ϕptq ´ ϕpsq|L2 “ 0. + +ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL +67 +We now consider a filtered probability space pΩ, F, Pq with filtration F :“ tFtutě0 satisfying +the usual hypotheses. Let pM, d1q be a complete, separable metric space and pynqnPN be a +sequence of F-adapted and M-valued processes. We recall from [20] the following definition. +Definition A.6. A sequence pynqnPN satisfies the Aldous condition in the space M if and +only if +@ǫ ą 0 @ζ ą 0 Dδ ą 0 such that for every sequence pτnqnPN of F-stopping times with +τn ď T one has sup +nPN +sup +0ďθďδ +P t|ynpτn ` θq ´ ynpτnq|M ě ζu ď ǫ. +In Definition A.6, and throughout we understand that yn is extended to zero outside the +interval r0, Ts. +The following lemma is proved in [28, Appendix A, Lemma 6.3]. +Lemma A.7. Let pX, |.|Xq be a separable Banach space and let pynqnPN be a sequence of +X-valued random variables. Assume that for every pτnqnPN of F-stoppings times with τn ď T +and for every n P N and θ ě 0 the following condition holds +(A.5) +E |ynpτn ` θq ´ ynpτnq|α +X ď Cθβ, +for some α, β ą 0 and some constant C ą 0. Then the sequence pynqnPN satisfies the Aldous +condition in the space X. +In the view of Lemma A.4 and Lemma 4.2, in the next corollaries, we will state a +tightness criteria for stochastic processes with part in +˜Zu or in +˜Zc. +Corollary A.8. Let pvmqm be a sequence of continuous tFtutPr0,Ts-adapted V ˚-valued processes +satisfying +(a): there exists a constant K1 ą 0 such that +sup +m E sup +0ďsďT +|vmpsq|2 +L2 ď K1, +(b): there exists a constant K2 ą 0 such that +sup +m +ż T +0 +|∇vmpsq|2 +L2 ds ď K2, +(c): pvmqm satisfies the Aldous condition in V ˚. +Let Lmpvmq be the law of vm on +˜Zu. Then, the sequence pLmpvmqqm is tight in +˜Zu. +Corollary A.9. pvmqm be a sequence of continuous tFtutPr0,Ts-adapted L2pOq-valued processes +satisfying +(a): there exists a constant K1 ą 0 such that +sup +m E sup +0ďsďT +|vmpsq|2 +H1 ď K1, +(b): there exists a constant K2 ą 0 such that +sup +m +ż T +0 +|vmpsq|2 +H2 ds ď K2, +(c): pvmqm satisfies the Aldous condition in L2pOq. +Let Lmpvmq be the law of vm on +˜Zc. Then, the sequence pLmpvmqqm is tight in +˜Zc. + +68 +E. +HAUSENBLAS˚, +B. +JIDJOU +MOGHOMYE˚ +AND +P. +A. +RAZAFIMANDIMBY˚˚ +ACKNOWLEDGMENT +We acknowledge financial support provided by the Austrian Science Fund (FWF). 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Liu, Global martingale solution for the three-dimensional stochastic chemotatix-Navier-Stokes +system, arXiv:2209.06500v1, 2022. + diff --git a/dtAyT4oBgHgl3EQfwvmK/content/tmp_files/load_file.txt b/dtAyT4oBgHgl3EQfwvmK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b59e1c87fcc29dcf00174842acf0d0089d27f6ea --- /dev/null +++ b/dtAyT4oBgHgl3EQfwvmK/content/tmp_files/load_file.txt @@ -0,0 +1,2327 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf,len=2326 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='00654v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='AP] 2 Jan 2023 ON THE EXISTENCE AND UNIQUENESS OF SOLUTION TO A STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ ˚ Department of Mathematics and Information Technology, Montanuniversitaet Leoben, Leoben Franz Josef Strasse 18, 8700 Leoben, Austria ˚˚ School of Mathematical Science, Dublin City University, Collins Avenue Dublin 9, Ireland ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In this article, we study a mathematical system which models the dynamic of the collective behaviour of oxygen-driven swimming bacteria in an aquatic fluid flowing in a two dimensional bounded domain under stochastic perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This model can be seen as a stochastic version of Chemotaxis-Navier-Stokes model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We prove the existence of a unique (probabilistic) strong solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In addition, we establish some properties of the strong solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' More precisely, we prove that the unique solution is non-negative and satisfies the mass conservation property and an energy inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' INTRODUCTION The migration of bacteria cells to a higher concentration of a chemical has been observed in biological applications concerning aerobic bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This phenomenon, called chemotaxis, is presumed to have a deep impact on the time evolution of a bacteria population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' There are different concepts of chemotaxis depending on the kind of bacteria and the chemical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In the present article, we focus on the mathematical model describing an oxygen-driven bacteria suspension swimming in an incompressible fluid like water which was firstly proposed in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Mainly, the system consists of three coupled partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The first equation describes the fluid flow with field velocity u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The second equation describes the dynamic of the oxygen concentration c, and the last equation describes the dynamic of the population density n of the bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, the coupled model can be written as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) $ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ % du ` rpu ¨ ∇qu ` ∇P ´ η∆us dt “ n∇Φdt in r0, Ts ˆ O, dc ` u ¨ ∇cdt “ rµ∆c ´ nfpcqs dt in r0, Ts ˆ O, dn ` u ¨ ∇ndt “ rδ∆n ´ ∇ ¨ pnχpcq∇cqs dt in r0, Ts ˆ O, ∇ ¨ u “ 0 in r0, Ts ˆ O, np0q “ n0, cp0q “ c0, up0q “ u0 in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In addition to the unknows u, c, n, we have the scalar pressure P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The positive number T is the final observation time, and O Ă R2 is a domain where the cells and the fluid move and interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The positive constants η, µ and δ are the corresponding diffusion coefficients Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 35R60,35Q35,60H15,76M35,86A05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Navier-Stokes system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Chemotaxis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Stochastic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Probabilistic weak solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' strong solution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 1 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ for the fluid, the oxygen, and the bacteria, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The given functions χ and f denote the chemotactic sensitivity and the oxygen consumption rate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The symbol Φ denotes a given time-independent potential function representing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', the gravitational force or centrifugal force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The mathematical analysis of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) has been investigated by several authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The existence of weak solutions and the existence of a unique classical solution have been proven, see for instance [9, 10, 15, 16, 18, 25, 36, 37, 42, 43] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In the case d “ 2, the existence of a global weak solutions for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) without the nonlinear convective term pu ¨ ∇qu is obtained in [16, 36, 37] and in [18] with nonlinear diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The existence of weak global solutions under various assumptions on the data can be found in [15, 25];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' the global existence of smooth solutions has been proven in [10, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Results on the existence of classical solution are found in [9, 16, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Fix T ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In this paper, we are interested in the mathematical analysis of a stochastic version of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) in the two-dimensional bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' More precisely, for a given family of independent, identically distributed standard real-valued Brownian motions tβkuk“1,2, and a cylindrical Wiener processes W evolving on a fixed separable Hilbert space U defined on a filtered probability space, pΩ, F, pFtqtPr0,Ts, Pq, we consider the following system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) $ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ % du ` rpu ¨ ∇qu ` ∇P ´ η∆us dt “ n∇Φdt ` gpu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cqdWt in r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Ts ˆ O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' dc ` u ¨ ∇cdt “ rµ∆c ´ nfpcqs dt ` γ 2ÿ k“1 σk ¨ ∇c ˝ dβk t in r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Ts ˆ O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' dn ` u ¨ ∇ndt “ rδ∆n ´ ∇ ¨ pnχpcq∇cqs dt in r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Ts ˆ O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ∇ ¨ u “ 0 in r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Ts ˆ O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Bn Bν “ Bc Bν “ 0 on r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Ts ˆ BO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' u “ 0 on r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Ts ˆ BO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' np0q “ n0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cp0q “ c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' up0q “ u0 in O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' where O Ă R2 is a bounded domain with smooth boundary BO and the positive constant γ is the intensity of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The symbol ˝ means that the stochastic differential is understood in the Stratonovich sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The main difference between the deterministic model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) and the stochastic model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) is the presence of the terms gpu, cqdWt and γ ř2 k“1 σk ¨ ∇c ˝ dβk t called noise terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The presence of these noise terms weakened the regularity in time of the velocity field and the concentration of oxygen and so, make the mathematical analysis more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Our investigation is motivated by the need for a sound mathematical analysis for the understanding of the effect of small scale perturbations such as random pollution of water or air which are inherently present in nature (see [11, 29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The presence of these stochastic perturbations can lead to new and important phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In fact, in two-dimensional case, many models such as the Navier-Stokes equation, the Oldroy-B type model, the Landau-Lifshitz-Bloch equation, and magnetohydrodynamics model with sufficiently degenerate noise for example have a unique invariant measure and hence exhibit ergodic behavior in the sense that the time average of a solution is equal to the average over all possible initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Despite continuous efforts in the last 30 years, such property has so far not been found for the deterministic counterpart of these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This property could lead to profound understanding of the nature of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' To the best of our knowledge, the only papers that consider the ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 3 mathematical analysis of a stochastic version of chemotaxis-fluid interaction model are [44, 45] where the authors have proved the existence of both mild and weak solutions for the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) with γ “ 0 and gpu, cq “ gpuq in a two and three dimensional bounded domain under some strong assumptions on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The aim of this article is to study the global resolvability of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) with positive parameters η, µ γ and δ different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We prove the existence and uniqueness of a probabilistic strong solution in a two dimensional bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The proof is based on a Galerkin scheme and the Yamada-Watanabe Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let us recall that the presence of the noise on the c-equation makes the mathematical analysis of the model more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In fact, the noise term in c-equation makes impossible the application of the deterministic maximum principle method for the proof of the non-negativity of solution as is done in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Moreover, the stochastic version of maximum principle method where we learn from [14] need to be adapted in order to conserve the positivity of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The main difference between our work and that of [44] is that the model considered in [44] does not contain any noise on the c-equation and the noise term in the u equation depend only on the velocity field u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Therefore, the present paper can be seen as a generalization of [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The organisation of this article is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In Section 2, we define various functional spaces, and introduce assumptions which are used throughout in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In Section 3, we state and prove the main result which is the existence of a unique probabilistic strong solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In Section 4, we give a detailed proof of important ingredients which have been useful for the proof of the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In Section 5, we prove the mass conservation property and the non-negativity of the strong solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Besides that, we prove an energy inequality which may be useful for the study of the invariant measure in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' FUNCTIONAL SETTING OF THE MODEL AND ASSUMPTIONS Throughout the paper, we assume that O Ă R2 is a bounded domain with boundary BO of class C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The symbol LppOq denotes the Lp space with respect to the Lebesgue measure while W m,ppOq denotes the Sobolev space of functions whose distributional derivatives of order up to m belong to LppOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The spaces of functions φ : O Ñ R2 such that each component of φ belongs to LppOq or to W m,ppOq are denoted by LppOq or by Wm,ppOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We denote by |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='|Lp the norm on LppOq or LppOq and by }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' }W m,q the norm on W m,ppOq or Wm,ppOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For p “ 2 the function space W m,2pOq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Wm,2pOq) is denoted by HmpOq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HmpOq) and its norm will be denoted by |¨|Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By H1 0pOq we mean the space of functions in H1 that vanish on the boundary BO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The inner product on L2pOq will be denoted by p¨, ¨q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Following the notations using in [38] for the Navier-Stokes model, we introduce the following space V “ tv P C8 c pO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' R2q : such that ∇ ¨ v “ 0u, and define the spaces H and V as the closure of V in L2pOq and H1 0pOq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We endow H with the scalar product and norm of L2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' As usual, we equip the space V with the gradient-scalar product and the gradient-norm |∇¨|L2, which is equivalent to the H1 0pOq-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' As usual, P denotes the Helmholtz projection from L2pOq onto H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' It is also known that V is dense in H and that the embedding is continuous and compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Identifying H with its dual, we have the Gelfand triple V ãÑ H ãÑ V ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We define the Newmann Laplacian operator on L2pOq by A1φ “ ´∆φ for all φ P DpA1q where DpA1q “ tφ P H2pOq : Bφ Bν “ 0, on BOu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ It is known that A1 is a non-negative self-adjoint operator in L2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' As we are working on a bounded domain, A1 has compact resolvent, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, there exists an orthonormal basis tϕiu8 i“1 Ă C8pOq of L2pOq consisting of the eigenfunctions of the Neumann Laplacian A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Also we have the dense and compact embeddings H2pOq ãÑ H1pOq ãÑ L2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now we define the Hilbert space H by H “ H ˆ H1pOq, endowed with the scalar product whose associated norm is given by |pu, cq|2 H “ |u|2 L2 ` |c|2 H1 , pu, cq P H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We introduce the bilinear operators B0, B1 and R2 and their associated trilinear forms b0, b1 and r2 respectively as follows: pB0pu, vq, wq “ ż O rpupxq ¨ ∇qvpxqs ¨ wpxqdx “ b0pu, v, wq, @u P V, v P V, w P V, pB1pu, cq, ψq “ ż O upxq ¨ ∇cpxqψpxqdx “ b1pu, c, ψq, @u P V, c P H1pOq, ψ P H1pOq, pR2pn, cq, ψq “ ż O ∇ ¨ pnpxq∇cpxqqψpxqdx “ ´ ż O npxq∇cpxq ¨ ∇ψpxqdx “ r2pn, c, ψq, @n P L2pOq, c P H1pOq, ψ P H3pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' It is well known in [38, Chapter II, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2] that the operator B0 is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The operator B1 is well-defined for u P V , c P H1pOq and ψ P H1pOq since by the H¨older inequality and the Sobolev embedding of H1pOq into L4pOq, we have pB1pu, cq, ψq ď |u|L4 |∇c|L2 |ψ|L4 ď K |∇u|L2 |c|H1 |ψ|H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In a similar way, we can also check that the operator R2 is well-defined for n P L2pOq, c P H1pOq and ψ P H1pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In fact, in addition to the H¨older inequality, by using the Sobolev embedding of H2pOq into L8pOq, we see that pR2pn, cq, ψq ď |n|L2 |∇c|L2 |∇ψ|L8 ď |n|L2 |c|H1 |ψ|H3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We also introduce the following coupling mappings R0 and R1 pR0pn, Φq, vq “ ż O npxq∇Φpxq ¨ vpxqdx, @n P L2pOq, v P H, Φ P W 1,8pOq, pR1pn, cq, ψq “ ż O npxqfpcpxqqψpxqdx, @n P L2pOq, c P L8pOq, ψ P L2pOq, f P L8pRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We note that the operators R0 and R1 are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Indeed, for n P L2pOq, v P H and Φ P W 1,8pOq we see that pR0pn, Φq, vq ď |Φ|W 1,8 |n|L2 |v|L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Further, for n P L2pOq, c P L8pOq, ψ P L2pOq and f P L8pRq, we also see that pR1pn, cq, ψq ď |fpcq|L8 |n|L2 |ψ|L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hereafter, A :“ pΩ, F, pFtqtPr0,Ts, Pq will be a complete probability space equipped with a filtration pFtqtPr0,Ts satisfying the usual conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' the filtration is right-continuous and ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 5 all null sets of F are elements of F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let U be a separable Hilbert space with basis teku8 k“1 and W be a cylindrical Wiener process over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In particular, according to [12, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3] the Wiener process t ÞÑ Wt can be expressed as Wt “ 8 ÿ k“1 W k t ek, for all t P r0, Ts, where tW k : k P Nu is a family of mutually independent standard R-valued Brownian motion over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For any Hilbert space X, we will denote by L2pU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Xq the separable Hilbert space of Hilbert-Schmidt operators from U into X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For a separable Banach space X, p P r1, 8q and T ą 0 we denote by Mp Ap0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Xq the space of all processes ψ P LppΩˆp0, Tq, dPbdt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Xq over A, being tFtutPr0,Ts-progressively measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We denote by LppΩ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Xqq, 1 ď p ă 8, the space of all continuous and tFtutPr0,Ts-progressively measurable X-valued processes tψt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 0 ď t ď Tu over A satisfying E « sup tPr0,Ts }ψt}p X ff ă `8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' If Y is a Banach space, we will denote by LpX, Y q the space of bounded linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' From the theory of stochastic integration on infinite dimensional Hilbert space (see [12, Chapter 4]), for any process ρ P M2 Ap0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hqq, the stochastic integral of ρ with respect to the Wiener process t ÞÑ Wt is denoted by ż t 0 ρpsqdWs, 0 ď t ď T, and is defined as the unique continuous H-valued martingale over A, such that for all h P H, we have ˆż t 0 ρpsqdWs, h ˙ H “ 8 ÿ k“1 ż t 0 pρpsqek, hqHdW k s , 0 ď t ď T, where the integral with respect to dW k s is understood in the sense of Itˆo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We introduce now the following conditions on the parameters and functions involved in the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For the parameter functions χ, f and Φ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2), we assume that χpcq is a non-negative constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' χpcq “ χ ą 0 and require that f and Φ satisfy f P C1pr0, 8qq, fp0q “ 0, and f ą 0, f 1 ą 0 in p0, 8q, Φ is time-independent and Φ P W 1,8pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) Throughout this paper, we set (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) Kf :“ χ2 2δ min 0ďcď|c0|L8 f 1 ` 1 min 0ďcď|c0|L8 f 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Furthermore, we consider a family of vector fields tσ1, σ2u satisfying the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (A1) For k P t1, 2u, σk :“ pσ1 k, σ2 kq P W 1,8pOq ˆ W 1,8pOq and σk “ 0 on BO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (A2) σk is a divergence free vector fields, that is ∇ ¨ σk “ 0, for k “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ (A3) The matrix-valued function q : O ˆ O Ñ R2 b R2 defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3) qi,jpx, yq “ 2ÿ k“1 σi kpxqσj kpyq, @i, j “ 1, 2 and @x, y P O, satisfies qpx, xq “ IdR2 for any x P O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Before introducing the other standing assumptions used in this paper, we shall make few important remarks and observations on Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2 and the noise 2ÿ k“1 σk ¨ ∇c ˝ dβk t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Setting for k “ 1, 2, σkpxq “ $ & % gk if x P ¯OzBO, 0 if x P BO, where tg1, g2u is the canonical basis of R2, the family of vector fields tσ1, σ2u satisfies (A1), (A2) and (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hereafter we will use the following notation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4) |σ|L8 “ ˜ 2ÿ k“1 |σk|2 L8 ¸1{2 and |σ|W 1,8 “ ˜ 2ÿ k“1 |σk|2 W 1,8 ¸1{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Owing to [17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 65, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1], the Stratonovich integral γ şt 0 σk ¨ ∇cpsq ˝ dβk s can be expressed as the Itˆo integral with a correction term as follows: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5) γ ż t 0 σk ¨ ∇cpsq ˝ dβk s “ 1 2 ż t 0 Dcpγσk ¨ ∇cpsqqpγσk ¨ ∇cpsqqds ` γ ż t 0 σk ¨ ∇cpsqdβk s , where, Dcpγσk ¨ ∇cq denotes the Fr´echet derivative of γσk ¨ ∇c with respect to c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' If Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2 holds, then for all t P r0, Ts, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6) 1 2 ż t 0 2ÿ k“1 Dcpγσk ¨ ∇cpsqqpγσk ¨ ∇cpsqqds “ γ2 2 ż t 0 ∆cpsqds, c P H2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let c P H2pOq and t P r0, Ts be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then for all s P r0, ts and k “ 1, 2, 2ÿ k“1 Dcpγσk ¨ ∇cqpγσk ¨ ∇cq “ γ 2ÿ k“1 σk ¨ ∇pγσk ¨ ∇cq “ γ2 2ÿ k“1 σk ¨ ∇pσk ¨ ∇cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since ∇ ¨ σk “ 0, we remark that σk ¨ ∇c “ ∇ ¨ pcσkq and therefore, γ2 2ÿ k“1 σk ¨ ∇pσk ¨ ∇cq “ γ2 2ÿ k“1 σk ¨ ∇p∇ ¨ pcσkqq “ γ2 2ÿ k“1 ∇ ¨ pσk∇ ¨ pcσkqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7) For the second equality we have used once more the fact that ∇ ¨ σk “ 0 for all k “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since σk “ pσ1 k, σ2 kq P W 1,8pOq ˆ W 1,8pOq and c P H2pOq ãÑ L8pOq, we can apply the differentiation of product formula given in [2, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 269] to obtain, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8) 2ÿ k“1 ∇ ¨ pσkp∇ ¨ pcσkqq “ 2ÿ i,j“1 B2 BxiBxj pqijpx, xqcq ´ ∇ ¨ ˜˜ 2ÿ k“1 σk ¨ ∇σk ¸ c ¸ , ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 7 where σk ¨ ∇σk is the vector field with components pσk ¨ ∇σkqi “ 2ÿ j“1 σj k B Bxj σi k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Applying the differentiation of product formula once more, for j “ 1, 2, we see that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9) 2ÿ k“1 2ÿ j“1 p∇σk ¨ σkqi “ 2ÿ j“1 B Bxj qijpx, xq ´ 2ÿ k“1 σi k∇ ¨ σk “ 2ÿ j“1 B Bxj δij “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9), we have used the fact that ∇ ¨ σk “ 0 and also the fact that qij “ δij (see (A3) of assumption 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9), we infer that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10) 2ÿ k“1 ∇ ¨ pσk∇ ¨ pcσkqq “ 2ÿ i,j“1 B2 BxiBxj pqijpx, xqcq “ 2ÿ i,j“1 B2 BxiBxj pδijcq “ ∆c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7), we derive (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6) which completes the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Define for k P t1, 2u, a map φk : H1pOq Ñ L2pOq by φkpcq “ σk ¨ ∇c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then, the map φ : H1pOq Ñ L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqq given by φpcqphq “ 2ÿ k“1 φkpcqhk, c P H1pOq, h “ ph1, h2q P R2, is well defined under the condition (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let tg1, g2u be the orthonormal basis of R2 then φpcqpgkq “ φkpcq, for all c P H1pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let β “ pβ1, β2q be a standard two dimensional Brownian motion over A, independent of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We will repeatedly use the following notation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='11) φpcqdβs “ 2ÿ k“1 φkpcqdβk s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We recall that throughout this paper, the symbols K, KGN and Ki, i P N will denote positive constants which may change from one line to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let g : H Ñ L2pU, Hq be a continuous mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In particular, there exists a positive constant Lg such that for any pu, cq P H, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12) |gpu, cq|L2pU,Hq ď Lgp1 ` |pu, cq|Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let g : H Ñ L2pU, Hq be a Lipschitz-continuous mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In particular, there exists a positive constant LLip such that for all pui, ciq P H, i “ 1, 2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13) |gpu1, c1q ´ gpu2, c2q|L2pU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Hq ď LLip |pu1 ´ u2, c1 ´ c2q|H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using the previous notations, setting ξ “ η ` γ2 2 , and taking into account Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2, the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) can formally be written in the following abstract form uptq ` ż t 0 rηA0upsq ` B0pupsq, upsqsds “ u0 ` ż t 0 R0pnpsq, Φqds ` ż t 0 gpupsq, cpsqqdWs, cptq ` ż t 0 rξA1cpsq ` B1pupsq, cpsqqsds “ c0 ´ ż t 0 R1pnpsq, cpsqqds ` γ ż t 0 φpcpsqqdβs, nptq ` ż t 0 rδA1npsq ` B1pupsq, npsqqsds “ n0 ´ ż t 0 R2pnpsq, cpsqqds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14) 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ These equations are understood being valid in V ˚, H´2pOq and H´3pOq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We end this section by introduce some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let Y be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By Cpr0, Ts : Y q we denote the space of continuous functions v : r0, Ts Ñ Y with the topology induced by the norm defined by |v|Cpr0,Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Y q :“ sup 0ďsďT }vpsq}Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' With L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Y q we denote the space of measurable functions v : r0, Ts Ñ Y with the topology generated by the norm |v|L2p0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Y q :“ ˆż T 0 }vpsq}2 Y ds ˙1{2 , while by L2 wp0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Y q we denote the space of measurable functions v : r0, Ts Ñ Y with weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For a Hilbert space X, we denote by Xw the space X endowed with the weak topology and by Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Xwq we denote the space of functions v : r0, Ts Ñ Xw that are weakly continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' THE MAIN RESULT: EXISTENCE OF PROBABILISTIC STRONG SOLUTIONS This section is devoted to the statement of the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Before proceeding further, let us state the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A probabilistic strong solution of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) is a HˆH1pOqˆL2pOq-valued stochastic process pu, c, nq such that i): We have P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' u P Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hq X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V q, c P Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H2pOqq, n P Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2 wpOqq X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H´3pOqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ii): pu, c, nq : r0, Ts ˆ Ω Ñ H ˆ H1pOq ˆ L2pOq is progessively measurable and for all p ě 1 E sup 0ďsďT |upsq|p L2 ` E ˆż T 0 |∇upsq|2 L2 ds ˙p ă 8, E ˆż T 0 |npsq|2 L2 ds ˙p ă 8, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) and E sup 0ďsďT |cpsq|p H1 ` E ˆż T 0 |cpsq|2 H2 ds ˙p ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' iii): for all t P r0, Ts the following identity holds P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' uptq ` ż t 0 rηA0upsq ` B0pupsq, upsqqsds “ u0 ` ż t 0 R0pnpsq, Φqds ` ż t 0 gpupsq, cpsqqdWs, cptq ` ż t 0 rξA1cpsq ` B1pupsq, cpsqqsds “ c0 ´ ż t 0 R1pnpsq, cpsqqds ` γ ż t 0 φpcpsqqdβs, nptq ` ż t 0 rδA1npsq ` B1pupsq, npsqqsds “ n0 ´ ż t 0 R2pnpsq, cpsqqds, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 9 in V ˚, H´2pOq and H´3pOq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let us now present the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1, Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2, Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3, and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4 be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let us assume that the initial data pu0, c0, n0q belong to H ˆ L8pOq X H1pOq ˆ L2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In addition, let us assume that c0pxq ą 0, n0pxq ą 0 for all x P O and ż O n0pxq ln n0pxqdx ă 8, as well as 4Kf max 0ďcď|c0|L8 f 2 min 0ďcď|c0|L8 f 1 ď δ, γ2 ď min ´ ξ, ξ 2K0 ¯ 6 |σ|2 L8 , and γ2p ď 3pξp 22p`1 |σ|2p L8 8p , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3) for all p ě 2, where K0 is positive constant such that |ψ|2 H2 ď K0p|∆ψ|2 L2 ` |ψ|2 H1q, for all ψ P H2pOq (see [35, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 404] for the existence of such constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then, there exists a unique probabilistic strong solution to the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) in the sense of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We note that in the case where fpcq “ c, then we have Kf “ χ2`2δ 2δ , and the first inequality of the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3) is satisfied if |c0|L8 ď δ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 2 2 a χ2 ` 2δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Furthermore, the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3) have been introduced in order to control the cell term in the inequality (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=') and the higher regularity of the noise term on the c-equation in the inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='36) and (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' However, it is known in [24, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1] that, for the two-dimensional deterministic chemotaxis system, there exists a critical mass phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' When the total initial mass of cells ş O n0pxqdx above a critical mass mcrit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ş O n0pxqdx ą mcrit), solutions blow-up in finite time, otherwise, all solutions remain bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' While, for the two-dimensional stochastic chemotaxis system, it is shown in [26] that, if the chemotaxis sensibility χ is sufficiently large, then blow-up occurs with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For the coupled system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2), despite the rapid flow of fluid, we also expect some phenomenons to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then, it is important to ask oneself what will happen if the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3) is violated?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The answer to this question will be given by the study of the blow-up criterion of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In order to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2, we will first show that problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) has a probabilistic weak solution, see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2, then prove the non-negativity property and the L8-stability property of weak solution, which give us the possibility to prove the pathwise uniqueness, and finally apply the Yamada-Watanabe Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' But before proceeding further, we now introduce the concept of a probabilistic weak solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A weak probabilistic solution of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) is a system p¯Ω, ¯F, ¯F, ¯P, pu, c, nq, p ¯ W , ¯βqq, where i): p¯Ω, ¯F, ¯F, ¯Pq is a filtered probability space, 10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ ii): p ¯W, ¯βq is a cylindrical Wiener processes on U ˆ R2 over p¯Ω, ¯F, ¯F, ¯Pq, iii): and pu, c, nq : r0, Ts ˆ ¯Ω Ñ H ˆ L2pOq is a strong solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) with driving noise p ¯W, ¯βq on the filtered probability space p¯Ω, ¯F, ¯F, ¯Pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The existence of weak solution to our problem is given in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let us assume that Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1, Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2 and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let pu0, c0, n0q P H ˆ L8pOq X H1pOq ˆ L2pOq, such that c0pxq ą 0, n0pxq ą 0 for all x P O and ż O n0pxq ln n0pxqdx ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We also assume that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then, there exists at least one probabilistic weak solution to the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) in the sense of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5, which is very technical is postponed to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Next, we prove some properties of probabilistic weak solutions to the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) such as the non-negativity and the L8-stability which will be useful for the proof of the pathwise uniqueness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In fact, the main ingredient for the pathwise uniqueness is the L8-stability property but to obtain this property we will need the non-negativity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1 and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let p¯Ω, ¯F, ¯F, ¯P, pu, c, nq, p ¯ W , ¯βqq be a probabilistic weak solution to the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' If c0 ą 0 and n0 ą 0, then the following inequality hold ¯P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4) nptq ą 0, and cptq ą 0, for all t P r0, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We will follow the idea developed in [19, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1] combined with the idea of [14, Lemma 14] and [5, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let t P r0, Ts arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We then define n´ptq :“ maxp´nptq, 0q and remark that n´ptq P W 2,2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, we multiply equation p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14q3 by n´ptq, integrate over O, and use an integration-by-parts to obtain ¯P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 1 2 d dt |n´ptq|2 L2 “ ´ ż O upt, xq ¨ ∇n´pt, xqn´pt, xqdx ´ δ |∇n´ptq|2 L2 ´ χ ż O npt, xq∇cpt, xq∇n´pt, xqdx “ 1 2 ż O n2 ´pt, xq∇ ¨ upt, xqdx ´ δ |∇n´ptq|2 L2 ` χ ż O n´pt, xq∇cpt, xq∇n´pt, xqdx (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5) ď ´δ |∇n´ptq|2 L2 ` χ |n´ptq|L4 |∇cptq|L4 |∇n´ptq|L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By the Gagliardo-Nirenberg-Sobolev inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7) and the Young inequality, we note that χ |n´|L4 |∇c|L4 |∇n´|L2 ď Kp|n´|1{2 L2 |∇n´|1{2 L2 ` |n´|L2q |∇c|L4 |∇n´|L2 ď K |n´|1{2 L2 |∇c|L4 |∇n´|3{2 L2 ` K |n´|L2 |∇c|L4 |∇n´|L2 ď δ 2 |∇n´|2 L2 ` K |n´|2 L2 p|∇c|4 L4 ` |∇c|2 L4q (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6) ď δ 2 |∇n´|2 L2 ` K |n´|2 L2 p|∇c|4 L4 ` 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 11 Owing to the fact that ¯P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' c P Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H2pOqq, by the following Gagliardo-Niremberg inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7) |f|L4 ď KGNp|f|1{2 L2 |∇f|1{2 L2 ` |f|L2q, f P W 1,2pOq, we note that for all t P r0, Ts and ¯P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ż t 0 p|∇cpsq|4 L4 ` 1qds ď ż t 0 |∇cpsq|4 L4 ds ` t ď K ż T 0 |∇cpsq|2 L2 |cpsq|2 H2 ds ` ż T 0 |∇cpsq|4 L2 ds ` T ď K sup 0ďsďT |∇cpsq|2 L2 ż T 0 |cpsq|2 H2 ds ` sup 0ďsďT |∇cpsq|2 L2 ż T 0 |∇cpsq|2 L2 ds ` T ď K sup 0ďsďT |cpsq|2 H1 ż T 0 |cpsq|2 H2 ds ` T ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, integrating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5) over r0, Ts, and using the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6), we infer that ¯P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' |n´ptq|2 L2 ď |n´p0q|2 L2 ` K ż t 0 p|∇cpsq|4 L4 ` |∇cpsq|2 L4q |n´psq|2 L2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to Gronwall’s inequality, we derive that |n´ptq|2 L2 ď |pn0q´|2 L2 exp ˆ K ż t 0 p|∇cpsq|4 L4 ` |∇cpsq|2 L4qds ˙ , which implies that ¯P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s, n´ptq “ 0 and the non-negativity of nptq follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For the proof of the non-negativity property of cptq, the main idea is to apply the Itˆo formula to the function Ψ : H2pOq Ñ R defined by Ψpzq “ ş O z2 ´pxqdx where z´ “ maxp´z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since the function Ψ is not twice Fr´echet differentiable, we will follow the idea of [14, Lemma 14] (see also [5, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7]) by introducing the following approximation of Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let ϕ : R Ñ r´1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 0s be a C8 class increasing function such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8) ϕpsq “ # ´1 if s P p´8, ´2s 0 if s P r´1, `8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let tψhuhPN be a sequence of smooth functions defined by ψhpyq “ y2ϕphyq, for all y P R and h P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For any h P N, we consider the following sequence of function Ψh : H2pOq Ñ R defined by Ψhpcq “ ż O ψhpcpxqqdx, for c P H2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We note that the mapping Ψh is twice Fr´echet-differentiable and Ψ1 hpcqpkq “ 2 ż O cpxqϕphcpxqqkpxqdx ` h ż O c2pxqϕ1phcpxqqkpxqdx, @c, k P H2pOq, as well as Ψ 2 hpcqpz, kq “ m2 ż O c2pxqϕ 2phcpxqqzpxqkpxqdx ` 4h ż O cpxqϕ1phcpxqqzpxqkpxqdx ` 2 ż O ϕphcpxqqzpxqkpxqdx, @c, z, k P H2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ By applying the Itˆo formula to t ÞÑ Ψhpcptqq, we obtain ¯P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Ψhpcptqq ´ Ψhpcp0qq “ ż t 0 Ψ1 hpcpsqq pupsq ¨ ∇cpsq ` ξ∆cpsq ´ npsqfpcpsqqq ds ` 1 2 ż t 0 2ÿ k“1 Ψ 2 hpcpsqq pγφkpcpsqq, γφkpcpsqqq ds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9) ` γ 2ÿ k“1 ż t 0 Ψ1 hpcpsqqpφkpcpsqqqd¯βk s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, we will find a simpler representation of the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For a fixed k “ 1, 2, we remark that for all h ě 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10) hϕ1phcqσk ¨ ∇c “ σk ¨ phϕ1phcq∇cq “ σk ¨ ∇pϕphcqq, and also that 2cσk ¨ ∇c “ σk ¨ ∇c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, for any h ě 1 thanks to an integration-by-parts and the fact that σk “ 0 on BO, we have that for any h P N, Ψ1 hpcqpφkpcqq “ 2 ż O cpxqϕphcpxqqσkpxq ¨ ∇cpxqdx ` h ż O c2pxqϕ1phcpxqqσkpxq ¨ ∇cpxqdx “ ż O ϕphcpxqqσkpxq ¨ ∇c2pxqdx ` ż O c2pxqσkpxq ¨ ∇pϕphcpxqqqdx “ ´ ż O c2pxq∇ ¨ pϕphcpxqqσkpxqqdx ` ż BO c2pσqϕphcpσqqσkpσq ¨ νdσ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='11) ` ż O c2pxqσkpxq ¨ ∇pϕphcpxqqqdx “ ´ ż O c2pxq∇ ¨ pϕphcpxqqσkpxqqdx ` ż O c2pxqσkpxq ¨ ∇pϕphcpxqqqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Owing to the fact that ∇ ¨ σk “ 0, we derive that Ψ1 hpcqpφkpcqq “ ´ ż O c2pxqϕphcpxqq∇ ¨ σkpxqdx ´ ż O c2pxqσkpxq ¨ ∇pϕphcpxqqqdx ` ż O c2pxqσkpxq ¨ ∇pϕphcpxqqqdx (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12) “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We note that 2ÿ k“1 σk ¨ ∇cσk ¨ ∇c “ 2ÿ k“1 2ÿ i,j“1 σi kσj k Bc Bxi Bc Bxj “ 2ÿ i,j“1 qijpx, xq Bc Bxi Bc Bxj (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13) “ 2ÿ i,j“1 δij Bc Bxi Bc Bxj “ 2ÿ i“1 Bc Bxi Bc Bxi “ |∇c|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 13 Therefore, 2ÿ k“1 Ψ 2 hpcq pγφkpcq, γφkpcqq “ γ2h2 ż O c2pxqϕ 2phcpxqq |∇cpxq|2 dx ` 4hγ2 ż O cpxqϕ1phcpxqq |∇cpxq|2 dx ` 2γ2 ż O ϕphcpxqq |∇cpxq|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' On the other hand, by integration-by-parts, we get γ2Ψ1 hpcqp∆cq “ 2γ2 ż O cpxqϕphcpxqq∆cpxqdx ` hγ2 ż O c2pxqϕ1phcpxqq∆cpxqdx “ ´2γ2 ż O ∇cpxq ¨ ∇pcpxqϕphcpxqqqdx ´ hγ2 ż O ∇cpxq ¨ ∇pc2pxqϕ1phcpxqqqdx ` 2γ2 ż BO Bcpσq Bν ϕphcpσqq∆cpσqdσ ` 2hγ2 ż BO Bcpσq Bν cpσqϕ1phcpσqq∆cpσqdσ “ ´2γ2 ż O ϕphcpxqq |∇cpxq|2 dx ´ 2hγ2 ż O cpxqϕ1phcpxqq |∇cpxq|2 dx (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14) ´ 2hγ2 ż O cpxqϕ1phcpxqq |∇cpxq|2 dx ´ γ2h2 ż O c2pxqϕ 2phcpxqq |∇cpxq|2 dx “ ´ 2ÿ k“1 Ψ 2 hpcq pγφkpcq, γφkpcqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In the equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14), we have used the fact that Bc Bν vanishes on BO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Therefore, recalling that ξ “ η ` γ2 2 and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14), the equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9) is equivalent to ż O ψhpcpt, xqqdx ´ ż O ψhpc0pxqqdx “ ż t 0 Ψ1 hpcpsqq pupsq ¨ ∇cpsq ` η∆cpsq ´ npsqfpcpsqqq ds, from which along with the passage to the limit as h Ñ 8 we infer that ´ ż O c2 ´pt, xqdx ` ż O pc0pxqq2 ´dx “ ´2 ż t 0 ż O ppups, xq ¨ ∇cps, xq ` η∆cps, xq ´ nps, xqfpcps, xqqqq cps, xq1tcps,xqă0udxds “ 2 ż t 0 ż O ´ η |∇cps, xq|2 ` nps, xqfpcps, xqqcps, xq ¯ 1tcps,xqă0udxds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We note that, in the last line, we used an integration-by-parts and the fact that ∇ ¨ u “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By the mean value theorem, the fact that fp0q “ 0, and f 1 ą 0 as well as 1tcă0u ą 0, c2 ą 0, and n ą 0, we deduce that |c´ptq|2 L2 ď |pc0q´|2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This implies that c´ptq “ 0 ¯P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' and end the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ With the non-negativity of probabilistic weak solutions in hand, we are able now to state and prove the L8-stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Under the same assumptions as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6, if p¯Ω, ¯F, ¯F, ¯P, pu, c, nq, p ¯ W , ¯βqq is a probabilistic weak solution to the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2), then for all t P r0, Ts (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='15) |cptq|L8 ď |c0|L8 , ¯P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 14 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The idea of the proof comes from [19, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We apply the Itˆo formula to the process t ÞÑ Ψpcptqq :“ ş O cppt, xqdx, for any p ě 2 and evaluate the limit as p tends to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let Ψ : H2pOq Ñ R be the functional defined by Ψpcq “ ş O cppxqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Note that this mapping is twice Fr´echet-differentiable and Ψ1pcqphq “ p ż M cp´1pxqhpxqdx, @c, h P H2pOq, Ψ 2pcqph, kq “ ppp ´ 1q ż M cp´2pxqhpxqkpxqdx, @c, h, k P H2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Applying the Itˆo formula to the process t ÞÑ Ψpcptqq, yields Ψpcptqq ´ Ψpcp0qq “ ż t 0 Ψ1pcpsqq pupsq ¨ ∇cpsq ` ξ∆cpsq ´ npsqfpcpsqqq ds ` 1 2 ż t 0 2ÿ k“1 Ψ 2pcpsqq pγφkpcpsqq, γφkpcpsqqq ds ` γ 2ÿ k“1 ż t 0 Ψ1pcpsqqpφkpcpsqqqd¯βk s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='16) By integration-by-parts, the divergence free property of σk and the fact that σk “ 0 on BO, we remark that for all k ě 1, Ψ1pcqpφkpcqq “ p ż O cp´1pxqσkpxq ¨ ∇cpxqdx “ ż O σkpxq ¨ ∇cppxqdx (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='17) “ ´ ż O cppxq∇ ¨ σkpxqdx ` ż BO cppσqσkpσq ¨ νdσ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This implies that the stochastic term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='16) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='18) ż O ∆cpxqcp´1pxqdx “ ´pp ´ 1q ż O |∇cpxq|2 cpxqp´2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since ∇ ¨ u “ 0, by integration by part, we infer that ż O upxq ¨ ∇cpxqcp´1pxqdx “ 1 p ż O upxq ¨ ∇cppxqdx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='19) Using the equalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='17), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='19), we deduce from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='17) that Ψpcptqq ´ Ψpc0q “ ż t 0 ż O ´ ´ppp ´ 2qξ |∇cps, xq|2 cp´2ps, xq ´ pnps, xqfpcps, xqqcp´1ps, xq ¯ dxds ` ppp ´ 1q 2 ż t 0 ż O cp´2psq 2ÿ k“1 σkpxq ¨ ∇cps, xqσkpxq ¨ ∇cps, xqdxds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='20) From the equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13), we get ř2 k“1 σk ¨ ∇cσk ¨ ∇c “ |∇c|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, the equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='20) becomes Ψpcptqq ´ Ψpc0q “ ż t 0 ż O ´ ´ppp ´ 2q |∇cps, xq|2 cp´2ps, xq ´ pnps, xqfpcps, xqqcp´1ps, xq ¯ dxds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using the non-negative property of n and c proved in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6 combined with the non-negativity of the function f, we infer from the last equality that for all p ě 2 and t P r0, Ts, |cptq|Lp ď |c0|Lp, which along with the passage to the limit p Ñ `8 completes the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1 (see [1, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14] for a detailed proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 15 We now proceed with the statement and proof of the pathwise uniqueness of the weak solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We assume that the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' If pΩ, F, tFtutPr0,Ts, P, pu1, c1, n1q, p ¯ W , ¯βqq and pΩ, F, tFtutPr0,Ts, P, pu2, c2, n2q, p ¯W, ¯βqq are two weak probabilistic solutions of system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14) with the same initial data pu0, c0, n0q, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='21) pu1ptq, c1ptq, n1ptqq “ pu2ptq, c2ptq, n2ptqq P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' for all t P r0, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For t P r0, Ts, let pwptq, ψptq, ϕptqq “ pu1ptq ´ u2ptq, c1ptq ´ c2ptq, n1ptq ´ n2ptqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then this process satisfies pwp0q, ψp0q, ϕp0qq “ 0 and for all t P r0, Ts, we have wptq ` ż t 0 rηA0wpsq ` B0pwpsq, u1psqq ` B0pu2psq, wpsqqsds “ ż t 0 R0pϕpsq, Φqds ` ż t 0 rgpu1psq, c1psqq ´ gpu2psq, c2psqqsdWs, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='22) ψptq ` ż t 0 rξA1ψpsq ` B1pwpsq, c1psqq ` B1pu2psq, ψpsqqsds “ ´ ż t 0 rR1pn1psq, c1psqq ´ R1pn2psq, c2psqqsds ` γ ż t 0 φpψpsqqdβs, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='23) ϕptq ` ż t 0 rδA1ϕpsq ` B1pwpsq, n1psqq ` B1pu2psq, φpsqqsds “ ´ ż t 0 rR2pn1psq, c1psqq ´ R2pn2psq, c2psqqsds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='24) Using the fact that pB0pu2, wq, wq “ 0, we get by applying the Itˆo formula to t ÞÑ |wptq|2 L2 that |wptq|2 L2 ` 2η ż t 0 |∇wpsq|2 L2 ds “ ´2 ż t 0 pB0pwpsq, u1psqq, wpsqqds ` 2 ż t 0 pR0pϕpsq, Φq, wpsqqds ` ż t 0 |gpu1psq, c1psqq ´ gpu2psq, c2psqq|2 L2pU,Hq ds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='25) ` 2 ż t 0 pgpu1psq, c1psqq ´ gpu2psq, c2psqq, wpsqqdWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using the continuous embeddings V ãÑ H and H1pOq ãÑ L4pOq as well as the H¨older inequality and the Young inequality, we derive that 2 |pB0pw, u1q, wq| ď 2 |w|L4 |u1|L4 |w|L2 ď η 5 |∇w|2 L2 ` K |∇u1|2 L2 |w|2 L2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='26) 16 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ and 2 |pR0pϕ, Φq, wq| ď 2 |∇Φ|L8 |ϕ|L2 |w|L2 ď K |∇Φ|L8 |ϕ|L2 |∇w|L2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='27) ď η 5 |∇w|2 L2 ` K |Φ|2 W 1,8 |ϕ|2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='28) |gpu1, c1q ´ gpu2, c2q|2 L2pU,Hq ď L2 Lipp|w|2 L2 ` |ψ|2 H1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since ∇ ¨ σ1 “ ∇ ¨ σ2 “ 0, we obtain pφpψq, ψq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Futhermore, by the fact that ∇ ¨ u2 “ 0, we derive that pB1pu2, ψq, ψq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Next, we recall that (A3) implies |φpψq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q “ 2ÿ k“1 ż O |σkpxq ¨ ∇ψpxq|2 dx “ |∇ψ|2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, by applying the Itˆo formula to t ÞÑ |ψptq|2 H1, we see that |ψptq|2 H1 ` 2 ż t 0 ´ µ |∇ψpsq|2 L2 ` ξ |A1ψpsq|2 L2 ¯ ds “ ´2 ż t 0 pB1pwpsq, c1psqq, ψpsqqds ´ 2 ż t 0 pR1pn1psq, c1psqq ´ R1pn2psq, c2psqq, ψpsqqds ` 2 ż t 0 pB1pwpsq, c1psqq ` B1pu2psq, ψpsqq, A1ψpsqqds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='29) ´ 2 ż t 0 pR1pn1psq, c1psqq ´ R1pn2psq, c2psqq, A1ψpsqqds ` γ2 ż t 0 |∇φpψpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ds ` 2γ ż t 0 p∇φpψpsqq, ∇ψpsqqdβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Taking the L2-inner product of the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='24) with ϕ and adding the result to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='29), yield |ϕptq|2 L2 ` |ψptq|2 H1 ` 2 ż t 0 pµ |∇ψpsq|2 L2 ` ξ |A1ψpsq|2 L2 ` δ |∇ϕpsq|2 L2qds “ ´2 ż t 0 pB1pwpsq, c1psqq, ψpsqqds ´ 2 ż t 0 pR1pn1psq, c1psqq ´ R1pn2psq, c2psqq, ψpsqqds ` 2 ż t 0 pB1pwpsq, c1psqq ` B1pu2psq, ψpsqq, A1ψpsqqds ´ 2 ż t 0 pR1pn1psq, c1psqq ´ R1pn2psq, c2psqq, A1ψpsqqds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='30) ´ 2 ż t 0 rr2pϕpsq, c1psq, ϕpsqq ` r2pn2psq, ψpsq, ϕpsqqsds ` γ2 ż t 0 |∇φpψpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ds ´ 2 ż t 0 pB1pwpsq, n1psqq, ϕpsqqds ` 2γ ż t 0 p∇φpψpsqq, ∇ψpsqqdβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 17 Now, we give an estimate for the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Similarly to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='26), we have 2 |pB1pw, c1q, ψq| ď 2 |w|L4 |∇c1|L2 |ψpsq|L4 ď K |∇w|L2 |∇c1|L2 |ψ|H1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='31) ď η 5 |∇w|2 L2 ` K |∇c1|2 L2 |ψ|H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to the continuous embedding H1pOq ãÑ L4pOq and the L8-stability property proved in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7, we have 2pR1pn1, c1q ´ R1pn2, c2q, ψq ď 2 |R1pn1, c1q ´ R1pn2, c2q|L2 |ψ|L2 ď 4 |pfpc1q ´ fpc2qqn1|2 L2 ` 4 |fpc2qψ|2 L2 ` 2 |ψ|2 L2 ď 4 sup 0ďrď|c0|L8 pf 1prqq2 |n1ψ|2 L2 ` 4 sup 0ďrď|c0|L8 fprq |ψ|2 L2 ` 2 |ψ|2 L2 ď K |ψ|2 L4 |n1|2 L4 ` Kf |ψ|2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Applying the Galiardo-Nirenberg-Sobolev inequality, we arrive at 2pR1pn1, c1q ´ R1pn2, c2q, ψq ď K |ψ|2 H1 ´ |∇n1|L2 |n1|L2 ` |n1|2 L2 ¯ ` Kf |ψ|2 L2 ď K ´ |∇n1|L2 |n1|L2 ` |n1|2 L2 ¯ |ψ|2 H1 ` Kf |ψ|2 H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='32) Thanks to the Ladyzhenskaya, Galiardo-Nirenberg-Sobolev, and Young inequalities, we find that 2 |pB1pw, c1q, A1ψq| ď 2 |w|L4 |∇c1|L4 |A1ψ|L2 ď ξ 6 |A1ψ|2 L2 ` K |w|L2 |∇w|L2 ´ |c1|H2 |∇c1|L2 ` |∇c1|2 L2 ¯ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='33) ď ξ 6 |A1ψ|2 L2 ` η 5 |∇w|2 L2 ` K ´ |c1|2 H2 |∇c1|2 L2 ` |∇c1|4 L2 ¯ |w|2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We recall that there exist a positive constant K0, such that |ψ|2 H2 ď K0p|A1ψ|2 ` |ψ|2 H1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, using also the continuous embedding V ãÑ H, we obtain 2 |pB1pu2, ψq, A1ψq| ď 2 |u2|L4 |∇ψ|L4 |A1ψ|L2 ď ξ 6 |A1ψ|2 L2 ` K |u2|L2 |∇u2|L2 ´ |ψ|H2 |∇ψ|L2 ` |∇ψ|2 L2 ¯ ď ξ 6 |A1ψ|2 L2 ` K´1 0 ξ 6 |ψ|2 H2 ` K |u2|2 L2 |∇u2|2 L2 |∇ψ|2 L2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='34) ` K |u2|L2 |∇u2|L2 |∇ψ|2 L2 ď ξ 3 |A1ψ|2 L2 ` ξ 6 |ψ|2 H1 ` K ´ |u2|2 L2 |∇u2|2 L2 ` |∇u2|2 L2 ¯ |ψ|2 H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using a similarly argument as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='32), we arrive at 2 |pR1pn1, c1q ´ R1pn2, c2q, A1ψ| ď ξ 6 |A1ψ|2 L2 ` K |R1pn1, c1q ´ R1pn2, c2q|2 L2 ď ξ 6 |A1ψ|2 L2 ` K |ψ|2 L4 |n1|2 L4 ` Kf |ψ|2 L2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='35) ď ξ 6 |A1ψ|2 L2 ` Kf |ψ|2 H1 ` K ´ |∇n1|L2 |n1|L2 ` |n1|2 L2 ¯ |ψ|2 H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 18 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ By using an integration-by-parts and H¨older, and the Galiardo-Nirenberg-Sobolev inequalities, we see that 2 |pB1pw, n1q, ϕq| ď 2 ˇˇˇˇ ż O n1pxqwpxq ¨ ∇ϕpxqdx ˇˇˇˇ ď 2 |n1|L4 |w|L4 |∇ϕ|L2 ď δ 4 |∇ϕ|2 L2 ` K |w|L2 |∇w|L2 ´ |∇n1|L2 |n1|L2 ` |n1|2 L2 ¯ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='36) ď δ 4 |∇ϕ|2 L2 ` η 5 |∇w|2 L2 ` K ´ |∇n1|2 L2 |n1|2 L2 ` |n1|4 L2 ¯ |w|2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By applying the Young and Galiardo-Nirenberg-Sobolev inequalities we obtain 2 |r2pϕ, c1, ϕq| ď 2 |ϕ|L4 |∇c1|L4 |∇ϕ|L2 ď δ 4 |∇ϕ|2 L2 ` K |ϕ|2 L4 |∇c1|2 L4 ď δ 4 |∇ϕ|2 L2 ` K ´ |∇ϕ|L2 |ϕ|L2 ` |ϕ|2 L2 ¯ ´ |c1|H2 |∇c1|L2 ` |∇c1|2 L2 ¯ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='37) ď δ 2 |∇ϕ|2 L2 ` K ´ |c1|2 H2 |∇c1|2 L2 ` |∇c1|4 L2 ` |c1|H2 |∇c1|L2 ` |∇c1|2 L2 ¯ |ϕ|2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In a similarly way we have that 2 |r2pn2, ψ, ϕq| ď 2 |n2|L4 |∇ψ|L4 |∇ϕ|L2 ď δ 4 |∇ϕ|2 L2 ` K |n2|2 L4 ´ |ψ|H2 |∇ψ|L2 ` |∇ψ|2 L2 ¯ ď δ 4 |∇ϕ|2 L2 ` K´1 0 ξ 6 |ψ|2 H2 ` K |n2|4 L4 |∇ψ|2 L2 ` K |n2|2 L4 |∇ψ|2 L2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='38) ď δ 4 |∇ϕ|2 L2 ` ξ 6 |A1ψ|2 L2 ` ξ 6 |ψ|2 H1 ` K |n2|4 L2 |ψ|2 H1 ` K ´ |∇n2|L2 |n2|L2 ` |n2|2 L2 |∇n2|2 L2 ` |n2|2 L2 ¯ |ψ|2 H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3) we derive that γ2 |∇φpψq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q “ γ2 2ÿ k“1 ż O |∇pσkpxq ¨ ∇ψpxqq|2 dx ď 2γ2 2ÿ k“1 |σk|2 W 1,8 |∇ψ|2 L2 ` 2γ2 2ÿ k“1 |σk|2 L8 |ψ|2 H2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='39) ď p1 ` K0q2γ2 |σ|2 W 1,8 |∇ψ|2 L2 ` 2γ2K0 |σ|2 L8 |A1ψ|2 L2 ď ξ 6 |A1ψ|2 L2 ` p1 ` K0q2γ2 |σ|2 W 1,8 |ψ|2 H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, for t P r0, Ts and s P r0, ts, let us set Yptq :“ |uptq|2 L2 ` |cptq|2 H1 ` |ϕptq|2 L2 , ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 19 Zpsq :“ K |∇u1psq|2 L2 ` K |∇c1psq|2 L2 ` K ´ |∇n1psq|L2 |n1psq|L2 ` |n1psq|2 L2 ¯ ` K ´ |c1psq|2 H2 |∇c1psq|2 L2 ` |∇c1psq|4 L2 ¯ ` K ´ |u2psq|2 L2 |∇u2psq|2 L2 ` |∇u2psq|2 L2 ¯ ` K ´ |∇n1psq|L2 |n1psq|L2 ` |n1psq|2 L2 ¯ ` K ´ |∇n1psq|2 L2 |n1psq|2 L2 ` |n1psq|4 L2 ¯ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='40) ` K ´ |c1psq|2 H2 |∇c1psq|2 L2 ` |∇c1psq|4 L2 ` |c1psq|H2 |∇c1psq|L2 ` |∇c1psq|2 L2 ¯ ` K ´ |∇n2psq|L2 |n2psq|L2 ` |n2psq|2 L2 |∇n2psq|2 L2 ` |n2psq|2 L2 ` |n2psq|4 L2 ¯ , and θptq :“ exp ˆ ´ ż t 0 Zpsqds ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Applying the Itˆo formula to t ÞÑ θptq |uptq|2 L2, we derive that θptq |wptq|2 L2 ` 2η ż t 0 θpsq |∇wpsq|2 L2 ds ď 2 ż t 0 θpsqpB0pwpsq, u1psqq, wpsqqds ` 2 ż t 0 θpsqpR0pϕpsqq, wpsqqds ` ż t 0 θ1psq |wpsq|2 L2 ds ` ż t 0 θpsq |gpu1psq, c1psqq ´ gpu2psq, c2psqq|2 L2pU,Hq ds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='41) ` 2 ż t 0 θpsqpgpu1psq, c1psqq ´ gpu2psq, c2psqq, wpsqqdWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Applying the Itˆo formula once more to t ÞÑ θptqp|ϕptq|2 L2 ` |ψptq|2 H1q and adding the result with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='41) after taking into account the estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='26)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='31)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='39), we arrive at θptqYptq ` ż t 0 θpsq ´ η |∇wpsq|2 L2 ` µ |∇ψpsq|2 L2 ` ξ |A1ψpsq|2 L2 ¯ ds ď ˆ K |Φ|2 W 1,8 ` L2 Lip ` 2Kf ` ξ 3 ` p1 ` K0q2γ2 |σ|2 W 1,8 ˙ ż t 0 θpsqYpsqds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='42) ` 2γ ż t 0 θpsqp∇φpψpsqq, ∇ψpsqqdβs ` 2 ż t 0 θpsqpgpu1psq, c1psqq ´ gpu2psq, c2psqq, wpsqqdWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Next, taking the mathematical expectation yields EθptqYptq ` E ż t 0 θpsq ´ η |∇wpsq|2 L2 ` µ |∇ψpsq|2 L2 ` ξ |A1ψpsq|2 L2 ¯ ds ď ˆ K |Φ|2 W 1,8 ` L2 Lip ` 2Kf ` ξ 3 ` p1 ` K0q2γ2 |σ|2 W 1,8 ˙ E ż t 0 θpsqYpsqds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='43) From which along with the Gronwall inequality we infer that for any t P r0, Ts EθptqYptq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' It follows that for all t P r0, Ts, Yptq “ 0 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since the paths of pui, ci, niq, i “ 1, 2 are continuous P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', then pu1ptq, c1ptq, n1ptqq “ pu2ptq, c2ptq, n2ptqq, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', for all t P r0, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 20 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ □ With the existence and pathwise uniqueness results at hand we now prove the existence of strong solution stated in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The existence of a probabilistic weak solution to the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) is shown in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The pathwise uniqueness of probabilistic weak solutions is given by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thus, the existence and uniqueness of a probabilistic strong solution to the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) follows from the Yamada-Watanabe Theorem (see [30, Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8]), which states that the existence of weak probabilistic solution and the pathwise uniqueness imply the existence of a unique probabilistic strong solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' PROOF OF PROPOSITION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5 In this section, we will show Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We introduce a Galerkin approximation first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We then discuss the existence of the Galerkin approximation and prove the mass conservation property, the non-negativity property and the L8-norm satibility in finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using these properties, we prove priori estimates and by these a priori estimates, we show the tightness of the family of approximations, and pass in a second step, to the limit in the deterministic terms and the construction of the noise terms by exploiting the usual martingale representation theorem proved in [12, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Galerkin approximation and a priori uniform estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In this subsection, we will construct a family of approximations of the solutions and prove some crucial estimates satisfied uniformly by the approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For this propose, let us recall that there exists an orthonormal basis twiu8 i“1 of H consisting of the eigenfunctions of the Stokes operator A0 and an orthonormal basis tϕiu8 i“1 Ă C8pOq of L2pOq consisting of the eigenfunctions of the Neumann Laplacian operator A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For m P N, we will consider the following finite-dimensional spaces Hm “ spamtw1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', wmu, Hm “ spamtϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', ϕmu, Hm “ Hm ˆ Hm ˆ Hm, where we endow Hm with the following norm |pu, c, nq|2 Hm “ |u|2 L2 ` |c|2 L2 ` |n|2 L2 , pu, c, nq P Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Owing to the fact that Hm is a finite dimensional space, the L2pOq, H1pOq and H2pOq-norms are equivalent on this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We choose as in [44, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 335] nm 0 , cm 0 and um 0 such that nm 0 ą 0, nm 0 Ñ n0 in L2pOq, nm 0 ln nm 0 Ñ n0 ln n0 in L1pOq, cm 0 ą 0, |cm 0 |L8 ď |c0|L8 , cm 0 Ñ c0 in H1pOq, and um 0 Ñ u0 in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 21 We then consider on the filtered probability space pΩ, F, tFtutPr0,Ts, Pq the following finite dimensional problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For all t P r0, Ts umptq ` ż t 0 rηA0umpsq ` P1 mB0pumpsq, umpsqqsds “ um 0 ` ż t 0 P1 mR0pnmpsq, Φqds ` ż t 0 P1 mgpumpsq, cmpsqqdWs, cmptq ` ż t 0 rξA1cmpsq ` P2 mB1pumpsq, cmpsqqsds “ cm 0 ´ ż t 0 P2 mR1pnmpsq, cmpsqqds ` γ ż t 0 P2 mφpcmpsqqdβs, nmptq ` ż t 0 rδA1nmpsq ` P2 mB1pumpsq, nmpsqqsds “ nm 0 ´ ż t 0 P2 mR2pnmpsq, cmpsqqds, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) where P1 m and P2 m are the projection from H and L2pOq onto Hm and Hm, respectively, and their operator norms are equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For each m, we consider the following mapping Ψm : Hm Ñ Hm defined by Ψmpu, c, nq “ ¨ ˝ ηA0u ` P1 mB0pu, uq ´ P1 mR0pn, Φq ξA1c ` P2 mB1pu, cq ` P2 mR1pn, cq δA1n ` P2 mB1pu, nq ` P2 mR2pn, cq ˛ ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In the following lemma, we are going to state an important property of the mappings Ψm, m P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1 and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3 be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For each m P N, the mapping Ψm is locally Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' To be more precise, for each m P N and every r ą 0, there exists a constant Kr such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3) |Ψmpv1q ´ Ψmpv2q|Hm ď Kr |v1 ´ v2|Hm , for v1 “ pu1, c1, n1q, v2 “ pu2, c2, n2q P Hm with |vi|Hm ď r, i “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let v1 “ pu1, c1, n1q, v2 “ pu2, c2, n2q P Hm and v “ pu, c, nq P Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We assume that |vi|Hm ď r, i “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We have pΨmpv1q ´ Ψmpv2q, vqHm “ pηA0pu1 ´ u2q ` B0pu1, u1q ´ B0pu2, u2q ´ R0pn1, Φq ` R0pn2, Φq, uq ` pξA1pc1 ´ c2q ` B1pu1, c1q ´ B1pu2, c2q ` R1pn1, c1q ´ R1pn2, c2q, cq ` pδA1pn1 ´ n2q ` B1pu1, n1q ´ B1pu2, n2q ` R2pn1, c1q ´ R2pn2, c2q, nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4) Using the bilinearity of the operator B0, we see that |pB0pu1, u1q ´ B0pu2, u2q, uq| ď |pB0pu1 ´ u2, u1q, uq| ` |pB0pu2, u1 ´ u2q, uq| ď 2Kr |u1 ´ u2|L2 |u|L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By the H¨older inequality we also note that pR0pn1, Φq ´ R0pn2, Φq, uq ď ż O |n1 ´ n2| |∇Φ| |u| dx ď |∇Φ|L8 |n1 ´ n2|L2 |u|L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 22 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Since the space H1pOq is continuously embedded in the space LqpOq for any q ě 2, we have |pB1pu1, c1q ´ B1pu2, c2q, cq| ď |pB1pu1 ´ u2, c1q, cq| ` |pB1pu2, c1 ´ c2q, cq| ď |u1 ´ u2|L4 |∇c1|L2 |c|L4 ` |u2|L4 |∇pc1 ´ c2q|L2 |c|L4 ď p|∇pu1 ´ u2q|L2 |∇c1| ` |∇u2|L2 |∇pc1 ´ c2q|L2q |c|H1 ď Krp|∇pu1 ´ u2q|L2 ` |∇pc1 ´ c2q|L2q |c|H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In a similar way we show that |pB1pu1, n1q ´ B1pu2, n2q, nq| ď Krp|∇pu1 ´ u2q|L2 ` |∇pn1 ´ n2q|L2q |n|H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Owing to the fact that Hm Ă C8pOq and fp0q “ 0 as well as f P C1pr0, 8qq, we derive that |pR1pn1, c1q ´ R1pn2, c2q, cq| ď ż O |n1 ´ n2| fpc1q |c| dx ` ż O |n2| |fpc1q ´ fpc2q| |c| dx ď max 0ďcď|c1|L8 fpcq ż O |n1 ´ n2| |c| dx ` max 0ďcďmaxp|c1|L8,|c2|L8q f 1pcq ż O |n2| |c1 ´ c2| |c| dx ď max 0ďcďr fpcq |n1 ´ n2|L2 |c|L2 ` max 0ďcďr f 1 |n2|L4 |c1 ´ c2|L4 |c|L2 ď Krp|n1 ´ n2|L2 ` |c1 ´ c2|H1q |c|L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Also, we note that |pR2pn1, c1q ´ R2pn2, c2q, nq| ď ż O |n1 ´ n2| |∇c1| |∇n| dx ` ż O |n2| |∇pc1 ´ c2q| |∇n| dx ď |n1 ´ n2|L2 |∇c1|L4 |∇n|L2 ` |n2|L4 |∇pc1 ´ c2q|L4 |∇n|L2 ď Krp|n1 ´ n2|L2 ` |c1 ´ c2|H2q |n|H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Taking into account the fact that all norms are equivalent in finite dimensional space, and the fact that the operators A0 and A1 are linear, we infer these previous inequalities and equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ The existence of solutions to the finite dimensional problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) is classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In fact, due to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1, the mapping Ψm is locally Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Also by the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12), P1 mgp¨, ¨q is locally Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' From the linearity of φp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='q, we can easily see that P2 mφp¨q is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, by well known theory for finite dimensional stochastic differential equations with locally Lipschitz coefficients (see [31, Theorem 38, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 303] for full details) there exists a local solution of system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) with continuous paths in Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' That is, there exists a stopping time τm, a process t ÞÑ pumptq, cmptq, nmptqq such that τm ą 0 P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', and the stopped process t ÞÑ pumpt ^ τmq, cmpt ^ τmq, nmpt ^ τmqq satisfies the system of Itˆo equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) and has continuous paths in Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Moreover, if a process t ÞÑ p¯umptq, ¯cmptq, ¯nmptqq, and a stopping time σm constitute another local solution, then pump¨q, cmp¨q, nmp¨qq “ p¯ump¨q, ¯cmp¨q, ¯nmp¨qq, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' on r0, τm ^ σms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 23 We will show in what follows that the solutions pum, cm, nmq exist almost surely for every t P r0, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For this goal, it will be enough to show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5) τmpωq ą T, for almost all ω P Ω, and all m P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' To this aim, we will use some idea from [33, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 132, Proof of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since for all m P N, the deterministic integrand Ψm and the stochastic integrand P1 mg are locally Lipschitz, for each N P N, we can define the integrands ΨN m and P1 mgN, agreeing respectively with Ψm and P1 mg on the ball BN Hm :“ ␣ pv, ϕ, ψq P Hm : |pv, ϕ, ψq|Hm ă N ( , such that ΨN m and P1 mgN are globally Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' As consequence, since P2 mφ is already globally Lipschitz, [33, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 128, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2] guarantees that there is a unique solution puN m, cN m, nN mq to a system associated to the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) with ΨN m and P1 mgN (instead of Ψm and P1 mg) and defined on r0, `8q almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We then define a sequence of stopping times as follows for all m, N P N (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6) τ m N :“ inftt ą 0 : b |nN mptq|2 L2 ` |uN mptq|2 L2 ` |cN mptq|2 H1 ě Nu ^ N, where a ^ b :“ minta, bu for any real numbers a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For any fixed m P N, the sequence tτ m N uNPN is obviously increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Moreover [33, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 131, Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10] implies that for all N P N, pum, cm, nmq “ puN m, cN m, nN mq on r0, τ m N s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' From this last equality, we infer that the solution pum, cm, nmq of system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) is defined on r0, τ m N s for all N P N and hence, τm ą τ m N almost surely for all N P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Therefore, τm ě sup NPN τ m N , P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In order to prove the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5), it is sufficient to prove that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7) sup NPN τ m N ą T, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Before proving this, in the following lemma, we prove some properties of the local solution pum, cm, nmq of system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1 and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then for all m, N P N, the following equality and inequalities hold P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8) ż O nmpt ^ τ m N , xqdx “ ż O nm 0 pxqdx, for all t P r0, Ts, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9) nmpt ^ τ m N q ą 0, and cmpt ^ τ m N q ą 0, for all t P r0, Ts, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10) |cmpt ^ τ m N q|L8 ď |c0|L8 , for all t P r0, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In order to prove the non-negativity of nmpt^τ m N q and cmpt^τ m N q, we will follow the idea of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' But, instead of the Gagliardo-Niremberg-Sobolev inequality, we will use the equivalence of the norms on finite dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let N, m P N and t P r0, Ts be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For all s P r0, ts define nm´ps^τ m N q :“ maxp´nmps ^ τ m N q, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 24 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ We remark that nm´ps ^ τ m N q P W 2,2pOq and nm´ps ^ τ m N q “ 0 ¨ 1tnmps^τ m N qě0u ´ nmps ^ τ m N q ¨ 1tnmps^τ m N qă0u, ∇nm´ps ^ τ m N q “ 0 ¨ 1tnmps^τ m N qě0u ´ ∇nmps ^ τ m N q ¨ 1tnmps^τ m N qă0u, ∆nm´ps ^ τ m N q “ 0 ¨ 1tnmps^τ m N qě0u ´ ∆nmps ^ τ m N q ¨ 1tnmps^τ m N qă0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We can easily see also that for all s P r0, ts, dnmps ^ τ m N q dt nm´ps ^ τ m N q “ ´dnm´ps ^ τ m N q dt nm´ps ^ τ m N q, nm´ps ^ τ m N q∇nmps ^ τ m N q “ ´nm´ps ^ τ m N q∇nm´ps ^ τ m N q, ∆nmps ^ τ m N qnm´ps ^ τ m N q “ ´∆nm´ps ^ τ m N qnm´ps ^ τ m N q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, we multiply equation p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14q3 by nm´ps ^ τ m N q for any s P r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' integrate over O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' and use an integration-by-parts with the fact that ∇ ¨ um “ 0 to obtain 1 2 d dt ˇˇnm´ps ^ τ m N q ˇˇ2 L2 “ ´ ż O umps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xq ¨ ∇nm´ps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xqnm´ps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xqdx ´ δ ˇˇ∇nm´ps ^ τ m N q ˇˇ2 L2 ´ χ ż O nmps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xq∇cmps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xq∇nm´ps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xqdx “ 1 2 ż O n2 m´ps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xq∇ ¨ umps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xqdx ´ δ ˇˇ∇nm´ps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xq ˇˇ2 L2 ` χ ż O nm´ps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xq∇cmps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xq∇n´ps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xqdx ď ´δ ˇˇ∇nm´ps ^ τ m N q ˇˇ2 L2 ` χ ˇˇnm´ps ^ τ m N q ˇˇ L4 |∇cmps ^ τ m N q|L4 ˇˇ∇nm´ps ^ τ m N q ˇˇ L2 ď K ˇˇnm´ps ^ τ m N q ˇˇ2 H1 |cmps ^ τ m N q|H2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In the last line we have used the continuous embedding of H1pOq into L4pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since the L2pOq, H1pOq and H2pOq-norms are equivalent on Hm, we then infer from this last inequality that for all s P r0, ts, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='11) 1 2 d dt ˇˇnm´ps ^ τ m N q ˇˇ2 L2 ď Kpmq ˇˇnm´ps ^ τ m N q ˇˇ2 L2 |cmps ^ τ m N q|L2 , where Kpmq is a constant depending of m which is the dimension of the space Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Owing to the fact that P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' the paths of cm are continuous, we derive that sup 0ďsďt |cmps ^ τ m N q|L2 ă 8, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='11) over r0, ts we arrive at (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12) ˇˇnm´pt ^ τ m N q ˇˇ2 L2 ď |pnm 0 q´|2 L2 ` K ż t 0 |cmps ^ τ m N q|L2 ˇˇnm´ps ^ τ m N q ˇˇ2 L2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to the Gronwall inequality, we derive from the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12) that ˇˇnm´pt ^ τ m N q ˇˇ2 L2 ď |pnm 0 q´|2 L2 exp ˆ K ż t 0 |cmps ^ τ m N q|L2 ds ˙ , which implies that P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s, nm´pt ^ τ m N q “ 0 for all t P r0, Ts since by the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1), nm 0 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 25 The non-negativity property of cmpt^τ m N q is quite similar to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We consider the function Ψ : Hm Ñ R defined by Ψpcq “ ş O c2 ´pxqdx where c´ “ maxp´c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let tψhuhPN be a sequence of smooth functions defined by ψhpyq “ y2ϕphyq, for all y P R and h P N, where the function ϕ is defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We consider for any h ě 1, the following sequence of function Ψh : Hm Ñ R defined by Ψh “ ş O ψhpcpxqqdx, for c P Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The mapping Ψh is twice (Fr´echet) differentiable and its first and second derivatives are given by Ψ1 hpcqpzq “ 2 ż O cpxqϕphcpxqqzpxqdx ` h ż O c2pxqϕ1phcpxqqzpxqdx, @c, z P Hm, and Ψ 2 hpcqpz, kq “ h2 ż O c2pxqϕ 2phcpxqqzpxqkpxqdx ` 4h ż O cpxqϕ1phcpxqqzpxqkpxqdx ` 2 ż O ϕphcpxqqzpxqkpxqdx, @c, z, k P Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Applying the Itˆo formula to t ÞÑ Ψhpcmpt ^ τ m N qq, we obtain for all t P r0, Ts, Ψhpcmpt ^ τ m N qq ´ Ψhpcmp0qq “ ż t^τ m N 0 Ψ1 hpcmpsqq pumpsq ¨ ∇cmpsq ` ξ∆cmpsq ´ nmpsqfpcmpsqqq ds ` 1 2 ż t^τ m N 0 2ÿ k“1 Ψ 2 hpcmpsqq pγφkpcmpsqq, γφkpcmpsqqq ds ` γ 2ÿ k“1 ż t^τ m N 0 Ψ1 hpcmpsqqpφkpcmpsqqqdβk s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Similarly to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='11), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14), we can infer from this last equality that ż O ψhpcmpt ^ τ m N , xqqdx ´ ż O ψhpcm 0 pxqqdx “ ż t^τ m N 0 Ψ1 hpcmpsqq pumpsq ¨ ∇cmpsq ` η∆cmpsq ´ nmpsqfpcmpsqqq ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13) Now, observe that from the assumptions on the function ϕ, we infer that for all y P R we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14) lim hÝÑ8 ψhpyq “ ´y2 ¨ 1tyă0u “ ´y2 ´ and lim hÝÑ8 2yϕphyq “ ´2y ¨ 1tyă0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We note that for any y P R, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='15) lim hÝÑ8 hϕ1phyq “ 0, and also that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='16) |ψhpyq| ď Ky2 and ˇˇhϕ1phyq ˇˇ ď K |y| , for any y P R and for all h ě 1, where K ą 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 26 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='16) and applying the Lebesgue Dominated Convergence Theorem, we can pass to the limit as h tends to infinity in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In this way, we derive that ´ ż O c2 m´pt ^ τ m N , xqdx ` ż O pcm 0 pxqq2 ´dx “ ´2 ż t^τ m N 0 ż O ppumps, xq ¨ ∇cmps, xq ` η∆cmps, xqqqq cmps, xq1tcmps,xqă0udxds ` 2 ż t^τ m N 0 ż O nmps, xqfpcmps, xqqcmps, xq1tcmps,xqă0udxds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='17) “ 2 ż t^τ m N 0 ż O ´ η |∇cmps, xq|2 ` nmps, xqfpcmps, xqqcmps, xq ¯ 1tcmps,xqă0udxds, where we have used integration-by-parts and the fact that ∇¨um “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By the mean value theorem we know that, for all x P O, there exists a number λmpxq P pminp0, cmpxqq, maxp0, cmpxqqq such that fpcmpxqq ´ fp0q “ cmpxqf 1pλmpxqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By the fact that fp0q “ 0, we infer from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='17) that ˇˇcm´pt ^ τ m N q ˇˇ2 L2 ´ |pcm 0 q´|2 L2 “ ´2 ż t^τ m N 0 ż O nmps, xqf 1pλmps, xqqc2 mps, xq1tcmps,xqă0udxds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since f 1 ą 0 and 1tcmă0u ą 0 as well as on r0, t ^ τ m N s, c2 m ą 0 and nm ą 0, we deduce that ˇˇcm´pt ^ τ m N q ˇˇ2 L2 ď |pcm 0 q´|2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Owing to the fact that by the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) we have cm 0 ą 0, we derive that pcm 0 q´ “ 0 and therefore ˇˇcm´pt ^ τ m N q ˇˇ2 L2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This gives cm´pt ^ τ m N q “ 0 and implies that for all t P r0, Ts, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s, cmpt ^ τ m N q ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' It remains to prove the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The proof is similar to the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let p ě 2 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let Ψ : Hm Ñ R be the functional defined by Ψpcq “ ş O cppxqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Note that the mapping Ψ is twice (Fr´echet) differentiable and its first and second derivatives are given by Ψ1pcqpzq “ p ż O cp´1pxqzpxqdx, @c, z P Hm, Ψ 2pcqpz, kq “ ppp ´ 1q ż O cp´2pxqzpxqkpxqdx, @c, z, k P Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By applying the Itˆo formula to the process t ÞÑ Ψpcmpt ^ τ m N qq, we derive that for all t P r0, Ts, Ψpcmpt ^ τ m N qq ´ Ψpcmp0qq “ ż t^τ m N 0 Ψ1pcmpsqq pupsq ¨ ∇cmpsq ` ξ∆cmpsq ´ nmpsqGpcmpsqqq ds ` 1 2 ż t^τ m N 0 2ÿ k“1 Ψ 2pcmpsqq pγφkpcmpsqq, γφkpcmpsqqq ds ` γ 2ÿ k“1 ż t^τ m N 0 Ψ1pcmpsqqpφkpcmpsqqqdβk s , ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 27 from which and calculations similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='18), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='20) we derive from the last equality that Ψpcmpt ^ τ m N qq ´ Ψpcm 0 q “ ż t^τ m N 0 ż O ´ ´ppp ´ 2q |∇cmps, xq|2 cp´2 m ps, xq ´ pnmps, xqfpcmps, xqqcp´1 m ps, xq ¯ dxds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since for all s P r0, ts the quantities nmps ^ τ m N q, fpcmps ^ τ m N qq and cmps ^ τ m N q are positive P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s, we infer from the last equality that for all t P r0, Ts, Ψpcmpt ^ τ m N qq ď Ψpcm 0 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This implies that |cmpt ^ τ m N q|Lp ď |cm 0 |Lp for all p ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using the fact that |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='|Lp Ñ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='|L8 as p Ñ `8 and the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1), we obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Next, we introduce for any t P r0, Ts and m, N P N, the following Lyapunov functional Epnm, cm, umqpt ^ τ m N q “ ż O nmpt ^ τ m N q ln nmpt ^ τ m N qdx ` Kf |∇cmpt ^ τ m N q|2 L2 ` K4 η |umpt ^ τ m N q|2 L2 ` e´1 |O| , where K4 is some positive constant to be given later and Kf is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since x ln x ě ´e´1 for any x ą 0, we can easily see that for all t P r0, Ts, Epnm, cm, umqpt^τ m N q ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' As in [44] the property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) implies that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='18) Epnm 0 , cm 0 , um 0 q ď Epn0, c0, u0q, for all m ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In addition, taking into account the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10) and setting K “ minpKf, K4 η q the following holds for all t P r0, Ts, |pumptq, cmpt ^ τ m N q|2 H ď K´1Epnm, cm, umqpt ^ τ m N q ` K´1 |cmpt ^ τ m N q|2 L2 ď K´1Epnm, cm, umqpt ^ τ m N q ` K´1 |O| |c0|2 L8 , P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='19) We now proceed to establish some uniform bounds for um, cm, and nm in some suitable spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For this purpose, we recall that hereafter, K will denote a positive constant independent of m and N, which may change from one term to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Under the same assumptions as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5, there exists a positive constant K such that for all m P N and N P N, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='20) sup 0ďsďT |cmps ^ τ m N q|2 L2 ` 2η ż T^τ m N 0 |∇cmpsq|2 L2 ds ď |O| |c0|2 L8 , P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' E sup 0ďsďT Epnm, cm, umqps ^ τ m N q ď K, E ż T^τ m N 0 ˆˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ` |∆cmpsq|2 L2 ` |∇umpsq|2 L2 ˙ ds ď K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='21) 28 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let t P r0, Ts be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We start by proving the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' To do this, we take pm, Nq P N2 arbitrary and apply the Itˆo formula to t ÞÑ |cmpt ^ τ m N q|2 L2 to get |cmpt ^ τ m N q|2 L2 ` 2ξ ż t^τ m N 0 |∇cmpsq|2 L2 ds “ |cm 0 |2 L2 ´ 2 ż t^τ m N 0 pB1pumpsq, cmpsqq, cmpsqqds ´ 2 ż t^τ m N 0 pR1pnmpsq, cmpsqq, cmpsqqds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='22) ` γ2 ż t^τ m N 0 |φpcmpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ` 2γ ż t^τ m N 0 pφpcmpsqq, cmpsqqdβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By integration by part, we derive that pB1pum, cmq, cmq “ 1 2 ż O umpxq ¨ ∇c2 mpxqdx “ ´1 2 ż O c2 mpxq∇ ¨ umpxqdx “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By the free divergence property of σk and the fact that σk “ 0 on BO, k “ 1, 2, we get pφpcmq, cmq “ 2ÿ k“1 ż O σkpxq ¨ ∇cmpxqcmpxqdx “ 1 2 2ÿ k“1 ż O σkpxq ¨ ∇c2 mpxqdx “ ´1 2 2ÿ k“1 ż O c2 mpxq∇ ¨ σkpxqdx ` 1 2 2ÿ k“1 ż BO c2 mpσqσkpσq ¨ νdσ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Taking into account the equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13), we infer that |φpcmq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q “ 2ÿ k“1 ż O |σkpxq ¨ ∇cmpxq|2 L2 dx “ |∇cm|2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using these three last equalities and the fact that |cm 0 |2 L2 ď |O| |c0|2 L8 (since by the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1), |cm 0 |2 L8 ď |c0|2 L8), we infer from the equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='22) that for all t P r0, Ts, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='23) |cmpt ^ τ m N q|2 L2`2η ż t^τ m N 0 |∇cmpsq|2 L2 ds`2 ż t^τ m N 0 ż O nmps, xqfpcmps, xqqcmps, xqdxds ď |O| |c0|2 L8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to the non-negativity of nmps ^ τ m N q, cmps ^ τ m N q and f over the interval r0, ts given in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2 and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1, we can deduce from the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='23) that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='24) sup 0ďtďT |cmpt ^ τ m N q|2 L2 ` 2η ż T^τ m N 0 |∇cmpsq|2 L2 ds ď |O| |c0|2 L8 , P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let us now move to the proof of the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Multiplying equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14)3 by 1 ` ln nmps ^ τ m N q for s P r0, ts and integrate the resulting equation in O and using an integration-by-parts as well as the divergence free property of um, we have d dt ż O nmps ^ τ m N , xq ln nps ^ τ m N , xqdx ` δ ż O |∇nmps ^ τ m N , xq|2 nmps ^ τ m N , xq dx “ χ ż O ∇nmps ^ τ m N , xq ¨ ∇cmps ^ τ m N , xqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='25) ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 29 In the equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='25), we have used the fact that um “ Bnm Bν “ 0 on BO and the fact that ´ ż O ∆nmpxq lnpnmpxqqdx “ ż O ∇nmpxq ¨ ∇ lnpnmpxqqdx ´ ż BO Bnmpσq Bν lnpnmpσqqdσ “ ż O ∇nmpxq ¨ ∇nmpxq nmpxq dx, as well as ż O umpxq ¨ ∇nmpxq lnpnmpxqqdx “ ´ ż O nmpxq∇ ¨ pumpxq lnpnmpxqqqdx ` ż BO nmpσq lnpnmpσqqumpσq ¨ νdσ “ ´ ż O nmpxqumpxq ¨ ∇ lnpnmpxqqdx ´ ż O nmpxq lnpnmpxqq∇ ¨ umpxqdx “ ´ ż O umpxq ¨ ∇nmpxqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' It follows from the Young inequality and the Cauchy-Schwarz inequality that χ ż O ∇nmpxq ¨ ∇cmpxqdx ď δ 2 ż O |∇nmpxq|2 nmpxq dx ` χ2 2δ ż O nmpxq |∇cmpxq|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since ż O |∇nmpxq|2 nmpxq dx “ 4 ż O ˇˇˇ∇ a nmpxq ˇˇˇ 2 dx, we may combine the last inequality with equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='25) to obtain ż O nmpt ^ τ m N , xq ln nmpt ^ τ m N , xqdx ` 2δ ż t^τ m N 0 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ds ď ż O nm 0 pxq ln nm 0 pxqdx ` χ2 2δ ż t^τ m N 0 ˇˇˇ a nmpsq∇cmpsq ˇˇˇ 2 L2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='26) Applying the Itˆo formula once more to t ÞÑ |∇cmpt ^ τ m N q|2 L2, yields |∇cmpt ^ τ m N q|2 L2 ` 2ξ ż t^τ m N 0 |∆cmpsq|2 L2 ds “ |∇cm 0 |2 L2 ´ 2 ż t^τ m N 0 p∇B1pumpsq, cmpsqq, ∇cmpsqqds ´ 2 ż t^τ m N 0 p∇R1pnmpsq, cmpsqq, ∇cmpsqqds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='27) ` γ2 ż t^τ m N 0 |∇φpcmpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ` 2γ ż t^τ m N 0 p∇φpcmpsqq, ∇cmpsqqdβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 30 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Since um is solenoidal and vanishes on BO, we derive that p∇B1pum, cmq, ∇cmq “ ż O ∇pumpxq ¨ ∇cmpxqq ¨ ∇cmpxqdx “ ż O ∇umpxq∇cmpxq ¨ ∇cmpxqdx ` ż O ∇cmpxq ¨ D2cmpxqumpxqdx ď ż O |∇umpxq| |∇cmpxq|2 dx ` 1 2 ż O umpxq ¨ ∇ |∇cmpxq|2 dx (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='28) ď |∇um|L2 |∇cm|2 L4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We use the Gagliardo-Niremberg inequality to obtain |∇cm|4 L4 ď KGN |cm|2 L8 ˇˇD2cm ˇˇ2 L2 ` KGN |cm|4 L8 , To cancel ˇˇD2cm ˇˇ L2, we invoke the pointwise identity |∆cm|2 “ ∇ ¨ p∆cm∇cmq ´ ∇cm ¨ ∇∆cm, and ∆ |∇cm|2 “ 2∇cm ¨ ∇∆cm ` 2 ˇˇD2cm ˇˇ2, as well as the integration-by-parts to rewrite |∆cm|2 L2 as |∆cm|2 L2 “ ´ ż O ∇cmpxq ¨ ∇∆cmpxqdx “ ˇˇD2cm ˇˇ2 L2 ´ 1 2 ż O ∆ |∇cmpxq|2 dx (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='29) “ ˇˇD2cm ˇˇ2 L2 ´ 1 2 ż BO B |∇cmpσq|2 Bν dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Invoking [27, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2] we obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='30) 1 2 ż BO B |∇cmpσq|2 Bν dσ ď κpOq ż BO |∇cmpσq|2 dσ, where κpOq is an upper bound for the curvatures of BO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to the trace theorem (see [21, (ii) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='22 with (i) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='24]), it holds that ż BO |∇cmpσq|2 dσ ď KpO, ςq |cm|2 H 3`ς 2 for any ς P p0, 1q, where KpO, ςq ą 0 depends only on O and ς, which can be fixed for instance ς “ 1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' On the other hand, the interpolation inequality, the Young inequality and the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2 imply the existence of K1 and K2 depending on O such that κpOqKpO, ςq |cm|2 H 7 4 ď K1p ˇˇD2cm ˇˇ7{4 L2 |cm|1{4 L2 ` |cm|2 L2q ď 1 4 ˇˇD2cm ˇˇ2 L2 ` K2 |c0|2 L8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using this previous inequality and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='30), we infer from the equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='29) that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='31) ˇˇD2cm ˇˇ2 L2 ď 4 3 |∆cm|2 L2 ` 4K2 3 |c0|2 L8 , ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 31 and therefore |∇cm|4 L4 ď 4KGN |c0|2 L8 3 |∆cm|2 L2 ` ˆ4K2 3 ` 1 ˙ KGN |c0|4 L8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='28) and the Young inequality, we infer that p∇B1pum, cmq, ∇cmq ď |∇um|L2 |∇cm|2 L4 ď 3ξ 16KGN |c0|2 L8 |∇cm|4 L4 ` 4KGN |c0|2 L8 3ξ |∇um|2 L2 ď ξ 4 |∆cm|2 L2 ` 4KGN |c0|2 L8 3ξ |∇um|2 L2 ` ξp4K2 ` 3q 16 |c0|2 L8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Due to the Assumption 1 and the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2, we note that ´p∇R1pnm, cmq, ∇cmq “ ´ ż O ∇pnmpxqfpcmpxqqq ¨ ∇cmpxqdx “ ´ ż O f 1pcmpxqq |∇cmpxq|2 nmpxqdx ´ ż O fpcmpxqq∇cmpxq ¨ ∇nmpxqdx ď ´ min 0ďcď|c0|L8 f 1pcq 2 ż O nmpxq |∇cmpxq|2 dx ` 1 2 min 0ďcď|c0|L8 f 1pcq ż O f 2pcmpxqq|∇nmpxq|2 nmpxq dx ď ´ min 0ďcď|c0|L8 f 1pcq 2 |?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='nm∇cm|2 L2 ` 2 max 0ďcď|c0|L8 f 2pcq min 0ďcď|c0|L8 f 1pcq |∇?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='nm|2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Combining these two last inequalities, we derive from equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='27) that |∇cmpt ^ τ m N q|2 L2 ` 3ξ 2 ż t^τ m N 0 |∆cmpsq|2 L2 ds ` min 0ďcď|c0|L8 f 1pcq ż t^τ m N 0 ˇˇˇ a nmpsq∇cmpsq ˇˇˇ 2 L2 ds ď |∇cm 0 |2 L2 ` ξp4K2 ` 3q 8 |c0|2 L8 t ` 8KGN |c0|2 L8 3ξ ż t^τ m N 0 |∇umpsq|2 L2 ds ` 4 max 0ďcď|c0|L8 f 2pcq min 0ďcď|c0|L8 f 1pcq ż t^τ m N 0 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ds ` γ2 ż t^τ m N 0 |∇φpcmpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ds ` 2γ ż t^τ m N 0 p∇φpcmpsqq, ∇cmpsqqdβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 32 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Multiplying this last inequality by Kf and adding the result with inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='26) we obtain ż O nmpt ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xq ln nmpt ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xqdx ` Kf |∇cmpt ^ τ m N q|2 L2 ` ξKf 4 ż t^τ m N 0 |∆cmpsq|2 L2 ds ` 2δ ż t^τ m N 0 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ds ` ż t 0 ˇˇˇ a nmpsq∇cmpsq ˇˇˇ 2 L2 ds ď Kf |∇cm 0 |2 L2 ` ż O nm 0 pxq ln nm 0 pxqdx ` Kfξp4KfK2 ` 3q 8 |c0|2 L8 t ` 8KfKGN |c0|2 L8 3ξ ż t^τ m N 0 |∇umpsq|2 L2 ds ` 4Kf max 0ďcď|c0|L8 f 2pcq min 0ďcď|c0|L8 f 1pcq ż t^τ m N 0 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ds ` γ2Kf ż t^τ m N 0 |∇φpcmpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ds ` 2γKf ż t^τ m N 0 p∇φpcmpsqq, ∇cmpsqqdβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By using the first inequality of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3), we see that the previous inequality reduces to ż O nmpt ^ τ m N , xq ln nmpt ^ τ m N , xqdx ` Kf |∇cmpt ^ τ m N q|2 L2 ` 3ξKf 2 ż t^τ m N 0 |∆cmpsq|2 L2 ds ` 2δ ż t^τ m N 0 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ds ` ż t^τ m N 0 ˇˇˇ a nmpsq∇cmpsq ˇˇˇ 2 L2 ds ď Kf |∇cm 0 |2 L2 ` ż O nm 0 pxq ln nm 0 pxqdx ` Kfξp4KfK2 ` 3q 8 |c0|2 L8 t ` 8KfKGN |c0|2 L8 3ξ ż t^τ m N 0 |∇umpsq|2 L2 ds ` γ2Kf ż t^τ m N 0 |∇φpcmpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='32) ` 2γKf ż t^τ m N 0 p∇φpcmpsqq, ∇cmpsqqdβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, we use the equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2 and the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7) to obtain that |nm|L2 ď KGN ´ |?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='nm|L2 |∇?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='nm|L2 ` |?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='nm|2 L2 ¯ ď KGN ˆ |nm 0 | 1 2 L1 |∇?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='nm|L2 ` |nm 0 |L1 ˙ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='33) By the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1), we have nm 0 Ñ n0 in L2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to the continuous embedding of L2pOq into L1pOq, we derive that nm 0 Ñ n0 in L1pOq and therefore the sequence tnm 0 umě1 is bounded in L1pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This implies that the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='33) can be controlled as follows |nm|L2 ď KGNK1{2 |∇?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='nm|L2 ` K, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='34) where K is a constant independent of m and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 33 Next, applying the Itˆo formula to t ÞÑ |umpt ^ τ m N q|2 L2 and using the estimation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='34), we infer the existence of K3 ą 0 such that |umpt ^ τ m N q|2 L2 ` 2η ż t^τ m N 0 |∇umpsq|2 L2 ds ď 2 ż t^τ m N 0 |∇Φ|L8 |nmpsq|L2 |umpsq|L2 ds ` ż t^τ m N 0 |gpumpsq, cmpsqq|2 L2pU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Hq ds ` 2 ż t^τ m N 0 pgpumpsq, cmpsqq, umpsqqdWs ď |um 0 |2 L2 ` δη K4 ż t^τ m N 0 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ds ` K3 |∇Φ|2 L8 ż t^τ m N 0 |umpsq|2 L2 ds ` 1 2t ` 1 2 |∇Φ|2 L8 K2 ż t^τ m N 0 |umpsq|2 L2 ds ` ż t^τ m N 0 |gpumpsq, cmpsqq|2 L2pU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Hq ds ` 2 ż t^τ m N 0 pgpumpsq, cmpsqq, umpsqqdWs, with K4 “ 8Kf KGN|c0|2 L8 3ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Multiplying this inequality by K4 η , and adding the result with inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='32) after using the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='18), we see that there exists positive constants K5 and K6 such that for all t P r0, Ts, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Epnm, cm, umqpt ^ τ m N q ` δ ż t^τ m N 0 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ds ` ż t^τ m N 0 „3ξKf 2 |∆cmpsq|2 L2 ` K4 |∇umpsq|2 L2 ` ˇˇˇ a nmpsq∇cmpsq ˇˇˇ 2 L2 \uf6be ds ď Epn0, c0, u0q ` K5T ` K6 ż t^τ m N 0 |umpsq|2 L2 ds ` γ2Kf ż t^τ m N 0 |∇φpcmpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='35) ` 2K4 η ż t^τ m N 0 pgpumpsq, cmpsqq, umpsqqdWs ` K4 η ż t^τ m N 0 |gpumpsq, cmpsqq|2 L2pU,Hq ds ` 2γKf ż t^τ m N 0 p∇φpcmpsqq, ∇cmpsqqdβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, since γ satisfies the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3), taking into account the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='31), we note that γ2Kf |∇φpcmq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ď 2γ2Kf 2ÿ k“1 ż O |∇σkpxq∇cpxq|2 dx ` 2γ2Kf 2ÿ k“1 ż O ˇˇD2cpxqσkpxq ˇˇ2 dx ď 2γ2Kf |∇c|2 L2 2ÿ k“1 |σk|2 W 1,8 ` |∆c|2 L2 8γ2Kf 3 2ÿ k“1 |σk|2 L8 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='36) ` 8γ2KfK2 3 |c0|L8 2ÿ k“1 |σk|2 L8 ď K |∇c|2 L2 ` ξKf 2 |∆c|2 L2 ` K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 34 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ By the inequalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='19), we also note that |gpum, cmq|2 L2pU,Hq ď 2L2 g |pum, cmq|2 H ` 2L2 g ď KEpnm, cm, umq ` K |c0|2 L8 ` 2L2 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='37) From the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='35) until (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='37), we derive that E sup 0ďsďT Epnm, cm, umqps ^ τ m N q ` δE ż T^τ m N 0 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ds ` E ż T^τ m N 0 „ ξKf |∆cmpsq|2 L2 ` K4 |∇umpsq|2 L2 ` ˇˇˇ a nmpsq∇cmpsq ˇˇˇ 2 L2 \uf6be ds ď Epn0, c0, u0q ` KT ` KE ż T^τ m N 0 Epnmpsq, cmpsq, umpsqqds ` 2L2 gT ` 2γKfE sup 0ďsďT ˇˇˇˇ ż s^τ m N 0 p∇φpcmpsqq, ∇cmpsqqdβs ˇˇˇˇ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='38) ` 2K4 η E sup 0ďsďT ˇˇˇˇˇ 8 ÿ k“1 ż s^τ m N 0 pgpumpsq, cmpsqqek, umpsqqdW k s ˇˇˇˇˇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, by making use of the Burholder-Davis-Gundy, Cauchy-Schwarz, Young inequalities and the fact that γ satisfies the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3), we infer that 2γKfE sup 0ďsďT ˇˇˇˇ ż s^τ m N 0 p∇φpcmpsqq, ∇cmpsqqdβs ˇˇˇˇ ď KE ˆż T^τ m N 0 |p∇φpcmpsqq, ∇cmpsqq|2 ds ˙1{2 ď KE ˆż T^τ m N 0 |∇φpcmpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q |∇cmpsq|2 L2 ds ˙1{2 ď Kf 4 E sup 0ďsďT |∇cmps ^ τ m N q|2 L2 ` KE ż T^τ m N 0 |∇φpcmpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ds ď 1 4E sup 0ďsďT Epnmpsq, cmpsq, umpsqqps ^ τ m N q ` ξKf 2 E ż T^τ m N 0 |∆cmpsq|2 L2 ds ` KE ż T^τ m N 0 |∇cmpsq|2 L2 ds ` KT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Similarly, 2K4 η E sup 0ďsďT ˇˇˇˇˇ 8 ÿ k“1 ż s^τ m N 0 pgpumpsq, cmpsqqek, umpsqqdW k s ˇˇˇˇˇ ď 1 4E sup 0ďsďT Epnm, cm, umqps ^ τ m N q ` KE ż T^τ m N 0 |gpumpsq, cmpsqq|2 L2pU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Hq ds ď 1 4E sup 0ďsďT Epnm, cm, umqps ^ τ m N q ` KE ż T^τ m N 0 |pumpsq, cmpsqq|2 H ds ` KTL2 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 35 It follows from the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='38) that E sup 0ďsďT Epnm, cm, umqps ^ τ m N q ` E ż T^τ m N 0 „ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ` Kf |∆cmpsq|2 L2 ` |∇umpsq|2 L2 ` ˇˇˇ a nmpsq∇cmpsq ˇˇˇ 2 L2 \uf6be ds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='39) ď KEpn0, c0, u0q ` KT ` KE ż T^τ m N 0 Epnm, cm, umqpsqds ` K, where K is a constant depending on the initial data and T but independent of m and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, the Gronwall lemma yields E sup 0ďsďT Epnm, cm, umqps ^ τ m N q ` E ż T^τ m N 0 „ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ` |∆cmpsq|2 L2 ` |∇umpsq|2 L2 ` ˇˇˇ a nmpsq∇cmpsq ˇˇˇ 2 L2 \uf6be ds ď K, from which we deduce the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='21) and hence completing the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Under the same assumptions as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3, for all p ě 1, there exists a positive constant K such that we have for all m P N and N P N, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='40) sup 0ďsďT |cmps ^ τ m N q|2p L2 ` ˆż T^τ m N 0 |∇cmpsq|2 L2 ds ˙p ď |O|p |c0|2p L8 , P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', E sup 0ďsďT Eppnm, cm, umqps ^ τ m N q ` E ˆż T^τ m N 0 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ds ˙p ď K, and E ˆż T^τ m N 0 |∆cmpsq|2 L2 ds ˙p ` E ˆż T^τ m N 0 |∇umpsq|2 L2 ds ˙p ď K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='41) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='40) follows directly from the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='20) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Next, we are going to derive estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We start with the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='38) and invoke the Jensen inequality to derive that for all p ě 2, E sup 0ďsďT Eppnm, cm, umqps ^ τ m N q ` E ˆż T^τ m N 0 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ds ˙p ` E ˆż T^τ m N 0 ξKf |∆cmpsq|2 L2 ds ˙p ` E ˆż T^τ m N 0 K4 |∇umpsq|2 L2 ds ˙p ď Eppn0, c0, u0q ` KT p ` KE ˆż T^τ m N 0 Epnm, cm, umqpsqds ˙p (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='42) ` Kp ` 2pγpKp fE sup 0ďsďT ˇˇˇˇ ż s^τ m N 0 p∇φpcmpsqq, ∇cmpsqqdβs ˇˇˇˇ p ` KE sup 0ďsďT ˇˇˇˇˇ 8 ÿ k“1 ż s^τ m N 0 pgpumpsq, cmpsqqek, umpsqqdW k s ˇˇˇˇˇ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Invoking the H¨older inequality, we see that KE ˆż T^τ m N 0 Epnm, cm, umqpsqds ˙p ď KT p p´1E ż T^τ m N 0 Eppnm, cm, umqpsqds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 36 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Thanks to the Burkholder-Davis-Gundy inequality, we see that KE sup 0ďsďT ˇˇˇˇˇ 8 ÿ k“1 ż s^τ m N 0 pgpumpsq, cmpsqqek, umpsqqdW k s ˇˇˇˇˇ p ď KE 8 ÿ k“1 ˆż T^τ m N 0 |pgpumpsq, cmpsqqek, umpsqq|2 ds ˙p{2 ď KE sup 0ďsďT |umps ^ τ m N q|p L2 ˆż T^τ m N 0 |gpumpsq, cmpsqq|2 L2pU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Hq ds ˙p{2 ď 1 4E sup 0ďsďT Eppnm, cm, umqps ^ τ m N q ` KE ˆż T^τ m N 0 |gpumpsq, cmpsqq|2 L2pU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Hq ds ˙p ď 1 4E sup 0ďsďT Eppnm, cm, umqps ^ τ m N q ` KE ż T^τ m N 0 |pumpsq, cmpsqq|2p H ds ` KT p p´1L2p g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Taking into account the fact that γ is sufficiently small such that the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3) is satisfied, we also arrive at 2pγpKp fKE sup 0ďsďT ˇˇˇˇ ż s^τ m N 0 p∇φpcmpsqq, ∇cmpsqqdβs ˇˇˇˇ p ď 2pγpKp fE ˆż T^τ m N 0 |∇φpcmpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q |∇cmpsq|2 L2 ds ˙p{2 ď 1 4E sup 0ďsďT |∇cmps ^ τ m N q|2p L2 ` 22pγ2pK2p f E ˆż T^τ m N 0 |∇φpcmpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ds ˙p ď 1 4E sup 0ďsďT Eppnm, cm, umqps ^ τ m N q ` 1 2E ˆż T^τ m N 0 ξKf |∆cmpsq|2 L2 ds ˙p ` KT p p´1E ż T^τ m N 0 |∇cmpsq|2p L2 ds ` KT p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' It follows from the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='42) that E sup 0ďsďT Eppnm, cm, umqps ^ τ m N q ` E ˆż T^τ m N 0 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ds ˙p ` E ˆż T^τ m N 0 |∆cmpsq|2 L2 ds ˙p ` E ˆż T^τ m N 0 |∇umpsq|2 L2 ds ˙p ď KEppn0, c0, u0q ` KT p ` KE ż T^τ m N 0 Eppnm, cm, umqpsqds ` K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, the Gronwall lemma yields E sup 0ďsďT Eppnm, cm, umqps ^ τ m N q ` E ˆż T^τ m N 0 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ds ˙p ` E ˆż T^τ m N 0 |∆cmpsq|2 L2 ds ˙p ` E ˆż T^τ m N 0 |∇umpsq|2 L2 ds ˙p ď K, and the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='41) follows directly from this last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This completes the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ In order to control the process t ÞÑ nmpt ^ τ m N q, we prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 37 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Under the same assumptions as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3, there exists a positive constant η0 ą 1 such that for all m P N, N P N and P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', sup 0ďsďT |nmps ^ τ m N q|2 L2 ` ż T^τ m N 0 |nmpsq|2 H1 ds ď η0 exp ˆ K ż T^τ m N 0 |∇cmpsq|4 L4 ds ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='43) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let t P r0, Ts be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Multiplying the last equation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) by nmps^τ m N q for 0 ď s ď t, and using the fact that ∇ ¨ um “ 0 and the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7) as well as the H¨older inequality and the Young inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' we obtain 1 2 d dt |nmps ^ τ m N q|2 L2 ` δ |∇nmps ^ τ m N q|2 L2 “ ξ ż O nmps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xq∇cmps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xq ¨ ∇nmps ^ τ m N ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' xqdx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='ď ξ |nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|L4 |∇cmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|L4 |∇nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|L2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='ď Kp|nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|1{2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2 |∇nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|1{2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2 ` |nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|L2q |∇cmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|L4 |∇nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|L2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='ď K |nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|1{2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2 |∇cmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|L4 |∇nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|3{2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='` K |nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|L2 |∇cmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|L4 |∇nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|L2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='ď δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2 |∇nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2 ` K |nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2 p|∇cmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L4 ` |∇cmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L4q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='ď δ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2 |∇nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2 ` K |nmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='|∇cmps ^ τ m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='N q|4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L4 ` 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This implies that for all t P r0, Ts, sup 0ďsďt |nmps ^ τ m N q|2 L2 ` δ ż t^τ m N 0 |∇nmpsq|2 L2 ds ď |nm 0 |2 L2 ` K ż t^τ m N 0 |nmpsq|2 L2 ´ |∇cmpsq|4 L4 ` 1 ¯ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since n0 m Ñ n0 in L2pOq, |nm 0 |2 L2 is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thus, applying the Gronwall lemma, we obtain that sup 0ďsďt |nmps ^ τ m N q|2 L2 ` ż t^τ m N 0 |nmpsq|2 H1 ds ď Kδ exp ˆ K ż t^τ m N 0 ´ |∇cmpsq|4 L4 ` 1 ¯ ds ˙ ď pKδ ` 1qeKT exp ˆ K ż t^τ m N 0 |∇cmpsq|4 L4 ds ˙ , and complete the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Under the same assumptions as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3, for any p ě 1, there exists a positive constant K such that for all m P N and N P N, E sup 0ďsďT |umps ^ τ m N q|2p L2 ` E ˆż T^τ m N 0 |∇umpsq|2 L2 ds ˙p ď K, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='44) E ˆż T^τ m N 0 |nmpsq|2 L2 ds ˙p ď K, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='45) E sup 0ďsďT |cmps ^ τ m N q|2p H1 ` E ˆż T^τ m N 0 |cmpsq|2 H2 ds ˙p ď K, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='46) E ż T^τ m N 0 |∇cmpsq|4 L2 ds ď K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='47) 38 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='44) is a consequence of the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' From the inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='34), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='41), we infer that E ˆż T^τ m N 0 |nmpsq|2 L2 ds ˙p ď E ˆż T^τ m N 0 ˆ KGNK1{2 ˇˇˇ∇ a nmpsq ˇˇˇ 2 L2 ` K ˙ ds ˙p ď K, which proves the second estimate of inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' According to [35, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 404], we have |cm|2 H2 ď Kp|∆cm|2 L2 ` |cm|2 H1q, from which along with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='41) we deduce (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By applying the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7), we obtain that |∇cm|4 L4 ď Kp|cm|2 H2 |∇cm|2 L2 ` |∇cm|4 L2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Therefore, E ż T^τ m N 0 |∇cmpsq|4 L4 ds ď KE ż T^τ m N 0 |cmpsq|2 H2 |∇cmpsq|2 L2 ds ` KE ż T^τ m N 0 |∇cmpsq|4 L2 ds ď KE sup 0ďsďT |cmps ^ τ m N q|2 H1 ż T 0 |cmpsq|2 H2 ds ` KTE sup 0ďsďT |cmps ^ τ m N q|4 H1 ď KE sup 0ďsďT |cmps ^ τ m N q|4 H1 ` KE ˆż T^τ m N 0 |cmpsq|2 H2 ds ˙2 , from which along with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='46) we deduce (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This completes the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ In the following lemma, we state and prove a result concerning the stopping time τ m N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' More precisely, we prove that sup NPN τ N m ě 2T with probability 1 such that the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let τ m N , m, N P N be the stopping times defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then, under the same assumptions as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3, it holds that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='48) P " ω P Ω : sup NPN τ N m pωq ě 2T “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Consequently, the solutions pum, cm, nmq of system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) exist almost surely for every t P r0, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We notice that the inequalities of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6 hold for every T ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, for a fixed T ą 0, we set ˜T “ 2T and note that for all J P N, " ω P Ω : sup NPN τ N m pωq ă ˜T Ă !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω P Ω : τ J mpωq ă ˜T ) , which implies that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='49) P " ω P Ω : sup NPN τ N m pωq ă 2T ď lim NÝÑ8 P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω P Ω : τ N m pωq ă ˜T ) , and therefore, it is enough to show that the second term of the right hand side of this last equality converges to zero as N Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' To this end, let AN “ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω P Ω : τ N m ă ˜T ) ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 39 and BN “ " ω P Ω : ˇˇˇnmp ˜T ^ τ N m q ˇˇˇ 2 L2 ` ˇˇˇump ˜T ^ τ N m q ˇˇˇ 2 L2 ` ˇˇˇcmp ˜T ^ τ N m q ˇˇˇ 2 H1 ě N 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then, we have AN Ă BN for N ą ˜T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Indeed, let ω P AN, then ˜T ^ τ N m pωq “ τ N m pωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thus, by the definition of the stopping time τ N m , we see that for N ą ˜T, ˇˇˇnmp ˜T ^ τ N m q ˇˇˇ 2 L2 ` ˇˇˇump ˜T ^ τ N m q ˇˇˇ 2 L2 ` ˇˇˇcmp ˜T ^ τ N m q ˇˇˇ 2 H1 “ ˇˇnmpτ N m q ˇˇ2 L2 ` ˇˇumpτ N m q ˇˇ2 L2 ` ˇˇcmpτ N m q ˇˇ2 H1 ě N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We then conclude that ω P BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, for N ą ˜T, using the inclusion AN Ă BN we derive that P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω P Ω : τ N m ă ˜T ) ď P " ω P Ω : ˇˇˇnmp ˜T ^ τ N m q ˇˇˇ 2 L2 ` ˇˇˇump ˜T ^ τ N m q ˇˇˇ 2 L2 ` ˇˇˇcmp ˜T ^ τ N m q ˇˇˇ 2 H1 ě N 2 ď P " ω P Ω : ˇˇˇnmp ˜T ^ τ N m q ˇˇˇ 2 L2 ě N 2 3 ` P " ω P Ω : ˇˇˇump ˜T ^ τ N m q ˇˇˇ 2 L2 ě N 2 3 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='50) ` P " ω P Ω : ˇˇˇcmp ˜T ^ τ N m q ˇˇˇ 2 H1 ě N 2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' According to the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='44) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='46) of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6 as well as the Markov inequality, we derive that for N ą ˜T P " ω P Ω : ˇˇˇcmp ˜T ^ τ N m q ˇˇˇ 2 H1 ě N 2 3 ď P # ω P Ω : sup 0ďsď ˜T |cmps ^ τ m N q|2 H1 ě N 2 3 + ď 3 N 2 E sup 0ďsď ˜T |cmps ^ τ m N q|2 H1 ď K N 2 , and P " ω P Ω : ˇˇˇump ˜T ^ τ N m q ˇˇˇ 2 L2 ě N 2 3 ď P # ω P Ω : sup 0ďsď ˜T |umps ^ τ m N q|2 L2 ě N 2 3 + ď 3 N 2 E sup 0ďsď ˜T |umps ^ τ m N q|2 L2 ď K N 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Also for N ą maxp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3η0, ˜Tq (where η0 is a constant obtained in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5), we use the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='43) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5 to infer that P " ω P Ω : ˇˇˇnmp ˜T ^ τ N m q ˇˇˇ 2 L2 ě N 2 3 ď P # ω P Ω : sup 0ďsď ˜T |nmps ^ τ m N q|2 L2 ě N 2 3 + ď P # ω P Ω : η0 exp ˜ K ż ˜T^τ m N 0 |∇cmpsq|4 L4 ds ¸ ě N 2 3 + ď P # ω P Ω : ż ˜T^τ m N 0 |∇cmpsq|4 L4 ds ě lnp N2 3η0 q K + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 40 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Invoking the Markov inequality and using the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='47) of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6, we see that P " ω P Ω : ˇˇˇnmp ˜T ^ τ N m q ˇˇˇ 2 L2 ě N 2 3 ď K lnp N2 3η0 q E ż ˜T^τ m N 0 |∇cmpsq|4 L4 ds ď K 2 lnpNq ´ lnp3η0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Plugging these inequalities into the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='50), we arrive at P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω P Ω : τ N m ă ˜T ) ď K N 2 ` K 2 lnpNq ´ lnpη0q ´ lnp3q, for all for N ą maxp?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3η0, ˜Tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Letting N to infinity in this last inequality we get lim NÝÑ8 P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω P Ω : τ N m ă ˜T ) “ 0, which along with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='49) imply (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By the equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='48) we infer the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7) and therefore, the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5) hold and the lemma is then proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Since pT ^ τ N m qNPN is increasing, we have T ^ τ N m Ñ T a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', as N Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' With this almost surely convergence in hand, we are going to give some consequences of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Under the same assumptions as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3, for any p ě 1, there exists a positive constant K such that for all m P N, sup 0ďsďT |nmpsq|2 L2 ` ż T 0 |nmpsq|2 H1 ds ď η0 exp ˆ K ż T 0 |∇cmpsq|4 L4 ds ˙ , P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='51) E sup 0ďsďT |umpsq|2p L2 ` E ˆż T 0 |∇umpsq|2 L2 ds ˙p ď K, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='52) E ˆż T 0 |nmpsq|2 L2 ds ˙p ď K, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='53) E sup 0ďsďT |cmpsq|2p H1 ` E ˆż T 0 |cmpsq|2 H2 ds ˙p ď K, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='54) E ż T 0 |∇cmpsq|4 L2 ds ď K, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='55) where η0 ą 1 is a constant obtained in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since T ^ τ N m Ñ T a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', as N Ñ 8, by the path continuity of the process t ÞÑ pumptq, cmptq, nmptqq, we can let N Ñ 8 in the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='43) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5 and derive the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In addition to the almost surely convergence of T ^ τ N m to T and the path continuity of the process t ÞÑ pumptq, cmptq, nmptqq, we invoke the Fatou lemma and pass to the limit as N Ñ 8 in the inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='44), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='45), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='46) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='47) and derive the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='52), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='53), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='54) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Under the same assumptions as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3, there exists a positive constant K such that for all m P N, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='56) E |nm|2 C1{2pr0,Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='H´3q ď K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 41 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let v P H3pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We recall that |∇v|L8 ď K |v|H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' So, using an integration by part and the H¨older inequality, we derive that |pA1nm, vq| “ |pnm, ∆vq| ď |nm|L2 |∆v|L2 ď |nm|L2 |v|H3 , ˇˇpP1 mB1pum, nmq, vq ˇˇ “ ˇˇpB1pum, nmq, P1 mvq ˇˇ “ ˇˇpnmum, ∇P1 mvq ˇˇ ď K |nm|L2 |um|L2 ˇˇ∇P1 mv ˇˇ L8 ď K |nm|L2 |um|L2 |v|H3 , and ˇˇpP1 mR2pnm, cmq, vq ˇˇ “ ξ ˇˇpnm∇cm, ∇P1 mvq ˇˇ ď K |nm|L2 |∇cm|L2 ˇˇ∇P1 mv ˇˇ L8 ď K |nm|L2 |∇cm|L2 |v|H3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Due to the continuous Sobolev embeddings W 1,2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H´3pOqq ãÑ C1{2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H´3pOqq, and L2pOq ãÑ H´3pOq, we have E |nm|2 C1{2p0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='H´3q ď E |nm|2 W 1,2p0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='H´3q “ E ż T 0 |nmpsq|2 H´3 ds ` E ż T 0 ˇˇˇˇ d dtnmpsq ˇˇˇˇ 2 H´3 ds ď KE ż T 0 |nmpsq|2 L2 ds ` E ż T 0 ˇˇˇˇ d dtnmpsq ˇˇˇˇ 2 H´3 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='52), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='53) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='54), we arrive at E |nm|2 C1{2p0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='H´3qq ď K ` KE ż T 0 |A1nmpsq|2 H´3 ds ` KE ż T 0 ”ˇˇP1 mB1pumpsq, nmpsqq ˇˇ2 H´3 ` ˇˇP1 mR2pnmpsq, cmpsqq ˇˇ2 H´3 ı ds ď K ` KE ż T 0 ” |nmpsq|2 L2 ` |umpsq|2 L2 |nmpsq|2 L2 ` |nmpsq|2 L2 |∇cmpsq|2 L2 ı ds ď K ` KE sup 0ďsďT |umpsq|2 L2 ż T 0 |nmpsq|2 L2 ds ` KE sup 0ďsďT |∇cmpsq|2 L2 ż T 0 |nmpsq|2 L2 ds ď K ` KE sup 0ďsďT |umpsq|4 L2 ` KE sup 0ďsďT |∇cmpsq|4 L2 ` KE ˆż T 0 |nmpsq|2 L2 ds ˙2 ď K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Under the same assumptions as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3, there exists a positive constant K such that for all m P N, E ż T 0 ” |A1cmpsq|2 L2 ` ˇˇP2 mB1pumpsq, cmpsqq ˇˇ2 L2 ` ˇˇP2 mR1pnmpsq, cmpsqq ˇˇ2 L2 ı ds ď K, E ż T 0 ” |A0umpsq|2 V ˚ ` ˇˇP2 mB0pumpsq, umpsqq ˇˇ2 V ˚ ` ˇˇP2 mR0pnmpsq, Φq ˇˇ2 V ˚ ı ds ď K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='57) 42 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to the inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='52), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='53) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='54) once more,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' we note that E ż T 0 |A1cmpsq|2 L2 ds “ E ż T 0 |∆cmpsq|2 L2 ds ď KE ż T 0 |cmpsq|2 H2 ds ď K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' and E ż T 0 ˇˇP2 mB1pumpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cmpsqq ˇˇ2 L2 ds ď KE ż T 0 |umpsq ¨ ∇cmpsq|2 L2 ds ď KE sup 0ďsďT |umpsq|2 L2 ż T 0 |∇cmpsq|2 L2 ď KE sup 0ďsďT |umpsq|4 L2 ` KE ˆż T 0 |∇cmpsq|2 L2 ds ˙2 ď K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' as well as E ż T 0 ˇˇP2 mR1pnmpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cmpsqq ˇˇ2 L2 ds ď KE ż T 0 |nmpsqfpcmpsqq|2 L2 ds ď K sup 0ďsď|c0|L8 f 2psqE ż T 0 |nmpsq|2 L2 ď K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' and E ż T 0 |A0umpsq|2 V ˚ ds ď E ż T 0 |∇umpsq|2 L2 ds ď K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In the same way, E ż T 0 ˇˇP2 mB0pumpsq, umpsqq ˇˇ2 V ˚ ds ď KE ż T 0 |um|2 L2 |∇umpsq|2 L2 ds ď KE sup 0ďsďT |umpsq|2 L2 ż T 0 |∇umpsq|2 L2 ds ď KE sup 0ďsďT |umpsq|4 L2 ` KE ˆż T 0 |∇umpsq|2 L2 ds ˙2 ď K, and E ż T 0 |R0pnmpsq, Φq|2 V ˚ ds ď |Φ|2 W 1,8 E ż T 0 |nmpsq|2 L2 ds ď K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Combining all these inequalities, we obtain the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Tightness result and passage to the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This subsection is devoted to the study of the tightness of the approximations solutions and the proof of several convergences which will enable us to pass to the limit and construct a weak probabilistic solution to our problem via the martingale representation theorem given in [12, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For this purpose, we consider the following spaces: Zn “ L2 wp0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H´3pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2 wpOqq, Zu “ L2 wp0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V q X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V ˚q X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hwq, Zc “ L2 wp0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H2pOqq X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1 wpOqq, Z “ Zn ˆ Zu ˆ Zc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='58) By making appropriate use of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3, Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8, and Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9, we will now show that the sequence of probability law Lm “ Lpnmq ˆ Lpumq ˆ Lpcmq is tight in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 43 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We suppose that the hypotheses of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then the family of probability laws pLmqmPN is tight on the space Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We firstly prove that pLpnmqqm is tight on Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For any ε ą 0 we set Kε “ η0eK{ε ą η0 where η0 ą 1 is given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' From the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='51), we deduce that sup m P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω P Ω : |nm|2 L8p0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ą Kε ) ď sup m P " ω P Ω : η0 exp ˆ K ż T 0 |∇cmpsq|4 L4 ds ˙ ą Kε ď sup m P " ω P Ω : K ż T 0 |∇cmpsq|4 L4 ds ą ln ˆKε η0 ˙* .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using the Markov inequality and inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='55), we infer that sup m P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω P Ω : |nm|2 L8p0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ą Kε ) ď 1 ln ´ Kε η0 ¯E ˆ K ż T 0 |∇cmpsq|4 L4 ds ˙ ď ε KE ˆ K ż T 0 |∇cmpsq|4 L4 ds ˙ ď ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Similarly, we can also prove that sup m P !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω P Ω : |nm|2 L2p0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='H1q ą Kε ) ď sup m P " ω P Ω : η0 exp ˆ K ż T 0 |∇cmpsq|4 L4 ds ˙ ą Kε ď ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='56) we derive that sup m P " ω P Ω : |nm|2 C1{2pr0,Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='H´3q ą K ε ď ε KE |nm|2 C1{2pr0,Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='H´3q ď ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since these three last inequalities hold, we can apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3 and conclude that the law of nm form a family of probability measures which is tight on Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Secondly, we will prove that the laws of um and cm are tight on Zu ˆ Zc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' From inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='52) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='54), we obtain the first two conditions of Corollaries A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9 for um and cm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, it is sufficient to prove that the sequences pumqm and pcmq satisfy the Aldous condition in the spaces V ˚ and L2pOq respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let θ ą 0 pτℓqℓě1 be a sequence of stopping times such that 0 ď τℓ ď T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' From the second equation of system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) we have cmpτℓ ` θq ´ cmpτℓq “ ξ ż τℓ`θ τℓ A1cmpsqds ´ ż τℓ`θ τℓ P2 mB1pumpsq, cmpsqqds ` ż τℓ`θ τℓ P2 mR1pnmpsq, cmpsqqds ` γ ż τℓ`θ τℓ P2 mφpcmpsqqdβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='59) By the Fubini theorem, the H¨older inequality and inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='57), we have the following estimates E ˇˇˇˇξ ż τℓ`θ τℓ A1cmpsqds ˇˇˇˇ 2 L2 ď ξ2θ1{2E ż τℓ`θ τℓ |A1cmpsq|2 L2 ds ď ξ2θ1{2E ż T 0 |A1cmpsq|2 L2 ds ď Kθ1{2, 44 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ E ˇˇˇˇ ż τℓ`θ τℓ P2 mB1pumpsq, cmpsqqds ˇˇˇˇ 2 L2 ds ď θ1{2E ż τℓ`θ τℓ ˇˇP2 mB1pumpsq, cmpsqq ˇˇ2 L2 ds ď θ1{2E ż T 0 ˇˇP2 mB1pumpsq, cmpsqq ˇˇ2 L2 ds ď Kθ1{2, and E ˇˇˇˇ ż τℓ`θ τℓ P2 mR1pnmpsq, cmpsqqds ˇˇˇˇ 2 L2 ds ď θ1{2E ż τℓ`θ τℓ ˇˇP2 mR1pnmpsq, cmpsqq ˇˇ2 L2 ds ď θ1{2E ż T 0 ˇˇP2 mR1pnmpsq, cmpsqq ˇˇ2 L2 ds ď Kθ1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By the Itˆo isometry, we note that E ˇˇˇˇγ ż τℓ`θ τℓ P2 mφpcmpsqqdβs ˇˇˇˇ 2 L2 ď γ2E ż τℓ`θ τℓ |φpcmpsqq|2 L2pR2,L2q ď γ2 2ÿ k“1 |σk|2 L2 E ż τℓ`θ τℓ |∇cmpsq|2 L2 ds ď KθE sup 0ďsďT |∇cmpsq|2 L2 ď Kθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Combining these inequalities, we infer from equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='59) that the condition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5) is satisfies for pcmqmě1 in L2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7 the sequence pcmqmě1 satisfies the Aldous condition in the space L2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now we will consider the sequence pumqmě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We first observe that from the first equation of system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) we infer that umpτℓ ` θq ´ umpτℓq “ ´η ż τℓ`θ τℓ A0umpsqds ´ ż τℓ`θ τℓ P1 mB0pumpsq, umpsqqds ` ż τℓ`θ τℓ P1 mR0pnmpsq, Φqds ` ż τℓ`θ τℓ P1 mgpumpsq, cmpsqqdWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='60) Thanks to the H¨older inequality and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='57),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' we have the following estimates E ˇˇˇˇη ż τℓ`θ τℓ A0umpsqds ˇˇˇˇ 2 V ˚ ď η2θ1{2E ż τℓ`θ τℓ |A0umpsq|2 V ˚ ds ď η2θ1{2E ż T 0 |A0umpsq|2 V ˚ ds ď Kθ1{2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' and E ˇˇˇˇ ż τℓ`θ τℓ P2 mB0pumpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' umpsqqds ˇˇˇˇ 2 V ˚ ds ď θ1{2E ż τℓ`θ τℓ ˇˇP2 mB1pumpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' umpsqq ˇˇ2 V ˚ ds ď θ1{2E ż T 0 ˇˇP2 mB1pumpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' umpsqq ˇˇ2 V ˚ ds ď Kθ1{2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 45 as well as E ˇˇˇˇ ż τℓ`θ τℓ P2 mR0pnmpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Φqds ˇˇˇˇ 2 V ˚ ds ď θ1{2E ż τℓ`θ τℓ ˇˇP2 mR0pnmpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Φq ˇˇ2 V ˚ ds ď θ1{2E ż T 0 ˇˇP2 mR0pnmpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Φq ˇˇ2 V ˚ ds ď Kθ1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to the Itˆo isometry and the assumption on g we obtain E ˇˇˇˇ ż τℓ`θ τℓ P1 mgpumpsq, cmpsqqdWs ˇˇˇˇ 2 V ˚ ď KE ˇˇˇˇ ż τℓ`θ τℓ P1 mgpumpsq, cmpsqqdWs ˇˇˇˇ 2 L2 ď KE ż τℓ`θ τℓ ˇˇP1 mgpumpsq, cmpsqq ˇˇ2 L2pU,Hq ds ď KE ż τℓ`θ τℓ p1 ` |pumpsq, cmpsqq|2 Hqds ď K ˆ 1 ` E sup 0ďsďT |pumpsq, cmpsqq|2 H ˙ θ ď Kθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' From these inequalities and equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='60), we can conclude by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7 that the sequence pumqmě1 satisfies the Aldous condition in the space V ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, by applying Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8 and Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9, we see that the laws of cm and um are tight on Zc and Zu, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Since pLmqm is tight on Z, invoking [28, Corollary 2, Appendix B] (see also [7, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13]) there exists a probability space pΩ1, F1, P1q, and a subsequence of random vectors p¯umk, ¯cmk, ¯nmkq with values in Z such that i): p¯umk, ¯cmk, ¯nmkq have the same probability distributions as pumk, cmk, nmkq, ii): p¯umk, ¯cmk, ¯nmkq converges in the topology of Z to a random element pu, c, nq P Z with probability 1 on pΩ1, F1, P1q as k Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' To simplify the notation, we will simply denote these sequences by pum, cm, nmqmě1 and p¯um, ¯cm, ¯nmqmě1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Next, from the definition of the space Z, we deduce that P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', ¯um Ñ u in L2 wp0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V q X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V ˚q X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hwq, ¯cm Ñ c in L2 wp0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H2pOqq X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1 wpOqq, ¯nm Ñ n in L2 wp0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H´3pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2 wpOqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='61) According to [40, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4 and Addendum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5], a family of measurable map Ψm : Ω1 Ñ Ω can be constructed such that together with the new probability space pΩ1, F1, P1q satisfy the property ¯umpω1q “ um ˝ Ψmpω1q, ¯nmpω1q “ nm ˝ Ψmpω1q, ¯cmpω1q “ cm ˝ Ψmpω1q, and P “ P1 ˝ Ψ´1 m , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='62) 46 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ for all ω1 P Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Taking into account the fact that inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10) holds, we can derive that for almost every pt, ω1q P r0, Ts ˆ Ω1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='63) ˇˇ¯cmpt, ω1q ˇˇ L8 “ ˇˇcmpt, Ψmpω1qq ˇˇ L8 ď |c0|L8 , for all m ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since the laws of pum, cm, nmq and p¯um, ¯cm, ¯nmq are equal in the space Zu ˆ Zc ˆ Zn, we have the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='52), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='54) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='64) E1 ż T 0 |¯cmpsq|2 H2 ds ď K, E1 ż T 0 |∇¯umpsq|2 L2 ds ď K, as well as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='65) E1 ż T 0 |¯nmpsq|2 L2 ds ď K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='64) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='65) and the Banach-Alaoglu Theorem, we conclude that, there exists a sub- sequence of p¯umqmě1, p¯cmqmě1, and p¯nmqmě1 weakly convergent in L2pΩ1, F1, P1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V qq, L2pΩ1, F1, P1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H2pOqqq, and L2pΩ1, F1, P1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqqq respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' u P L2pΩ1, F1, P1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V qq, c P L2pΩ1, F1, P1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H2pOqqq, n P L2pΩ1, F1, P1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='66) On the other hand, from estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='52), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='53) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='54) of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8, and the equalities given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='62), we get for any p ě 1, E1 sup 0ďsďT |¯umpsq|2p L2 ` E1 ˆż T 0 |∇¯umpsq|2 L2 ds ˙p ď K, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='67) E1 ˆż T 0 |¯nmpsq|2 L2 ds ˙p ď K, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='68) E1 sup 0ďsďT |¯cmpsq|p H1 ` E1 ˆż T 0 |¯cmpsq|2 H2 ds ˙p ď K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='69) Then, invoking the Fatou lemma, we infer that for p ě 2, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='70) E1 sup 0ďsďT |upsq|p L2 ă 8, E1 sup 0ďsďT |cpsq|p H1 ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' and E1 ˆż T 0 |∇upsq|2 L2 ds ˙p ă 8, E1 ˆż T 0 |npsq|2 L2 ds ˙p ă 8, E1 ˆż T 0 |cpsq|2 H2 ds ˙p ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='71) Now, we prove three lemmata which show how convergence in Z given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='61) will be used for the convergence of the deterministic terms appearing in the Galerkin approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We start by noting that since nm 0 , cm 0 and um 0 have been chosen such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) holds, we can derive that for all ψ P H3pOq and pψ, vq P L2pOq ˆ V , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='72) lim mÝÑ8pnm 0 , ψq “ pn0, ψq, lim mÝÑ8pcm 0 , ψq “ pc0, ψq, and lim mÝÑ8pum 0 , vq “ pu0, vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 47 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For any r, t P r0, Ts with r ď t and ψ P H3pOq, the following convergences hold P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' lim mÝÑ8p¯nmptq, ψq “ pnptq, ψq, lim mÝÑ8 ż t r pA1¯nmpsq, ψqds “ ż t r pA1npsq, ψqds lim mÝÑ8 ż t r pP2 mB1p¯umpsq, ¯nmpsqq, ψqds “ ż t r pB1pupsq, npsqq, ψqds, lim mÝÑ8 ż t r pP2 mR2p¯nmpsq, ¯cmpsqq, ψqds “ ż t r pR2pnpsq, cpsqq, ψqds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='73) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let ψ P H3pOq and t P r0, Ts be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By the H¨older inequality we have |p¯nmptq, ψq ´ pnptq, ψq| ď |¯nmptq ´ nptq|H´3 |ψ|H3 ď |¯nm ´ n|Cpr0,Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='H´3q |ψ|H3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='74) which along with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='61) implies the first convergence in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, we also fix r P r0, Ts such that r ď t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By an integration-by-parts and the H¨older inequality we note that ˇˇˇˇ ż t r pA1¯nmpsq, ψqds ´ ż t r pA1npsq, ψqds ˇˇˇˇ dt ď ż T 0 |pA1¯nmpsq ´ A1npsq, ψq| ds ď ż T 0 |p¯nmpsq ´ npsq, A1ψq| ds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='75) ď T sup 0ďsďT |p¯nmpsq ´ npsq, A1ψq| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' From the convergence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='61) we infer that ¯nm Ñ n in Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2 wpOq, P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This means that sup0ďsďT |p¯nmpsq ´ npsq, ϕq| tends to zero for all ϕ P L2pOq as m goes to infinity with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We plug ϕ “ A1ψ and pass to the limit in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='75) and derive the second convergence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We have for all ω P Ω, ˇˇˇˇ ż t r pP2 mB1p¯umpsq, ¯nmpsqq, ψqds ´ ż t r pB1pupsq, npsqq, ψqds ˇˇˇˇ ď ż T 0 ˇˇpB1p¯umpsq, ¯nmpsqq, P2 mψ ´ ψq ˇˇ ` ż T 0 |pB1p¯umpsq, ¯nmpsqq ´ B1pupsq, npsqq, ψq| ds Since ¯um Ñ u in L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hq, and ¯nm Ñ n in L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqq P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', by integration-by-parts, we derive that ż T 0 ˇˇpB1p¯umpsq, ¯nmpsqq, P2 mψ ´ ψq ˇˇ ds ď ż T 0 ˇˇp¯nmpsq¯umpsq, ∇pP2 mψ ´ ψqq ˇˇ ď ˇˇ∇pP2 mψ ´ ψq ˇˇ L8 ż T 0 |¯nmpsq|L2 |¯umpsq|L2 ds ď ˇˇP2 mψ ´ ψ ˇˇ H3 ˆż T 0 |¯umpsq|2 L2 ds ˙1{2 ˆż T 0 |¯nmpsq|2 L2 ds ˙1{2 ď K ˇˇP2 mψ ´ ψ ˇˇ H3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 48 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ By using an integration-by-parts and the fact that ∇ ¨ u “ 0 we get ż T 0 |pB1p¯umpsq, ¯nmpsqq ´ B1pupsq, npsqq, ψq, ψq| ds ď ż T 0 |pp¯umpsq ´ upsqq∇¯nmpsq, ψq| ds ` ż T 0 |pupsq∇p¯nmpsq ´ npsqq, ψq| ds ď ż T 0 |p¯nmpsq, p¯umpsq ´ upsqq ¨ ∇ψq| ds ` ż T 0 |pp¯nmpsq ´ npsqq, upsq ¨ ∇ψq| ds ď |ψ|L8 ż T 0 |¯umpsq ´ upsq|L2 |¯nmpsq|L2 ds ` |∇ψ|L8 ż T 0 |¯nmpsq ´ npsq|L2 |upsq|L2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using the fact that |∇ψ|L8 ď |ψ|H3, we infer from the two last inequalities that ˇˇˇˇ ż t 0 pP2 mB1p¯umpsq, ¯nmpsqq, ψqds ´ ż t 0 pB1pupsq, npsqq, ψqds ˇˇˇˇ ď T |ψ|H3 ˆż T 0 |¯umpsq ´ upsq|2 L2 ds ˙1{2 ˆż T 0 |¯nmpsq|2 L2 ds ˙1{2 ` T |ψ|H3 ˆż T 0 |¯nmpsq ´ npsq|2 L2 ds ˙1{2 ˆż T 0 |upsq|2 L2 ds ˙1{2 ` K ˇˇP2 mψ ´ ψ ˇˇ H3 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='76) ď K ˆż T 0 |¯umpsq ´ upsq|2 L2 ds ˙1{2 ` K ˆż T 0 |¯nmpsq ´ npsq|2 L2 ds ˙1{2 ˆż T 0 |upsq|2 L2 ds ˙1{2 ` K ˇˇP2 mψ ´ ψ ˇˇ H3 , which upon letting n Ñ 8, implies the third convergence in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Similarly, we have ˇˇˇˇ ż t r pP2 mR2p¯nmpsq, ¯cmpsqq, ψqds ´ ż t r pR2pnpsq, cpsqq, ψqds ˇˇˇˇ ď ż T 0 |pR2p¯nmpsq, ¯cmpsqq ´ R2pnpsq, cpsqq, ψq| ds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='77) ` ż T 0 ˇˇpR2p¯nmpsq, ¯cmpsqq, P2 mψ ´ ψq ˇˇ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since p¯cm, ¯nmq Ñ pc, nq in Zc ˆ Zn, we see that P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s, ż T 0 ˇˇpR2p¯nmpsq, ¯cmpsqq, P2 mψ ´ ψq ˇˇ ds ď ˇˇ∇pP2 mψ ´ ψq ˇˇ L8 ż T 0 |¯nmpsq|L2 |∇¯cmpsq|L2 ds ď K ˇˇP2 mψ ´ ψ ˇˇ H3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 49 On the other hand, we obtain ż T 0 |pR2p¯nmpsq, ¯cmpsqq ´ R2pnpsq, cpsqq, ψq, ψq| ds ď ż T 0 |pp¯nmpsq ´ npsqq∇¯cmpsq, ∇ψq| ds ` ż T 0 |pnpsq∇p¯cmpsq ´ cpsqq, ∇ψq| ds ď |∇ψ|L8 ż T 0 |¯nmpsq ´ npsq|L2 |∇¯cmpsq|L2 ds ` |∇ψ|L8 ż T 0 |∇p¯cmpsq ´ cpsqq|L2 |npsq|L2 ds ď K ˆż T 0 |¯nmpsq ´ npsq|2 L2 ds ˙1{2 ` K ˆż T 0 |∇p¯cmpsq ´ cpsqq|2 L2 ds ˙1{2 ˆż T 0 |npsq|2 L2 ds ˙1{2 , which along with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='61) implies the fourth convergence in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For any r, t P r0, Ts with r ď t and ψ P H2pOq, the following convergences hold P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' lim mÝÑ8p¯cmptq, ψq “ pcptq, ψq, lim mÝÑ8 ż t r pA1¯cmpsq, ψqds “ ż t r pA1cpsq, ψqds, lim mÝÑ8 ż t r pP2 mB1p¯umpsq, ¯cmpsqq, ψqds “ ż t r pB1pupsq, cpsqq, ψqds, lim mÝÑ8 ż t r pP2 mR1p¯nmpsq, ¯cmpsqq, ψqds “ ż t r pR1pnpsq, cpsqq, ψqds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='78) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since ¯cm Ñ c in Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqq, P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', the first convergence is done exactly using a similarly inequality as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By an integration by part and the H¨older inequality we note that ˇˇˇˇ ż t r pA1¯cmpsq, ψqds ´ ż t r pA1cpsq, ψqds ˇˇˇˇ ď ż T 0 |pA1¯cmpsq ´ A1cpsq, ψq| ds ď ż T 0 |p∇p¯cmpsq ´ cpsqq, ∇ψq| ds ď T 1{2 |ψ|H1 ˆż T 0 |¯cmpsq ´ cpsq|2 H1 ds ˙1{2 , which altogether with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='61) implies the second convergence in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 50 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' using the Sobolev embedding H1pOq ãÑ L4pOq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' we get ˇˇˇˇ ż t r pP2 mB1p¯umpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmpsqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ψqds ´ ż t r pB1pupsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cpsqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ψqds ˇˇˇˇ ď ż T 0 |pB1p¯umpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmpsqq ´ B1pupsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cpsqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ψq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ψq| ds ` ż T 0 ˇˇpB1p¯umpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmpsqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' P2 mψ ´ ψq ˇˇ ds ď ż T 0 |pp¯umpsq ´ upsqq∇¯cmpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ψq| ds ` ż T 0 |pupsq∇p¯cmpsq ´ cpsqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ψq| ds ` T 1{2 ˇˇP2 mψ ´ ψ ˇˇ L2 ˆż T 0 |B1p¯umpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmpsqq|2 L2 ds ˙1{2 ď |ψ|L4 ż T 0 |¯umpsq ´ upsq|L2 |∇¯cmpsq|L4 ds ` |ψ|L4 ż T 0 |∇p¯cmpsq ´ cpsqq|L2 |upsq|L4 ds ` K ˇˇP2 mψ ´ ψ ˇˇ L2 ď T |ψ|H1 ż T 0 |¯umpsq ´ upsq|L2 |¯cmpsq|H2 ds ` T |ψ|H1 ż T 0 |∇p¯cmpsq ´ cpsqq|L2 |∇upsq|L2 ds ` K ˇˇP2 mψ ´ ψ ˇˇ L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since the convergence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='61) holds, we arrive at ˇˇˇˇ ż t r pP2 mB1p¯umpsq, ¯cmpsqq, ψqds ´ ż t r pB1pupsq, cpsqq, ψqds ˇˇˇˇ ď T |ψ|H1 ˆż T 0 |¯umpsq ´ upsq|2 L2 ds ˙1{2 ˆż T 0 |¯cmpsq|2 H2 ds ˙ 1 2 ` T |ψ|H1 ˆż T 0 |¯cmpsq ´ cpsq|2 H1 ds ˙ 1 2 ˆż T 0 |∇upsq|2 L2 ds ˙ 1 2 ` K ˇˇP2 mψ ´ ψ ˇˇ L2 , which along with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='61) implies the third convergence in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now we prove the last convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' To this purpose, we note that ˇˇˇˇ ż t r pP2 mR1p¯nmpsq, ¯cmpsqq, ψqds ´ ż t r pR1pnpsq, cpsqq, ψqds ˇˇˇˇ ď ż T 0 |pR1p¯nmpsq, ¯cmpsqq ´ R1pnpsq, cpsqq, ψq| ds ` ż T 0 ˇˇpR1p¯nmpsq, ¯cmpsqq, P2 mψ ´ ψq ˇˇ ds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='79) ď ż T 0 |pp¯nmpsq ´ npsqqfp¯cmpsqq, ψq| ds ` ż T 0 |npsqpfp¯cmpsqq ´ fpcpsqqq, ψq| ds ` K ˇˇP2 mψ ´ ψ ˇˇ L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 51 Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='63), we derive that ż T 0 |pp¯nmpsq ´ npsqqfp¯cmpsqq, ψq| ds ď |ψ|L8 ż T 0 ż O |¯nmpsq ´ npsq| |fp¯cmpsqq| dxds ď T 1{2 |O|1{2 |ψ|H2 sup 0ďsď|c0|L8 fpsq ˆż T 0 |¯nmpsq ´ npsq|2 L2 ds ˙1{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In a similar way, we see that ż T 0 |nps, xqpfp¯cmps, xqq ´ fpcps, xqqq, ψq| dsdx ď |ψ|H2 ż T 0 ż O |nps, xqfp¯cmps, xqq ´ nps, xqfpcps, xqq| dxds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='80) Since the strong convergence ¯cm Ñ c in L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq, P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', holds, we derive that up to a subsequence ¯cm Ñ c dt b dx-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='e Owing to the fact that f is continuous, we infer that P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', nfp¯cmq Ñ nfpcq a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='e in ˆ p0, Tq ˆ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We also note that P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', tnfp¯cmqumě1 is uniformly integrable over p0, Tq ˆ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Indeed, we have ż p0,TqˆO |nps, xqfp¯cmps, xqq|2 dxdsdP ď sup 0ďsď|c0|L8 f 2psq ż T 0 ż O |nps, xq|2 dxds ď K ż T 0 |npsq|2 L2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Therefore, by the Vitali Convergence Theorem, we derive that P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', the right answer of the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='80) tends to zero as m tends to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Owing to this result, we can pass to the limit in the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='79) and obtain the last convergence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Next we prove the following convergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For any r, t P r0, Ts with r ď t and v P V , the following convergences hold P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' lim mÝÑ8p¯umptq, vq “ puptq, vq, lim mÝÑ8 ż t r pA0¯umpsq, vqds “ ż t r pA0upsq, vqds, lim mÝÑ8 ż t r pP1 mB0p¯umpsq, ¯umpsqq, vqds “ ż t r pB0pupsq, upsqq, vqds, lim mÝÑ8 ż t r pP1 mR0p¯nmpsq, Φq, vqds “ ż t r pR0pnpsq, Φq, vqds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='81) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The proof is similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ 52 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ In what follows, we will combine the convergence result from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14 as well as martingale representation theorem to construct a probabilistic weak solution to the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In order to simplify the notation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' we define on the probability space pΩ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' F1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' P1q the processes N 1 m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' N 2 m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' and N 3 m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' by for t P r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' N 1 mptq :“ ´¯umptq ´ ż t 0 rηA0¯umpsq ` P1 mB0p¯umpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯umpsqqsds ` um 0 ` ż t 0 P1 mR0p¯nmpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Φqds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' N 2 mptq :“ ´¯cmptq ´ ż t 0 rξA1¯cmpsq ` P2 mB1p¯umpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmpsqqsds ` cm 0 ´ ż t 0 P2 mR1p¯nmpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmpsqqds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' and N 3 mptq :“ ´¯nmptq ´ ż t 0 rδA1¯nmpsq ` P2 mB1p¯umpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯nmpsqqsds ` nm 0 ´ ż t 0 P2 mR2p¯nmpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmpsqqds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For all m P N and for any t P r0, Ts, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='82) N 3 mptq “ 0, P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let m P N and t P r0, Ts be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' On the probability space pΩ, F, Pq, we define the processes M3 mptq by M3 mptq :“ ´nmptq ´ ż t 0 rδA1nmpsq ` P2 mB1pumpsq, nmpsqqsds ` nm 0 ´ ż t 0 P2 mR2pnmpsq, cmpsqqds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We also define the following subsets of Ω and Ω1 AN mptq :“ ␣ ω1 P Ω1 : N 3 mptq “ 0 ( and AM m ptq :“ ␣ ω P Ω : M3 mptq “ 0 ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We note that, since the last equation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) holds, PpAM m ptqq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Furthermore, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='62), we derive that for all ω1 P Ω, N 3 mpt, ω1q “ M3 mpt, Ψmpω1qq and therefore we observe that AN mptq “ Ψ´1 m pAM m ptqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Invoking (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='62) once more, we deduce that P1pAN mptqq “ P1pΨ´1 m pAM m ptqqq “ PpAM m ptqq “ 1, which completes the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Using the convergences (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='72) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='73) as well as Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='15 we see that for all t P r0, Ts, P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='83) nptq ` ż t 0 rδA1npsq ` B1pupsq, npsqqsds “ n0 ´ ż t 0 R2pnpsq, cpsqqds, in H´3pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, on the probability space pΩ1, F1, P1q we define a the Hm ˆ Hm-valued processes Nm by Nmptq “ pN 1 mptq, N 2 mptqq for all m ě 1 and t P r0, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='84) Hm ˆ Hm Ă H ˆ L2pOq ãÑ V ˚ ˆ H´2pOq, the process Nm can be seen as a V ˚ ˆ H´2pOq-valued process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Next, we collect the necessary ingredients for the application of the martingale representation theorem from [12, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' To this aim, we consider the following Gelfand triple V ãÑ H ãÑ V ˚ and H2pOq ãÑ L2pOq ãÑ H´2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let i1 : V ãÑ H be the usual embedding and i1˚ its Hilbert-space-adjoint such that pix, yq “ px, i1˚yqV for all x P V and y P H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In a very similar way, we denote the usual embedding H2pOq ãÑ L2pOq by i2 and by i˚2 its ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 53 Hilbert-space-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We define the embedding i : V ˆ H2pOq ãÑ H ˆ L2pOq and its adjoint i˚ : H ˆ L2pOq ÝÑ V ˆ H2pOq respectively by i “ ˆ i1 0 0 i2 ˙ , i˚ “ ˆ i1˚ 0 0 i2˚ ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Further, we set L1 “ pi1˚q1 : V ˚ ÝÑ H as the dual operator of i1˚ such that for all x P H and y P V ˚, pLy, xq “ xy, xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Similarly, the dual operator of i2˚ will be denoted by L2 : H´2pOq ÝÑ L2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We then define the following dual operator L :“ pi˚q1 : V ˚ ˆ H´2pOq ÝÑ H ˆ L2pOq by L “ ˆ L1 0 0 L2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' On the space Hm ˆ Hm, we define a mapping Gm by Gmpv, ψq “ ˆ L1P1 mgpv, ψq 0 0 L2P2 mφpψq ˙ , pv, ψq P Hm ˆ Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Here pP1 mgpv, ψq, P2 mφpψqq “ pP1 mgpv, ψq, P2 mφpψqq is seen as an element of V ˚ ˆ H´2pOq owing to the inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In the following lemma, we prove the martingale property of the process LNm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For each m ě 1, the process LNm is an H ˆ L2pOq-valued continuous square integrable martingale with respect to the filtration F 1m “ ␣ σ ` σ pp¯umpsq, ¯cmpsq, ¯nmpsqq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' s ď tq Y N 1˘( tPr0,Ts , where N 1 is the set of null sets of F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The quadratic variation of LNm is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='85) xxLNmyyt “ ż t 0 Gmp¯umpsq, ¯cmpsqqGmp¯umpsq, ¯cmpsqq˚ds, where Gmp¯um, ¯cmq˚ : H ˆ L2pOq Ñ U ˆ R2 is the adjoint of the operator Gmp¯um, ¯cmq and is given by Gmp¯um, ¯cmq˚v “ ˜ 8 ÿ k“1 pP1 mgp¯um, ¯cmqek, i1˚wqek, 2ÿ k“1 pP2 mφp¯cmqgk, i2˚ψqgk ¸ , for all v “ pw, ψq P H ˆ L2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For any m ě 1 we define the V ˚ ˆ H´2pOq-valued processes Mm by Mmptq “ pM1 mptq, M2 mptqq, t P r0, Ts, where M1 mptq :“ ´umptq ´ ż t 0 rηA0umpsq ` P1 mB0pumpsq, umpsqqsds ` um 0 ` ż t 0 P1 mR0pnmpsq, Φqds, M2 mptq :“ ´cmptq ´ ż t 0 rξA1cmpsq ` P2 mB1pumpsq, cmpsqqsds ` cm 0 ´ ż t 0 P2 mR1pnmpsq, cmpsqqds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let us set Ws :“ pWs, βsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then, since pum, cm, nmq is a solution of the finite dimensional system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2), we deduce that LMm can be represented as LMmptq “ ż t 0 Gmpumpsq, cmpsqqdWs, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' for all t P r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 54 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Using the continuity property of the operators L1 and L2 as well as Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8, the estimate E ż T 0 |Gmpumpsq, cmpsqq|2 L2pUˆR2,HˆL2q ds ď KE ż T 0 ˇˇP1 mgpumpsq, cmpsqq ˇˇ2 L2pU,Hq ds ` KE ż T 0 ˇˇP2 mφpcmpsqq ˇˇ2 L2pR2,L2q ds ď KE ż T 0 p1 ` |pumpsq, cmpsqq|2 Hqds ` γ2 2ÿ k“1 |σk|2 L2 E ż T 0 |∇cmpsq|2 L2 ds ď K ˆ 1 ` E sup 0ďsďT |pumpsq, cmpsqq|2 H ˙ ` KE sup 0ďsďT |∇cmpsq|2 L2 ă 8, yields that Mm is a square integrable continuous martingale over the probability space pΩ, F, pFtqtPr0,Ts, Pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Moreover, from the definition of Mm we derive that for each t P r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Ts, Mmptq is measurable with respect to the σ-field Fm “ tσ pσ ppumpsq, cmpsq, nmpsqq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' s ď tq Y NqutPr0,Ts , where N is the set of null sets of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence, invoking [12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='27] we infer that Mm is a Fm-martingale with quadratic variation xxMmyyt “ ż t 0 Gmpumpsq, cmpsqqGmpumpsq, cmpsqq˚ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This means that for all s, t P r0, Ts, s ď t, all vi “ pwi, ψiq P H ˆ L2pOq, i “ 1, 2, and all bounded and continuous real-valued functions h “ ph1, h2, h3q on Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H ˆL2pOqˆL2pOqq, we have E ” pLMmptq ´ LMmpsq, v1qHˆL2pOq h1pum|r0,ssqh2pcm|r0,ssqh3pnm|r0,ssq ı “ 0, and E ”´ pLMmptq, v1qHˆL2pOq pLMmptq, v2qHˆL2pOq ´ pLMmpsq, v1qHˆL2pOq pLMmpsq, v2qHˆL2pOq ´ ż t 0 pGmpumpsq, cmpsqq˚v1, Gmpumpsq, cmpsqq˚v2qUˆR2 ds ˙ ˆ ˆh1pum|r0,ssqh2pcm|r0,ssqh3pnm|r0,ssq ‰ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since pum, cm, nmq and p¯um, ¯cm, ¯nmq have the same laws on Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hmq, we deduce from these two last equalities that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='86) E1 ” pLNmptq ´ LNmpsq, v1qHˆL2pOq h1p¯um|r0,ssqh2p¯cm|r0,ssqh3p¯nm|r0,ssq ı “ 0, and E1 ”´ pLNmptq, v1qHˆL2pOq pLNmptq, v2qHˆL2pOq ´ pLNmpsq, v1qHˆL2pOq pLNmpsq, v2qHˆL2pOq ´ ż t 0 pGmp¯umpsq, ¯cmpsqq˚v1, Gmp¯umpsq, ¯cmpsqq˚v2qUˆR2 ds ˙ ˆ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='87) ˆh1p¯um|r0,ssqh2p¯cm|r0,ssqh3p¯nm|r0,ssq ‰ “ 0, for all s, t P r0, Ts, s ď t, all vi “ pwi, ψiq P H ˆL2pOq, i “ 1, 2, and all (real-valued) function hi, i “ 1, 2, 3 bounded and continuous on Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hmq, Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hmq, and Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hmq ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 55 respectively, This implies that LNm is a continuous square integrable martingale with respect to F 1m and the quadratic variation is given as claimed by equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ On the new probability space pΩ1, F1, P1q, we consider the V ˚ ˆH´2pOq-valued continuous process N defined by Nptq “ pN 1ptq, N 2ptqq for all t P r0, Ts, where N 1ptq :“ ´uptq ´ ż t 0 rηA0upsq ` B0pupsq, upsqqsds ` u0 ` ż t 0 R0pnpsq, Φqds, N 2ptq :“ ´cptq ´ ż t 0 rξA1cpsq ` B1pupsq, cpsqqsds ` c0 ´ ż t 0 R1pnpsq, cpsqqds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In the next lemma, we state that LN “ pL1N 1, L2N 2q is also an H ˆL2pOq-valued martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The process LN is an H ˆL2pOq-valued continuous square integrable martingale with respect to the filtration F1 “ tσ ppupsq, cpsq, npsqq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' s ď tqutPr0,Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The quadratic variation is given by xxLNyyt “ ż t 0 Gpupsq, cpsqqGpupsq, cpsqq˚ds, where Gpu, cq “ ˆ L1gpu, cq 0 0 L2φpcq ˙ , and Gpu, cq˚ : H ˆ L2pOq Ñ U ˆ R2 is the adjoint of the operator Gpu, cq given by Gpu, cq˚v “ ˜ 8 ÿ k“1 pL1gpupsq, cpsqqek, wqek, 2ÿ k“1 pL2φpcpsqqgk, ψqgk ¸ , for all v “ pw, ψq P H ˆ L2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let t P r0, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We first prove that LN is an H ˆL2pOq-valued square integrable random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to the continuity of L, it will be sufficient to prove that E |N|2 V ˚ˆH´2 ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14, we conclude that lim mÝÑ8 Nmptq “ Nptq P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' in V ˚ ˆ H´2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By the continuity of the injection H ˆ L2pOq ãÑ V ˚ ˆ H´2pOq, the Burkholder-Gundy-Davis inequality for continuous martingales and equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='85) as well as inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='67) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='69), we have E1 sup 0ďsďT |Nmpsq|4 V ˚ˆH´2 ď KE1 sup 0ďsďT |Nmpsq|4 L2ˆL2 ď KE1 ˆż T 0 |Gmp¯umpsq, ¯cmpsqq|2 L2pUˆR2,HˆL2q ds ˙2 “ 2KE1 ˆż T 0 ˇˇP1 mgp¯umpsq, ¯cmpsqq ˇˇ2 L2pU,Hq ds ˙2 ` 2KE1 ˆż T 0 ˇˇP2 mφp¯cmpsqq ˇˇ2 L2pR2,L2q ds ˙2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='88) ď K ˆ 1 ` E1 sup 0ďsďT |p¯umpsq, ¯cmpsqq|4 H ˙ ` KE1 sup 0ďsďT |∇¯cmpsq|4 L2 ă K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 56 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Hence, by the Vitali Theorem, we infer that Nptq P L2pΩ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V ˚ ˆ H´2pOqq and lim mÝÑ8 Nmptq “ Nptq in L2pΩ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V ˚ ˆ H´2pOqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Next, let v “ pw, ψq P V ˚ ˆH´2pOq, and hi, i “ 1, 2, 3 be a bounded and continuous function on Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V ˚q, Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H´2pOqq, and Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H´3pOqq respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let s, t P r0, Ts such that s ď t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let Fmpt, sq :“ pLNmptq ´ LNmpsq, vqHˆL2pOq h1p¯um|r0,ssqh2p¯cm|r0,ssqh3p¯nm|r0,ssq, Fpt, sq :“ pLNptq ´ LNpsq, vqHˆL2pOq h1pu|r0,ssqh2pc|r0,ssqh3pn|r0,ssq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We will prove that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='89) lim mÝÑ8 E1Fmpt, sq “ E1Fpt, sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' To this aim, we start by noting that by the P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='-convergence p¯um, ¯cm, ¯nmq Ñ pu, c, nq in Z and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13 as well as Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14, we infer that lim mÝÑ8 Fmpt, sq “ Fpt, sq, P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We will now show that the function tFmpt, squmě1 are uniformly integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We use the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='88) to derive that E1 |Fmpt, sq|4 ď K |h1|4 L8 |h2|4 L8 |h3|4 L8 |v|4 HˆL2 E1 ” |Nmptq|4 L2ˆL2 ` |Nmpsq|4 L2ˆL2 ı ď K |h1|4 L8 |h2|4 L8 |h3|4 L8 |v|4 HˆL2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Invoking the Vitali Theorem, we get the convergence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let 0 ď s ď t ď T and vi “ pwi, ψiq P H ˆ L2pOq, i “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let Qmpt, sq : “ ´ pLNmptq, v1qHˆL2pOq pLNmptq, v2qHˆL2pOq ´ pLNmpsq, v1qHˆL2pOq pLNmpsq, v2qHˆL2pOq ¯ h1p¯um|r0,ssqh2p¯cm|r0,ssqh3p¯nm|r0,ssq, Qpt, sq : “ ´ pLNptq, v1qHˆL2pOq pLNptq, v2qHˆL2pOq ´ pLNpsq, v1qHˆL2pOq pLNpsq, v2qHˆL2pOq ¯ h1pu|r0,ssqh2pc|r0,ssqh3pn|r0,ssq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Our purpose now is to prove that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='90) E1Qpt, sq “ lim mÝÑ8 E1Qmpt, sq, imitating the proof before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Indeed, by P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='-convergence p¯um, ¯cm, ¯nmq Ñ pu, c, nq in Z and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='13 as well as Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14 once more, we obtain lim mÝÑ8 Qmpt, sq “ Qpt, sq, P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We now prove the uniform integrability of Qmpt, sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For this purpose, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='88) we find that E1 |Qmpt, sq|2 ď K |h1|2 L8 |h2|2 L8 |h3|2 L8 E1 „ˇˇˇpNmptq, v1qHˆL2pOq pNmptq, v2qHˆL2pOq ˇˇˇ 2 ` ˇˇˇpNmpsq, v1qHˆL2pOq pNmpsq, v2qHˆL2pOq ˇˇˇ 2\uf6be ď K |h1|2 L8 |h2|2 L8 |h3|2 L8 |v1|2 HˆL2 |v2|2 HˆL2 E1 ” |Nmptq|4 L2ˆL2 ` |Nmpsq|4 L2ˆL2 ı ď K |h1|2 L8 |h2|2 L8 |h3|2 L8 |v|2 HˆL2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 57 As before, the Vitali Theorem yields equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Finally, we also define Rmpt, sq :“ ˆż t s pGmp¯umprq, ¯cmprqq˚v1, Gmp¯umprq, ¯cmprqq˚v2qUˆR2 dr ˙ ˆ ˆ h1p¯um|r0,ssqh2p¯cm|r0,ssqh3p¯nm|r0,ssq, and Rpt, sq :“ ˆż t s pGpuprq, cprqq˚v1, Gpuprq, cprqq˚v2qUˆR2 dr ˙ h1pu|r0,ssqh2pc|r0,ssqh3pn|r0,ssq, We claim that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='91) lim mÝÑ8 E1Rmpt, sq “ E1Rpt, sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In order to establish this claim we first show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='92) lim mÝÑ8 Rmpt, sq “ Rpt, sq, P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since h1p¯um|r0,ssqh2p¯cm|r0,ssqh3p¯nm|r0,ssq Ñ h1pu|r0,ssqh2pc|r0,ssqh3pn|r0,ssq P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', in order to prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='92), it is sufficient to prove that lim mÝÑ8 ż t s pGmp¯umprq, ¯cmprqq˚v1, Gmp¯umprq, ¯cmprqq˚v2qUˆR2 dr “ ż t s pGpuprq, cprqq˚v1, Gpuprq, cprqq˚v2qUˆR2 dr, P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='93) For all r P rs, ts, we set Jprq :“ pGmp¯umprq, ¯cmprqq˚v1, Gmp¯umprq, ¯cmprqq˚v2qUˆR2 ´ pLGpuprq, cprqq˚v1, LGpuprq, cprqq˚v2qUˆR2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then, we note that ż t s |Jprq| dz ď ż T 0 ˇˇpGmp¯umprq, ¯cmprqq˚v1 ´ LGpuprq, cprqq˚v1, Gmp¯umprq, ¯cmprqq˚v2qUˆR2 ˇˇ dr ` ż T 0 ˇˇpGpuprq, cprqq˚v1, Gmp¯umprq, ¯cmprqq˚v2 ´ Gpuprq, cprqq˚v2qUˆR2 ˇˇ dr (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='94) “ I1pmq ` I2pmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using the Cauchy-Schwarz inequality and the H¨older inequality, we derive that I1pmq ď ˆż T 0 |Gmp¯umprq, ¯cmprqq˚v1 ´ Gpuprq, cprqq˚v1q|2 UˆR2 dr ˙ 1 2 ˆ ˆ ˆż T 0 |Gmp¯umprq, ¯cmprqq˚v2|2 UˆR2 dr ˙ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Owing to the fact that P1 mgp¯um, ¯cmqek P H and P2 mφp¯cmqgk P L2pOq, we infer that pL1P1 mgp¯um, ¯cmqek, w1q :“ xP1 mgp¯um, ¯cmqek, i1˚w1y “ pgpu, cqek, i1˚w1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' and pL2P2 mφp¯cmqgk, ψ2q :“ xP2 mφp¯cmqgk, i2˚ψ2y “ pP2 mφp¯cmqgk, i2˚ψ2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 58 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Thus, using the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12) and the fact that tekukě1 and tgkuk“1,2 are orthonormal basis of U and R2 respectively, we derive that ż T 0 |Gmp¯umprq, ¯cmprqq˚v2|2 UˆR2 dr “ ż T 0 ¨ ˝ ˇˇˇˇˇ 8 ÿ k“1 pL1P1 mgp¯umprq, ¯cmprqqek, w2qek ˇˇˇˇˇ 2 U ` ˇˇˇˇˇ 2ÿ k“1 pL2P2 mφp¯cmprqqgk, ψ2qgk ˇˇˇˇˇ 2 R2 ˛ ‚dr ď ż T 0 8 ÿ k“1 ˇˇpP1 mgp¯umprq, ¯cmprqqek, i1˚w2q ˇˇ2 dr ` ż T 0 2ÿ k“1 ˇˇpP2 mφp¯cmprqqgk, i2˚ψ2q ˇˇ2 dr (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='95) ď ˇˇi1˚w2 ˇˇ2 L2 ż T 0 |gp¯umprq, ¯cmprqq|2 L2pU,Hq dr ` ˇˇi2˚ψ2 ˇˇ2 L2 ż T 0 |φp¯cmprqq|2 L2pR2,L2q dr ď K ż T 0 p1 ` |p¯umprq, ¯cmprqq|2 Hqdr ` K ż T 0 |∇¯cmprqq|2 L2 dr ď K, P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In the last line we used the fact that ¯cm Ñ c in L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq and ¯um Ñ u in L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hq P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' On the other hand, we note that ż T 0 |Gmp¯umprq, ¯cmprqq˚v1 ´ Gpuprq, cprqq˚v1q|2 UˆR2 dr ď ż T 0 ˇˇˇˇˇ « 8 ÿ k“1 pL1P1 mgp¯umprq, ¯cmprqqek, w1q ´ 8 ÿ k“1 pL1gpuprq, cprqqek, w1q ff ek ˇˇˇˇˇ 2 U dr ` ż T 0 ˇˇˇˇˇ « 2ÿ k“1 pL2P2 mφp¯cmprqqgk, ψ1q ´ 2ÿ k“1 pL2φpcprqqgk, ψ1q ff gk ˇˇˇˇˇ 2 R2 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then by this last inequality and the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='95), we infer that I2 1pmq ď K ż T 0 ˇˇˇˇˇ 8 ÿ k“1 pgp¯umprq, ¯cmprqqek, P1 mi1˚w1q ´ 8 ÿ k“1 pgpuprq, cprqqek, i1˚w1q ˇˇˇˇˇ 2 dr ` K ż T 0 ˇˇˇˇˇ 2ÿ k“1 pφp¯cmprqqgk, P2 mi2˚ψ1q ´ 2ÿ k“1 pφpcprqqgk, i2˚ψ1q ˇˇˇˇˇ 2 dr ď K ˇˇi1˚w1 ˇˇ2 L2 ż T 0 |gp¯umprq, ¯cmprqq ´ gpuprq, cprqq|2 L2pU,Hq dr (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='96) ` K ˇˇP1 mi1˚w1 ´ i1˚w1 ˇˇ2 L2 ż T 0 |gp¯umprq, ¯cmprqq|2 L2pU,Hq dr ` ˇˇi2˚ψ1 ˇˇ2 L2 ż T 0 |φp¯cmprqq ´ φpcprqq|2 L2pR2,L2q dr ` ˇˇP2 mi2˚ψ1 ´ i2˚ψ1 ˇˇ2 L2 ż T 0 |φp¯cmprqq|2 L2pR2,L2q dr :“ II1pmq ` II2pmq ` II3pmq ` II4pmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 59 By means of the continuity of g, the P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='-convergence p¯um, ¯cm, ¯nmq Ñ pu, c, nq in Z, the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12) and the Vitali Theorem, we can derive that limmÝÑ8 II1pmq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Furthermore, since ż T 0 |gp¯umprq, ¯cmprqq|2 L2pU,Hq dr ` ż T 0 |φp¯cmprqq|2 L2pR2,L2q dr ď K ż T 0 p1 ` |p¯umprq, ¯cmprqq|2 Hqdr ` K ż T 0 |∇¯cmprq|2 L2 dr ď K P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', we deduce that lim mÝÑ8 II2pmq “ lim mÝÑ8 II4pmq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, we study the II3pmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We see that II3pmq ď |ψ1|2 L2 γ2 |σ|2 L8 ż T 0 |∇¯cmprq ´ ∇cprq|2 L2 dr ď |ψ1|2 L2 γ2 |σ|2 L8 ż T 0 |¯cmprq ´ cprq|2 H1 dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By using the fact that ¯cm Ñ c in L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq, P1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', we can pass to the limit in this last inequality and infer that limmÝÑ8 II3pmq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hence passing to the limit in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='96) we get limmÝÑ8 I1pmq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In a similar fashion, we can also prove that limmÝÑ8 I2pmq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Therefore, passing to the limit in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='94), we obtain the convergence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='93) and completes the proof of the almost surely convergence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' To finish the proof of equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='91), it remains to prove the uniform integrability of Rmpt, sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' For this purpose, using the Young inequality, a similar calculations as in inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='95) and the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='67) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='69),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' we arrive at E1 |Rmpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' sq|2 ď 3 ź i“1 |hi|2 L8 E1 ˆż t s pGmp¯umprq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmprqq˚v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Gmp¯umprq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmprqq˚v2qUˆR2 dr ˙2 ď Kpt ´ sqE1 ż t s |Gmp¯umprq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmprqq˚v1|2 UˆR2 |Gmp¯umprq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmprqq˚v2|2 UˆR2 dr ď KE1 ż T 0 |Gmp¯umprq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmprqq˚v1|4 UˆR2 dr ` KE1 ż T 0 |Gmp¯umprq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmprqq˚v2|4 UˆR2 dr ď KE1 ż T 0 |gp¯umprq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmprqq|4 L2pU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Hq dr ` KE1 ż T 0 |φp¯cmprqq|4 L2pR2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q dr ď KE1 sup 0ďrďT p1 ` |p¯umprq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯cmprqq|4 Hq ` KE1 sup 0ďrďT |∇¯cmprqq|4 L2 ď K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' which prove the uniform integrability of Rmpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thus, invoking the Vitali Theorem, we obtain the convergence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Taking into account the convergences (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='89), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='90) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='91), we can pass to the limit in the equalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='86) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='87) to get E ” pLNptq ´ LNpsq, v1qHˆL2pOq h1pu|r0,ssqh2pc|r0,ssqh3pn|r0,ssq ı “ 0, 60 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ and E ”´ pLNptq, v1qHˆL2pOq pLNptq, v2qHˆL2pOq ´ pLNpsq, v1qHˆL2pOq pLNpsq, v2qHˆL2pOq ´ ż t 0 pGpupsq, cpsqq˚v1, Gpupsq, cpsqq˚v2qUˆR2 ds ˙ h1pu|r0,ssqh2pc|r0,ssqh3pn|r0,ssq \uf6be “ 0, which complete the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Thanks to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='17, we apply the usual martingale representation theorem proved in [12, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2] to the process LN and conclude that there exists a probability space p˜Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ˜F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ˜Pq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' a filtration ˜F and a U ˆ R2-cylindrical Wiener process ¯ Ws :“ p ¯Ws,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯βsq defined on the probability space p¯Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ¯Pq “ pΩ1 ˆ ˜Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' F1 b ˜F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' P1 b ˜Pq adapted to the filtration ¯F “ F1 b ˜F such that LNpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ˜ωq “ ż t 0 Gpups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ˜ωq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ˜ωqqd ¯ Wspω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ˜ωq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' t P r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' pω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ˜ωq P ¯Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' where LNpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ˜ωq “ LNpt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω1q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' pups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ˜ωq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ˜ωqq “ pups,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω1q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ω1qq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' t P r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' pω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ˜ωq P ¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' This implies that in the probability space p¯Ω, ¯F, ¯Pq, for t P r0, Ts and ¯P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='97) $ ’ ’ ’ & ’ ’ ’ % L1N 1ptq “ ż t 0 L1gpupsq, cpsqqd ¯ Ws, in H, L2N 2ptq “ ż t 0 L2φpcpsqqd¯βs, in L2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='70) the estimate ¯E ż T 0 |gpupsq, cpsqq|2 L2pU,V ˚q ds ď K¯E ż T 0 |gpupsq, cpsqq|2 L2pU,Hq ds ď K ˆ 1 ` E1 sup 0ďsďT |pupsq, cpsqq|2 H ˙ ă 8, and ¯E ż T 0 |φpcpsqq|2 L2pR2,H´2q ds ď K¯E ż T 0 |φpcpsqq|2 L2pR2,L2q ds ď K ˆ 1 ` E1 sup 0ďsďT |cpsq|2 H1 ˙ ă 8, yield that L1N 1 and L2N 2 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='97) are continuous martingale in H and L2pOq respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In a similar fashion as in [6, Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1], using the continuity of the operators L1 and L2, we get ż t 0 L1gpupsq, cpsqqd ¯ Ws “ L1 ˆż t 0 gpupsq, cpsqqd ¯ Ws ˙ and ż t 0 L2φpcpsqqd¯βs “ L2 ˆż t 0 φpcpsqqd¯βs ˙ , for all t P r0, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Combining these two last inequalities with the injectivity of the operators L1 and L2, we infer from the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='97) that for t P r0, Ts, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='98) $ ’ ’ ’ & ’ ’ ’ % N 1ptq “ ż t 0 gpupsq, cpsqqd ¯ Ws, in V ˚, N 2ptq “ ż t 0 φpcpsqqd¯βs, in H´2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 61 On the new probability space p¯Ω, ¯F, ¯Pq, we also extend the random variable nptq by npt, ω1, ˜ωq “ npt, ω1q, t P r0, Ts, pω1, ˜ωq P ¯Ω, and infer that the equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='83) also hods in p¯Ω, ¯F, ¯Pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using this, the definition of N 1 and N 2, and the system (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='98), we derive that p¯Ω, ¯F, ¯F, ¯P, pu, c, nq, p ¯ W , ¯βqq satisfies the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In particular, we have for all t P r0, Ts and ¯P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' $ ’ ’ ’ & ’ ’ ’ % uptq “ u0 ´ ż t 0 rηA0upsq ` B0pupsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' upsqq ` R0pnpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Φqsds ` ż t 0 gpupsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cpsqqd ¯ Ws,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' in V ˚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cptq “ c0 ´ ż t 0 rξA1cpsq ` B1pupsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cpsqq ´ R1pnpsq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cpsqqsds ` γ ż t 0 φpcpsqqd¯βs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' in H´2pOq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' which can be written as $ ’ ’ ’ & ’ ’ ’ % uptq “ u0 ´ ż t 0 G0psqds ` ż t 0 S0psqd ¯Ws,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' in V ˚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cptq “ c0 ´ ż t 0 G1psqds ` ż t 0 S1psqd¯βs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' in H´2pOq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' where for all t P r0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Ts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' G0ptq :“ ηA0uptq ` B0puptq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' uptqq ` R0pnptq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Φq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' G1ptq :“ ξA1cptq ` B1puptq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cptqq ´ R1pnptq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cptqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' S0ptq :“ gpuptq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' cptqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' and S1ptq :“ γφpcptqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Since the identities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='66), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='70) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='71) hold, following the idea of the proof of estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='57), we can see that G0 P L2pr0, Ts ˆ ¯Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V ˚q, G1 P L2pr0, Ts ˆ ¯Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqq, S0 P L2pr0, Ts ˆ ¯Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hq and S1 P L2pr0, Ts ˆ ¯Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Therefore, it follows from [23, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2] that there exists ¯Ω0 P ¯F such that ¯Pp¯Ω0q “ 1 and for all ω P ¯Ω0, the function u and c take values in H and in H1pOq respectively and are continuous in H and H1pOq with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Owing to the fact that pu, c, nq is Zu ˆ Zc ˆ Zn-valued random variable and progressively measurable over the filtration ¯F, we derive that p¯Ω, ¯F, ¯F, ¯P, pu, c, nq, p ¯ W , ¯βqq is a probabilistic weak solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We recall that the inequalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) directly follows from relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='66), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='70), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' PROPERTIES OF SOLUTION AND ENERGY INEQUALITY In this section we prove the mass conservation property, the non-negativity property and the L8-norm stability for the prrobabilistic strong solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By these properties, we also prove an energy inequality which may be useful for the study of the invariant measure of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) which is still an opened problem according to our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Non-negativity and mass conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The following theorem gives the conservation of the total mass property and the non-negativity of the strong solutions of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let A “ pΩ, F, tFtutPr0,Ts, Pq be a filtered probability space, U be a separable Hilbert space, W be cylindrical Wiener process on U over A, and β “ pβ1, β2q be a two dimensional standard Brownian motion over A independent of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' If pu, c, nq is a probabilistic strong solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2), then the following equality holds for all t P r0, Ts (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) ż O npt, xqdx “ ż O n0pxqdx, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 62 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ Furthermore, if c0 ą 0 and n0 ą 0, then the following inequality hold P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) nptq ą 0, and cptq ą 0, for all t P r0, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We Note that, the conservation of the total mass (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) follows straightforwardly from the fact that ∇ ¨ u “ 0 and the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) is very similar to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ The following theorem gives the L8-stability of the probabilistic strong solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let A “ pΩ, F, tFtutPr0,Ts, Pq be a filtered probability space, U be a separable Hilbert space, W be cylindrical Wiener process on U over A, and β “ pβ1, β2q be a two dimensional standard Brownian motion over A independent of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' If pu, c, nq is a probabilistic strong solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) in the filtered probability space A, then for all t P r0, Ts (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3) |cptq|L8 ď |c0|L8 , P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The proof is similar to the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Energy inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In this subsection, we will derive an energy inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The probabilistic strong solution pu, n, cq involving the following Lyapunov functional Epn, c, uqptq “ ż O nptq ln nptqdx`Kf |∇cptq|2 L2 ` 8KfKGN |c0|2 L8 3ξη |uptq|2 L2 `e´1 |O| , t P r0, Ts, where KGN is a constant given by the Gagliardo-Niremberg inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7) and Kf is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Suppose that Assumption 1, Assumption 2 and the following inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4) 4Kf max 0ďcď|c0|L8 f 2 min 0ďcď|c0|L8 f 1 ď δ, are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let A “ pΩ, F, tFtutPr0,Ts, Pq be a filtered probability space, U be a separable Hilbert space, W be cylindrical Wiener process on U over A, and β “ pβ1, β2q be a two dimensional standard Brownian motion over A independent of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then, any probabilistic strong solution pu, c, nq of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) in the filtered probability space A satisfies the following entropy functional relations for almost all t P r0, Ts, |cptq|2 L2 ` 2η ż t 0 |∇cpsq|2 L2 ds ` 2 ż t 0 pnpsqfpcpsqq, cpsqqds “ |c0|2 L2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5) Epn, c, uqptq ` ż t 0 « δ ˇˇˇ∇ a npsq ˇˇˇ 2 L2 ` 3ξKf 2 |∆cpsq|2 L2 ` 8KfKGN |c0|2 L8 3ξ |∇upsq|2 L2 ` ˇˇˇ a npsq∇cpsq ˇˇˇ 2 L2 ff ds ď Epn0, c0, u0q ` K5t ` K6 ż t 0 |upsq|2 L2 ds ` γ2Kf ż t 0 |∇φpcpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ds ` 8KfKGN |c0|2 L8 3ξη ż t 0 |gpupsq, cpsqq|2 L2pU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Hq ds ` 2γKf ż t 0 p∇φpcpsqq, ∇cpsqqdβs (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6) ` 16KfKGN |c0|2 L8 3ξη ż t 0 pgpupsq, cpsqq, upsqqdWs, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', where K5 and K6 are some positive constant to be given later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 63 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5) follows directly from the application of the Itˆo formula to t ÞÑ |cptq|2 L2 and the fact that pB1pu, cq, cq “ 1 2 ż O upxq ¨ ∇c2pxqdx “ ´1 2 ż O c2pxq∇ ¨ upxqdx “ 0, as well as pφpcq, cq “ 2ÿ k“1 ż O σkpxq ¨ ∇cpxqcpxqdx “ 1 2 2ÿ k“1 ż O σkpxq ¨ ∇c2pxqdx “ 0 and |φpcq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q “ |∇c|2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Next, we multiply equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='14)3 by 1 ` ln npsq for s P r0, ts and integrate the resulting equation in O to obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7) d dt ż O nps, xq ln nps, xqdx ` δ ż O |∇nps, xq|2 nps, xq dx “ χ ż O ∇nps, xq ¨ ∇cps, xqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thanks to the Young inequality and the Cauchy-Schwartz inequality we note that χ ż O ∇npxq ¨ ∇cpxqdx ď 2δ ż O ˇˇˇ∇ a npxq ˇˇˇ 2 dx ` χ2 2δ ż O npxq |∇cpxq|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Combining the last inequality with equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7) we arrive at ż O npt, xq ln npt, xqdx ` 2δ ż t 0 ˇˇˇ∇ a npsq ˇˇˇ 2 L2 ds ď ż O n0pxq ln n0pxqdx ` χ2 2δ ż t 0 ˇˇˇ a npsq∇cpsq ˇˇˇ 2 L2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8) By applying the Itˆo formula to t ÞÑ |∇cptq|2 L2, we find that |∇cptq|2 L2 ` 2ξ ż t 0 |∆cpsq|2 L2 ds “ |∇c0|2 L2 ´ 2 ż t 0 p∇B1pupsq, cpsqq, ∇cpsqqds ´ 2 ż t 0 p∇R2pnpsq, cpsqq, ∇cpsqqds ` γ2 ż t 0 |∇φpcpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ` 2γ ż t 0 p∇φpcpsqq, ∇cpsqqdβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9) Due to the Assumption 1 and the L8-norm stability obtained in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2, we obtain p∇B1pu, cq, ∇cq ď |∇u|L2 |∇c|2 L4 ď 3ξ 16KGN |c0|2 L8 |∇c|4 L4 ` 4KGN |c0|2 L8 3ξ |∇u|2 L2 ď ξ 4 |∆c|2 L2 ` 4KGN |c0|2 L8 3ξ |∇u|2 L2 ` ξp4K2 ` 3q 16 |c0|2 L8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 64 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ and ´p∇R2pn, cq, ∇cpsqqds ď ´ min 0ďcď|c0|L8 f 1pcq 2 ż O npxq |∇cpxq|2 dx ` 1 2 min 0ďcď|c0|L8 f 1 ż O f 2pcpxqq|∇npxq|2 npxq dx ď ´ min 0ďcď|c0|L8 f 1pcq 2 ˇˇ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='n∇c ˇˇ2 L2 ` 2 max 0ďcď|c0|L8 f 2 min 0ďcď|c0|L8 f 1pcq ˇˇ∇?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='n ˇˇ2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Thus, we see from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9) that |∇cptq|2 L2 ` 3ξ 2 ż t 0 |∆cpsq|2 L2 ds ` min 0ďcď|c0|L8 f 1 ż t 0 ˇˇˇ a psq∇cpsq ˇˇˇ 2 L2 ds ď |∇c0|2 L2 ` ξp4K2 ` 3q 8 |c0|2 L8 t ` 8KGN |c0|2 L8 3ξ ż t 0 |∇upsq|2 L2 ds ` 4 max 0ďcď|c0|L8 f 2 min 0ďcď|c0|L8 f 1 ż t 0 ˇˇˇ∇ a npsq ˇˇˇ 2 L2 ds ` γ2 ż t 0 |∇φpcpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ds ` 2γ ż t 0 p∇φpcpsqq, ∇cpsqqdβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Now, we multiply this last inequality by Kf, add the result with inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8), and use the inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4) to obtain ż O npt, xq ln npt, xqdx ` Kf |∇cptq|2 L2 ` 3ξKf 2 ż t 0 |∆cpsq|2 L2 ds ` 2δ ż t 0 ˇˇˇ∇ a npsq ˇˇˇ 2 L2 ds ` ż t 0 ˇˇˇ a npsq∇cpsq ˇˇˇ 2 L2 ds ď Kf |∇c0|2 L2 ` ż O n0pxq ln n0pxqdx ` Kfξp4KfK2 ` 3q 8 |c0|2 L8 t ` 8KfKGN |c0|2 L8 3ξ ż t 0 |∇upsq|2 L2 ds ` γ2Kf ż t 0 |∇φpcpsqq|2 L2pR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10) ` 2γKf ż t 0 p∇φpcpsqq, ∇cpsqqdβs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Using the equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) and the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7) we note that |n|L2 ď KGN ´ˇˇ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='n ˇˇ L2 ˇˇ∇?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='n ˇˇ L2 ` ˇˇ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='n ˇˇ2 L2 ¯ ď KGN ˆ |n0| 1 2 L1 ˇˇ∇?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='n ˇˇ L2 ` |n0|L1 ˙ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='11) ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 65 which altogether with the Itˆo formula to t ÞÑ |uptq|2 L2 implies the existence of K3 ą 0 such that |uptq|2 L2 ` 2η ż t 0 |∇upsq|2 L2 ds ď 2 ż t 0 |∇Φ|L8 |npsq|L2 |upsq|L2 ds ` ż t 0 |gpupsq, cpsqq|2 L2pU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Hq ds ` 2 ż t 0 pgpupsq, cpsqq, upsqqdWs ď |u0|2 L2 ` δη K4 ż t 0 ˇˇˇ∇ a npsq ˇˇˇ 2 L2 ds ` K3 |∇Φ|2 L8 |n0|L1 ż t 0 |upsq|2 L2 ds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12) ` 1 2t ` 1 2 |∇Φ|2 L8 |n0|2 L1 ż t 0 |upsq|2 L2 ds ` ż t 0 |gpupsq, cpsqq|2 L2pU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='Hq ds ` 2 ż t 0 pgpupsq, cpsqq, upsqqdWs, with K4 “ 8Kf KGN|c0|2 L8 3ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Multiplying the inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='12) by K4 η , and adding the result with inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='10), we obtain some positive constants K5 and K6 such that the inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' COMPACTNESS AND TIGHTNESS CRITERIA In this appendix we recall several compactness and tightness criteria that are frequently used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We start with the following lemma based on the Dubinsky Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let us consider the space (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1) ˜Z0 “ L2 wp0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H´3pOqq and ˜T0 be the supremum of the corresponding topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then a set ¯¯K0 Ă ˜Z0 is ˜T0-relatively compact if the following three conditions hold (a) sup ϕP ¯¯ K0 Tż 0 |ϕpsq|2 H1ds ă 8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', ¯¯K0 is bounded in L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq, (b) Dγ ą 0: sup ϕP ¯¯ K0 |ϕ|Cγpr0,Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='H´3q ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We note that the following embedding is continuous H1pOq ãÑ L2pOq ãÑ H´3pOq with H1pOq ãÑ L2pOq compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' By the Banach-Alaoglu Theorem condition (a) yields that ¯¯K0 is compact in L2 wp0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Moreover (b) implies that the functions ϕ P ¯¯K0 are equicontinuous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' for all ε ą 0, there exists δ ą 0 such that if |t ´ s| ă δ then |ϕptq ´ ϕpsq|H´3 ă ε for all ϕ P ¯¯K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We can then apply Dubinsky’s Theorem (see [41, Theorem IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='1]) since by condition (a), ¯¯K0 is bounded in L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' □ Following the same method as in [8, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3 ], we obtain the following compactness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let us consider the space (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2) ˜Zn “ L2 wp0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H´3pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2 wpOqq, and ˜T0 be the supremum of the corresponding topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then a set ¯¯K0 Ă ˜Zn is ˜T0-relatively compact if the following three conditions hold 66 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ (a) sup ϕP ¯¯ K0 |ϕ|L8p0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ă 8, (b) sup ϕP ¯¯ K0 Tż 0 |ϕpsq|2 H1ds ă 8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', ¯¯K0 is bounded in L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1pOqq, (c) Dγ ą 0: sup ϕP ¯¯ K0 |ϕ|Cγpr0,Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='H´3q ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' From this lemma we also get the following tightness criteria for stochastic processes with paths in ˜Zn where the proof is the same as the proof of [3, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3 (Tightness criterion for n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let γ ą 0 be a given parameters and pϕnqn be a sequence of continuous tFtutPr0,Ts-adapted H´3pOq-valued processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let Lm be the law of ϕn on ˜Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' If for any ε ą 0 there exists a constant Ki, i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', 3 such that sup m P ´ |ϕm|L8p0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='L2q ą K1 ¯ ď ε, sup m P ´ |ϕm|L2p0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='H1q ą K2 ¯ ď ε, sup m P ´ |ϕm|Cγp0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='H´3q ą K3 ¯ ď ε, then the sequence pLmqm is tight on ˜Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The following compactness results are due to [7, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5] (see also [28]), where we can see the details of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let us consider the space (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3) ˜Zu “ L2 wp0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V q X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V ˚q X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Hwq, and ˜T1 be the supremum of the corresponding topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then a set ¯¯K1 Ă ˜Zu is ˜T1-relatively compact if the following three conditions hold (a) sup vP ¯¯ K1 sup tPr0,Ts |vptq|L2 ă 8, (b) sup vP ¯¯ K1 Tż 0 |∇vpsq|2 L2ds ă 8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', ¯¯K2 is bounded in L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' V q, (c) lim δÑ0 sup vP ¯¯ K1 sup s,tPr0,Ts,|t´s|ďδ |vptq ´ vpsq|V ˚ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let us consider the space (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4) ˜Zc “ L2 wp0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H2pOqq X L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1 wpOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' L2pOqq X Cpr0, Ts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H1 wpOqq, and ˜T2 be the supremum of the corresponding topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then a set ¯¯K2 Ă ˜Zc is ˜T2-relatively compact if the following three conditions hold (a) sup ϕP ¯¯ K2 sup tPr0,Ts |ϕptq|H1 ă 8, (b) sup ϕP ¯¯ K2 Tż 0 |ϕpsq|2 H2ds ă 8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=', ¯¯K2 is bounded in L2p0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' H2pOqq, (c) lim δÑ0 sup ϕP ¯¯ K2 sup s,tPr0,Ts,|t´s|ďδ |ϕptq ´ ϕpsq|L2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' ON THE STOCHASTIC CHEMOTAXIS-NAVIER-STOKES MODEL 67 We now consider a filtered probability space pΩ, F, Pq with filtration F :“ tFtutě0 satisfying the usual hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let pM, d1q be a complete, separable metric space and pynqnPN be a sequence of F-adapted and M-valued processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' We recall from [20] the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A sequence pynqnPN satisfies the Aldous condition in the space M if and only if @ǫ ą 0 @ζ ą 0 Dδ ą 0 such that for every sequence pτnqnPN of F-stopping times with τn ď T one has sup nPN sup 0ďθďδ P t|ynpτn ` θq ´ ynpτnq|M ě ζu ď ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='6, and throughout we understand that yn is extended to zero outside the interval r0, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' The following lemma is proved in [28, Appendix A, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let pX, |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='|Xq be a separable Banach space and let pynqnPN be a sequence of X-valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Assume that for every pτnqnPN of F-stoppings times with τn ď T and for every n P N and θ ě 0 the following condition holds (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='5) E |ynpτn ` θq ´ ynpτnq|α X ď Cθβ, for some α, β ą 0 and some constant C ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then the sequence pynqnPN satisfies the Aldous condition in the space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In the view of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='4 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='2, in the next corollaries, we will state a tightness criteria for stochastic processes with part in ˜Zu or in ˜Zc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let pvmqm be a sequence of continuous tFtutPr0,Ts-adapted V ˚-valued processes satisfying (a): there exists a constant K1 ą 0 such that sup m E sup 0ďsďT |vmpsq|2 L2 ď K1, (b): there exists a constant K2 ą 0 such that sup m ż T 0 |∇vmpsq|2 L2 ds ď K2, (c): pvmqm satisfies the Aldous condition in V ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let Lmpvmq be the law of vm on ˜Zu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then, the sequence pLmpvmqqm is tight in ˜Zu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' pvmqm be a sequence of continuous tFtutPr0,Ts-adapted L2pOq-valued processes satisfying (a): there exists a constant K1 ą 0 such that sup m E sup 0ďsďT |vmpsq|2 H1 ď K1, (b): there exists a constant K2 ą 0 such that sup m ż T 0 |vmpsq|2 H2 ds ď K2, (c): pvmqm satisfies the Aldous condition in L2pOq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Let Lmpvmq be the law of vm on ˜Zc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Then, the sequence pLmpvmqqm is tight in ˜Zc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' 68 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' HAUSENBLAS˚, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' JIDJOU MOGHOMYE˚ AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' RAZAFIMANDIMBY˚˚ ACKNOWLEDGMENT We acknowledge financial support provided by the Austrian Science Fund (FWF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' In particular, Boris Jidjou Moghomye and partially Erika Hausenblas were supported by the Austrian Science Fund, project 32295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' REFERENCES [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' Adams, Sobolev Spaces, Academic Press, New York-London, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAyT4oBgHgl3EQfwvmK/content/2301.00654v1.pdf'} +page_content=' [2] H.' metadata={'source': 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mode 100644 index 0000000000000000000000000000000000000000..5ff8c5d3e4da8caba61e20b04889630df8db9dc4 --- /dev/null +++ b/iNA0T4oBgHgl3EQfIP_x/content/tmp_files/2301.02074v1.pdf.txt @@ -0,0 +1,2469 @@ +Test of Time: Instilling Video-Language Models with a Sense of Time +Piyush Bagad +University of Amsterdam +piyush.bagad@student.uva.nl +Makarand Tapaswi +IIIT Hyderabad +makarand.tapaswi@iiit.ac.in +Cees G. M. Snoek +University of Amsterdam +c.g.m.snoek@uva.nl +Abstract +Modeling and understanding time remains a challenge in +contemporary video understanding models. With language +emerging as a key driver towards powerful generalization, +it is imperative for foundational video-language models to +have a sense of time. In this paper, we consider a specific as- +pect of temporal understanding: consistency of time order +as elicited by before/after relations. We establish that six +existing video-language models struggle to understand even +such simple temporal relations. We then question whether +it is feasible to equip these foundational models with tem- +poral awareness without re-training them from scratch. To- +wards this, we propose a temporal adaptation recipe on top +of one such model, VideoCLIP, based on post-pretraining on +a small amount of video-text data. We conduct a zero-shot +evaluation of the adapted models on six datasets for three +downstream tasks which require a varying degree of time +awareness. We observe encouraging performance gains es- +pecially when the task needs higher time awareness. Our +work serves as a first step towards probing and instilling a +sense of time in existing video-language models without the +need for data and compute-intense training from scratch. +1. Introduction +Self-supervised pretraining at scale on multimodal web +corpora tied with powerful architectures [105] has led to +foundational models [12] for images [2, 48, 58, 83, 84] and +videos [2, 6, 25, 107, 117, 125]. +These models have en- +abled remarkable improvements on a plethora of down- +stream tasks, video-language tasks such as video-text re- +trieval, video question-answering, and action recognition. +Given the cost and difficulty of video annotations, even +for a small amount of downstream data, such foundational +models are emerging as the de-facto backbone for zero- +shot [117, 121, 127] and few-shot generalization [2]. How- +ever, it remains unclear if these video-language models cap- +ture essential properties of a video beyond what can be +learned from static images, most notably: time. +Many before us have shown that existing video-language +models [6,56,65,117] can achieve impressive performance +1. +2. +3. +4. +A. +C. +B. +D. +Dog runs away before it +brings a ball to the man +The baby eats food after it +looks into the camera +The dog brings a a ball to +the man before it runs away +The baby looks into the +camera after it eats food +Figure 1. Can you match the correct video-text pairs? Under- +standing the time order of events in both video and language is +necessary to be able to solve this task. See footnote on next page +for answers. +on several video benchmarks [21, 40, 118] without reli- +ably encoding time [13, 55, 58]. +For example, Buch et +al. [13] show that a model that uses a single (carefully +selected) frame often outperforms recent video-language +models [56, 117] on standard video benchmarks such as +MSR-VTT [118]. Lie et al. [55] report similar findings with +a single-frame pretraining approach. These findings hint at +a lack of time awareness in video models. However, it re- +mains unclear if these findings are caused, indeed, by the +lack of time in video models or whether the benchmarks +themselves do not mandate time awareness. Furthermore, +there is no clear definition of what it means for a model to +be time aware. In this paper we strive to shed light on all +these factors of time awareness in video-language models. +As a first step, we consider a simple notion of under- +standing time, i.e., understanding temporal relations such as +before and after [4]. Consider the task presented in Fig. 1. A +time invariant model shall be able to associate (A) with (1) +1 +arXiv:2301.02074v1 [cs.CV] 5 Jan 2023 + +or (2) and (B) with (3) or (4) based on static frames alone. +But to distinguish between (1) and (2), one needs to be able +to understand time order and connect it across video and +language1. Thus, the first question we ask in Section 3: do +the representations learnt by foundational video-language +models encode this sense of time? To reliably attribute lack +of time awareness to models and not existing benchmarks, +we design our own synthetic dataset to probe models for this +sense of time. We create video-language pairs that show a +sequence of two events. Then, we alter the order of events +either in the text or the video and check if models can con- +nect the order in video and language. We find that existing +video-language models indeed struggle to associate the time +order across video and language. +In light of these findings, the second question we ask in +Section 4 is: can we adapt a video-language model, with- +out expensive re-training from scratch, to instill this sense +of time? Towards this, we take inspiration from literature +on understanding time in natural language, where there has +been much work on developing time aware language mod- +els [19, 35, 36, 130, 131]. Our objective is to instill time +awareness in a video-language model without having to pre- +train from scratch. To do that, we propose TACT: Temporal +Adaptation by Consistent Time-ordering based on two key +components: (i) we artificially create samples that provide +temporal signal, for example, by flipping the order of events +in the video or the text, (ii) we introduce a modified con- +trastive loss to learn time order consistency based on these +samples. Instead of training from scratch, we adapt an ex- +isting video-language model, VideoCLIP [60], using the +paradigm of post-pretraining on a small amount of video- +text data. We demonstrate the effectiveness of TACT in con- +necting the time order in video and language on four diverse +datasets in Section 5. +Finally, in line with the original motive of video- +language models for zero-shot generalization, we evalu- +ate in Section 6 our TACT adapted model for three sets +of tasks on six downstream datasets which require vary- +ing degree of time awareness. On tasks that need higher +time awareness, with the appropriate choice of adaptation +dataset, TACT outperforms a strong baseline that is based +on post-pretraining on canonical clip-text pairs without con- +sideration of time-order. +In summary, our contributions are as follows: (i) We +show that existing video-language models struggle to as- +sociate time order in video and language through a con- +trolled experiment on synthetic data. (ii) Based on Video- +CLIP [117], we propose TACT, a method for temporal adap- +tation using this time order consistency without having to +pretrain from scratch. (iii) We demonstrate improved zero- +shot generalizability of our temporally adapted models on +tasks that require higher time awareness. +1Answers: (A)-(2), (B)-(1), (C)-(4), (D)-(3). +2. Background and Related Work +We briefly discuss recent advances in video-language +models followed by their consideration of time. +Foundational +video-language +models. +Large-scale +datasets, self-supervision, and the advent of Transform- +ers [105] have led to the emergence of powerful encoders +for images [20,38,101], videos [5,11,23,102,115], language +models [18,63,69,86] and even universal encoders [31,45]. +These encoders form the basis for several vision-language +foundational models. Popular image-language models such +as CLIP [83] and ALIGN [48] are trained on massive +datasets by using web images and alt-text. Similarly, video- +language models are catching up and can be categorised into +two broad directions: (i) adapting image-language mod- +els for videos [8, 22, 49, 50, 62, 65, 71, 108, 110, 119], and +(ii) pure video-based models that are learned using large +video-text datasets [3,7,26–28,30,57,61,64,67,68,95,117]. +Recently, a new paradigm of post-pretraining has emerged +where an existing image- or video-language model goes +through another stage of self-supervised pretraining on a +small amount of video data before it is evaluated on down- +stream tasks [65, 119]. This is promising as it circumvents +the prohibitive cost of pretraining on large datasets from +scratch. +In [65] the post-pretraining uses time-invariant +mean-pooling, while [119] strives to bridge the domain-gap +between image captions and video subtitles. In contrast, +our proposed temporal adaptation involves post-pretraining +of VideoCLIP [117] with a small amount of data that instills +the model to learn the time-order of events in a video. +Time in vision. Time separates videos from static images +or an unordered set of frames. While modeling time re- +mains a challenge, it also presents a natural source of su- +pervision that has been exploited for self-supervised learn- +ing. +For example, as a proxy signal by posing pretext +tasks involving spatio-temporal jigsaw [1, 43, 52], video +speed [10,16,47,94,109,123], arrow of time [78,80,112], +frame/clip ordering [24, 70, 90, 97, 116], video continu- +ity [60], or tracking [44,106,111]. Several works have also +used contrastive learning to obtain spatio-temporal repre- +sentations by (i) contrasting temporally augmented versions +of a clip [46, 77, 81], or (ii) encouraging consistency be- +tween local and global temporal contexts [9, 17, 85, 122]. +Nevertheless, it remains unclear if the learnt representations +actually encode time reliably. There has been some very +recent work on evaluating self-supervised video representa- +tions [87, 98] on their temporal recognition ability instead +of only relying on time as a guidance for training. +In the same spirit, a related direction pursues evaluation +and benchmarking of time awareness in video datasets [88], +models [13, 14, 29, 55, 89, 124] or both [42, 92]. Huang et +al. [42] measure the effect of motion on temporal action +recognition to find that only a subset of classes in UCF-101 +2 + +and Kinetics-400 require motion information. Ghodrati et +al. [29] propose new tasks to evaluate temporal asymme- +try, continuity and causality in video models. Our work +derives inspiration from these but applies more generally +to video-language models as language provides a basis for +open-world generalization. +Time in language. Time has also been extensively stud- +ied in the natural language literature. Early works identified +temporal structures in language such as temporal preposi- +tions and quantifiers [4,79]. More recent literature focuses +on tasks such as extracting temporal relations [34, 72–74], +as well as temporal reasoning [35,36,82,130,131]. For ex- +ample, Han et al. [35,36] and Zhou et al. [131] pretrain lan- +guage models specifically to focus on understanding tem- +poral relations such as before, after, during, etc. Emergence +of large language models has also spurred an increased in- +terest in developing benchmarks to test for time awareness +in these models [19, 75, 76, 100, 104, 129]. For example, +Ning et al. [75] propose a new benchmark of reading com- +prehension with questions involving before/after relations. +Since temporal relations in language are grounded in the +video, we draw inspiration from [35, 36, 131] and aim to +instill time awareness in video-language models. +Visual-linguistic compositionality has been explored for +image-language models [66, 99, 120, 126]. The composi- +tional nature of language allows the evaluation of various +aspects: meaning change due to change in word order [99], +relationship between objects [126], systematicity and pro- +ductivity [66], etc. Similar to the Winograd scheme pre- +sented in [99], we change the word order keeping the tem- +poral prepositions constant which changes the order of time +in language. This enables us to evaluate the temporal under- +standing of video-language models beyond static images. +Time in video-language models appears implicitly through +tasks like video-text alignment [37] and temporal ground- +ing [41,59]. In this work, we consider large self-supervised +video-language models. +We do not consider supervised +models designed for specific downstream tasks, e.g., tem- +poral grounding, question-answering. Some recent works +have shown the under-utilisation of time in classic video- +text benchmarks such as MSR-VTT [118], YouCook [132], +ActivityNet [21], and DiDeMo [40]. For example, [13, 55, +56] discover that on several benchmarks, using only one +or few frames or clips achieves competitive performance. +Adaptations of the popular CLIP architecture for videos +(e.g., CLIP4Clip [65]) show that weighted mean pooling of +a set of frames already achieves impressive performance on +retrieval benchmarks. +These raise some key questions: +do existing video- +language models truly understand time in the sense of cor- +rectly associating order of events in language and video? +If not, can we adapt them to instill time awareness? Our +A red circle appears before a yellow circle +A yellow circle appears before a red circle +A red circle appears +A yellow circle appears +Attractor +Distractor +Time-order Consistency +Control Task +Figure 2. +Overview of the proposed task to evaluate time- +order consistency across synthetic video-language pairs having be- +fore/after relations. We also define a control task to check if the +synthetic videos are considered out-of-distribution by the model. +work addresses these questions. There has been some work +in using time-order across video and language as a source +of self-supervision. Specifically, concurrent to our work, +both Sun et al. [96] and Cao et al. [15] propose fine-grained +temporal alignment between video and text as the pretrain- +ing objective. Different from these works, we consider the +notion of time order and we aim to adapt a given video- +language model using post-pretraining, which circumvents +the need for a new round of compute-intense pretraining. +3. Do Video-Language Models Sense Time? +Probing video-language models for temporal under- +standing is an open direction of research. In this work, we +consider a specific sense of temporal understanding: con- +sistency in the order of events in a video with the associated +textual description. For example, consider a text descrip- +tion: A red circle appears before a yellow circle. +This imposes an order constraint on the video stream to have +the event red circle appears happen before the event +yellow circle appears. Can existing video-language +models connect time-order in text with that in video? To +answer this, we design an experiment with synthetic data. +Synthetic dataset. We construct simple videos that com- +prise of a pair of events such as the ones mentioned above. +We generate N=180 video-language pairs as a combination +of C=6 colors, S=3 shapes, and |τ|=2 temporal relations: +before and after. The corresponding caption describes the +order of events connected with a before/after temporal rela- +tion. We call this caption as an attractor since it is consis- +tent with the time-ordering in the video. Likewise, we con- +struct a distractor where we flip the order of event descrip- +tions while retaining the temporal relation. An example pair +is illustrated in Fig. 2 (left). Ideally, a time aware video- +language model should be able to associate the video with +the temporally consistent text, or vice versa. We refer to this +task as time-order consistency check. In order to rule out +the possibility that synthetic videos are out-of-distribution, +we also perform the same experiment with canonical clips +3 + +Paradigm +Method +Video-to-Text +Text-to-Video +Chance +- +50.0 +50.0 +50.0 +50.0 +Image-Language +adapted to video +CLIP4Clip [65] +49.4 +51.1 +50.0 +49.4 +CLIP2Video [22] +100.0 +47.8 +97.8 +52.3 +CenterCLIP [128] +91.7 +46.1 +97.2 +51.1 +Video-Language +Contrastive +VideoCLIP [117] +87.1 +51.1 +66.7 +48.3 +Frozen in Time [6] +97.8 +49.4 +100.0 +50.6 +Video-Language +Masking +BridgeFormer [28] +100.0 +51.1 +97.2 +42.2 +Table 1. Results on synthetic control ( +) and time-order consis- +tency ( +) task as described in Fig. 2. Existing video-language +models show random performance on our time-order task. +where a video displays a single event and the text describes +that same event as shown in Fig. 2 (right). We refer to this +as the control task. +Choice of models. +We consider recent video-language +models, broadly categorized into three groups: (i) image- +language models like CLIP [83] that are adapted to +videos [22,65,128], (ii) pure video-language models trained +on a contrastive learning objective [6, 117], and (iii) pure +video-language models trained on a masking objective [28]. +Findings. We evaluate video-to-text and text-to-video re- +trieval on both time-order consistency and control tasks. +From Tab. 1, we observe that while most video-language +models perform well on the control task, all of them strug- +gle and perform on par with random chance on the temporal +task. This gap in performance deserves attention given the +importance of time in videos. +4. Adaptation by Consistent Time-Ordering +We describe a post-pretraining recipe for instilling a +sense of time into a video-language model. +We pro- +pose TACT: Temporal Adaptation by Consistency of Time- +order, that is based on two key components: (i) we artifi- +cially create samples that provide temporal signals, e.g., by +flipping the order of events; and (ii) we introduce a modi- +fied contrastive loss to learn temporal consistency based on +these samples. We start by defining the notation and then +describe the key components of our adaptation recipe. +Preliminaries. Let V be the space of video clips and T +be the space of text clips. Consider two non-overlapping +video clips vi, vj ∈ V. Let ζi, ζj ∈ T be their respective +captions. Let τ be a temporal relation, τ ∈ {before, after}. +Then, we denote a stitched and time-order consistent clip as +(uij, tij), where uij := [vi; vj], tij := [ζi; τ; ζi], and [·; ·] +denotes concatenation. Note that depending on τ, the order +of vi and vj may need to change in uij. For brevity, we drop +the subscripts and refer to the stitched pair as (u, t) unless +stated otherwise. +Time-order reversal. +The classical contrastive learning +paradigm for video-language models aligns components of +a video clip vi with it’s text counterpart ζi and contrasts +against other texts ζj that usually describe a completely dif- +ferent clip. This makes such models ignore the finer de- +tails of temporal understanding as it is easier to contrast the +negatives by simply focusing on the objects or the scene. +This is evident from simple bag-of-word like methods that +are shown to work well for contrastive learning, both on +the visual (e.g., CLIP4Clip [65]) and textual (e.g., MIL- +NCE [67]) modalities. We hypothesize that unless there +are negatives in a contrastive setup that contain the same +scenes and objects, models do not need to learn a sense of +time. Thus, we present a simple strategy to generate nega- +tives that force the learning process to focus on the temporal +order. +We define a time-order reversal function T that operates +on the stitched video clip or text description and temporally +swaps its constituents : +T(u) = T([vi; vj]) := [vj; vi], +and +(1) +T(t) = T([ζi; τ; ζj]) := [ζj; τ; ζi] . +(2) +An illustration of T is shown in Fig. 3. Note that T does +not reverse the actual video, i.e., time does not flow back- +wards, but only changes the order in which events happen +in the stitched clips. Our objective is to train a model that +is able to distinguish between the original pair (u, t) and +time-reversed versions (u, T(t)), and (T(u), t). +Loss function. We assume access to an existing pre-trained +video-language model with a visual encoder fθ and text en- +coder gφ. We obtain the video encoding zu := fθ(u) ∈ Rd +and the text encoding zt := gφ(t) ∈ Rd. Our goal is to +adapt Θ = {θ, φ} via post pre-training such that the re- +sulting model is time aware while maintaining its original +performance on tasks such as retrieval. As we aim to use a +small amount of data, we only update some parameters of +the model (Θ), such as the last few layers. +We now introduce our recipe for temporal adaptation +based on the InfoNCE loss [103] to learn time-order sen- +sitive video-text correspondence. For a positive (or time- +order consistent) video-text pair (u, t), we first define a for- +ward loss where the stitched pair is in its original time-order. +Lf = +� +(u,t)∈B +(TNCE(zu, zt) + TNCE(zt, zu)) , +(3) +where TNCE is the Noise Contrastive Estimation (NCE) +loss for temporal adaptation, defined as: +TNCE(zu, zt) := − log +exp(zu · zt) +� +t′∈Bt exp(zu · zt′) + Ctime , +(4) +where B is the batch of (u, t) pairs and Bt specifically refers +to other stitched text captions in the batch. Ctime accumu- +4 + +Usual Positives +Usual Negatives +Time-order reversed +Negatives (Cross sample) +Time-order reversed +Negatives (Same sample) +A red circle appears +before a yellow circle +A yellow circle appears +before a red circle +𝕋 +𝕋 +Time-order +Reversal +function +𝕋 +ℒ! +ℒ" +Figure 3. Overview of TACT. Along with the usual contrastive +loss, where negatives come from other samples in the batch, we +make use of time-order reversal within the same sample and +cross samples to generate additional negatives for both video and +text. We also extend the contrastive loss to time-order reversed +video and text corresponding to reverse consistency Lr. +lates negatives defined using time-order reversal as: +Ctime = αsame exp(zu·zT(t))+αcross +� +t′∈Bt\{t} +exp(zu·zT(t′)), (5) +where αsame controls the effect of contrasting text from the +same sample but with reversed text time-order, i.e., T(t), +and αcross encourages the model to contrast between re- +versed versions of other text captions, i.e., T(t′). Note that +when both αsame and αcross are 0, we revert back to the +standard NCE formulation, albeit on stitched pairs. While +Eq. (4) corresponds to the video-text loss TNCE(zu, zt), +the text-video loss TNCE(zt, zu) is defined symmetrically. +Furthermore, we also apply a reverse loss Lr to bring +time-order reversed versions of both the video and the text +together. Note that as we consider (u, t) as a positive pair, +(T(u), T(t)) also form a positive pair, +Lr = +� +(T(u),T(t))∈B +� +TNCE(zT(u), zT(t)) + TNCE(zT(t), zT(u)) +� +. +(6) +Here, the TNCE term operates on time-reversed clips and +Ctime contrasts (T(u), T(t)) with un-reversed text clips in +the batch (T(u), t). +The overall loss function is defined as a combination, +L = Lf + βLr . +(7) +Depending +on +the +choice +of +loss +coefficients +αsame, αcross, β +∈ +{0, 1}, we can vary properties of +the adapted model. For example, setting αsame=1 encour- +ages high sensitivity to time-order reversal. As we will see +empirically, the loss coefficients also provide the flexibility +to adapt the model to various datasets. +We illustrate this temporal extension of the contrastive +loss in Fig. 3 (best seen in color). T illustrates the time or- +der reversal function. The top half corresponds to Lf while +the bottom half visualizes Lr. +In particular, the top-left +quadrant alone corresponds to the standard contrastive loss. +While the green diagonal terms are positive pairs, the red di- +agonal terms are the strongest drivers for instilling temporal +understanding in the model. +5. Experiments: TACT Ablations +Base model. We demonstrate the effectiveness of TACT +as an adaptation recipe on top of VideoCLIP [117] ow- +ing to its simple architecture, contrastive objective, and use +of pre-computed S3D [114] features that enable faster ex- +perimentation and allow encoding a long temporal context +(∼32 secs). We initialize Θ from the model pretrained on +HowTo100M [68] and post-pretrain on multiple datasets. +Datasets. +One of our key objectives is to post-pretrain +on a small amount of data in contrast to massive pretrain- +ing datasets such as WebVid2M [7] or HowTo100M [68]. +We consider dense video captioning datasets that offer suf- +ficient diversity in terms of size, backgrounds, clip dura- +tions, viewpoints and activities. +Specifically, we experi- +ment with: (i) TEMPO [39]: the subset of stitched di- +verse third-person videos from DiDeMo [40] with text de- +scriptions for fixed 5s segments that contain before/after +relations; (ii) ActivityNet Captions [54]: a dense video +captioning dataset with human-centric actions; (iii) Cha- +rades [93]: a scripted indoor daily human activities video +dataset; and (iv) Charades-Ego [91]: similar to Charades, +scripted human activities from the egocentric viewpoint. To +construct stitched clips, we randomly sample any two non- +overlapping clip-text pairs in the video. Since we require +stitched clips instead of raw videos, we create new splits +for each dataset (see Tab. 2). We will release all the splits +publicly on our project page. +Evaluation metrics. We evaluate the post-pretrained model +using two sets of metrics: (i) standard retrieval metrics, +recall R@1, R@5, R@10 and median-rank evaluated on +stitched video-text clips; and (ii) time-order consistency, +i.e., the fraction of videos for which the model correctly +associates text that is time order consistent with the video: +Atime := +1 +|D| +� +(u,t)∈D +I[d(zu, zt) < d(zu, zT(t))], +(8) +where (u, t) are time-order consistent pairs, D is the dataset, +and d(·, ·) is a distance metric based on cosine similarity. +5 + +Dataset +Train +Validation +Test +Ego Length +NV +NC +NV +NC +NV +NC +(s) +TEMPO +3,904 +28,427 411 1,000 396 1,000 + +30 +ActivityNet +7,440 +95,474 453 +906 456 +912 + +120 +Charades +5,262 +99,928 500 1,000 500 1,000 + +30 +Charades-Ego 2,679 155,306 500 1,000 210 +420 + +31 +Table 2. Statistics of datasets we consider for temporal adapta- +tion. NV is the number of unique videos and NC is the number +of stitched clips. Based on NV, TEMPO and Charades-Ego are +smaller as compared to ActivityNet and Charades. +Post-pretraining details. We freeze the word embeddings +and layers 1 to 5 for both the video and text encoders in +VideoCLIP. For adaptation, we use the Adam optimizer [53] +with learning rate 5e−6, batch size 32 trained on a single +node with 4 GeForce GTX 1080 GPUs. On TEMPO, we +train for 60 epochs while on the other datasets, we train for +10 epochs and pick the checkpoint that maximizes the geo- +metric mean of R@1 and Atime on the respective validation +set. A typical training run takes about 1-3 hours. +Evaluation on the test set. Results in Tab. 3 show that +TACT⋆ with optimal loss coefficients outperforms TACT† +(all 0 loss coefficients) and the zero-shot baseline (no post- +pretraining), both on the retrieval and time-order consis- +tency tasks. This indicates the robustness of the adaptation. +Impact of loss coefficients. +Choosing appropriate +Dataset +Method +Retrieval +Time-order +R@1↑ +MedR ↓ +Atime↑ +Zero-shot +3.7 +49.0 +48.1 +TEMPO +TACT† +7.7 +13.0 +46.5 +TACT⋆ +9.3 +9.0 +66.5 +Zero-shot +1.1 +44.0 +49.6 +ActivityNet +TACT† +5.8 +34.0 +59.7 +TACT⋆ +5.8 +35.0 +85.7 +Zero-shot +1.3 +170.0 +47.1 +Charades +TACT† +5.3 +38.5 +73.5 +TACT⋆ +5.7 +35.0 +77.0 +Zero-shot +1.6 +64.0 +53.7 +Charades-Ego +TACT† +6.4 +35.0 +60.1 +TACT⋆ +10.1 +28.5 +68.2 +Table 3. Results for the best TACT model on test sets of various +datasets. TACT⋆ is the model with optimal loss coefficients and +TACT† is a baseline with all coefficients 0. On time order, TACT +generalizes well with TACT⋆ outperforming the baselines. On re- +trieval, for TEMPO and Charades-Ego, TACT⋆ outperforms the +baseline as their optimal models have β=1 which helps retrieval +with small amount of data. +5.0 +7.5 +10.0 +12.5 +15.0 +∆time between clips (sec) +0 +100 +200 +300 +400 +500 +600 +Frequency +TEMPO +Mean: 6.4 +0 +100 +200 +∆time between clips (sec) +0 +50 +100 +150 +200 +250 +300 +ActivityNet +Mean: 58.8 +10 +20 +30 +40 +∆time between clips (sec) +0 +50 +100 +150 +200 +Charades +Mean: 14.5 +0 +10 +20 +30 +∆time between clips (sec) +0 +50 +100 +150 +200 +250 +CharadesEgo +Mean: 13.3 +Figure 4. Time-distance between stitched clips in datasets for tem- +poral adaptation (∆time). TEMPO has stitched clips close to each +other while those in Charades-Ego are farthest apart suggesting a +correlation between ∆time and the difficulty of temporal adapta- +tion. +values for loss coefficients Θl:={αsame, αcross, β} al- +lows the model to learn various aspects and adapt us- +ing different datasets. +On each dataset, +we vary +Θl∈{0, 1}3 and find the best configuration based on the +GeometricMean(R@1, max(Atime − 50, 0)) on the valida- +tion sets. The above metric ensures the geometric mean is +not overpowered by Atime. The results are shown in Tab. 4. +As αsame is directly responsible for discriminating be- +tween original and time-reversed orders, unsurprisingly, +setting it to 1 is necessary to achieve the best results on Atime +for all the datasets. For TEMPO and Charades-Ego, using +all loss components (all 1) provides the best results, whereas +αcross=1 and β=0 achieves a better trade-off for Activi- +tyNet and Charades. Choosing β=1 leads to an improve- +ment in retrieval performance for TEMPO and Charades- +Ego but leads to a decline for ActivityNet and Charades. +We attribute this to the number of unique videos in the train +set for these datasets. As ActivityNet and Charades have +more videos than TEMPO or Charades-Ego (see train NV +Tab. 2) additional positives introduced by setting β=1 are +not necessary and in fact hurt performance. +What makes temporal adaptation hard? We observe a +large gap in Atime between TEMPO and ActivityNet. We +hypothesize that the distance (in seconds) between the two +clips (∆time) in a stitched clip is strongly correlated with +the difficulty of adaptation. Intuitively, it is easier to in- +fer the time-order consistency for a stitched clip u with the +text t that has distant constituent clips vi, vj since the ob- +jects and scene could be vastly different. +In contrast, it +is harder to discern the correct time-order when the con- +stituent clips are closer in time. Fig. 4 shows the distribu- +tion of ∆time for each dataset. Indeed, the mean distance +between clips in ActivityNet (58.8s) is much higher than +that in TEMPO (6.4s) making the task harder on TEMPO. +To further strengthen our hypothesis, we conduct a con- +trolled experiment where we gradually vary the distribution +of ∆time for Charades-Ego to match it to that of TEMPO. +We find a strong correlation (ρ=0.92) between ∆time and +hardness of adaptation. More details can be found in the +supplementary material. +6 + +Loss coefficients +TEMPO +ActivityNet +Charades +Charades-Ego +αsame +αcross +β +R@1 ↑ MedR ↓ Atime ↑ +R@1 ↑ MedR ↓ Atime ↑ +R@1 ↑ MedR ↓ Atime ↑ +R@1 ↑ MedR ↓ Atime ↑ +Chance +0.1 +500.0 +50.0 +0.1 +453.0 +50.0 +0.1 +500.0 +50.0 +0.1 +500.0 +50.0 +0 +0 +0 +8.3 +14.0 +49.4 +6.4 +30.0 +57.3 +5.7 +42.0 +71.5 +2.9 +44.0 +64.6 +0 +0 +1 +8.2 +14.0 +49.5 +5.6 +27.0 +47.0 +4.2 +58.0 +75.1 +3.2 +41.5 +65.2 +0 +1 +0 +8.2 +15.0 +49.3 +6.1 +29.0 +78.8 +5.2 +45.0 +78.9 +3.4 +38.0 +64.5 +0 +1 +1 +8.1 +14.0 +49.5 +5.8 +27.0 +48.3 +4.2 +58.0 +75.1 +3.1 +41.0 +67.0 +1 +0 +0 +6.4 +20.0 +60.6 +5.9 +28.0 +79.1 +6.1 +38.0 +76.3 +3.2 +42.0 +66.1 +1 +0 +1 +6.5 +24.0 +62.9 +5.6 +26.0 +63.1 +4.9 +51.0 +78.0 +3.3 +39.0 +70.7 +1 +1 +0 +5.9 +24.0 +59.7 +6.0 +29.0 +86.3 +6.6 +43.0 +77.8 +3.7 +40.5 +67.9 +1 +1 +1 +7.5 +17.0 +62.5 +5.7 +27.0 +63.8 +5.1 +51.0 +77.7 +3.8 +38.5 +68.3 +Table 4. Impact of loss coefficients for TACT post-pretraining on validation sets of various datasets. For each dataset, the corresponding +color-marked row denotes the best configuration based on the geometric mean of R@1 and Atime. TACT is able to connect time-order in +video and language while maintaining its retrieval capabilities across several datasets. +6. Experiments: Downstream Evaluation +The goal of foundational video-language models such +as VideoCLIP is to pretrain them on massive video-text +datasets and generalize in a zero- or few-shot manner to a +diverse range of downstream video understanding tasks. We +evaluate TACT adapted models on three sets of downstream +tasks that need a low-to-high time awareness. +Baseline: Standard post-pretraining. +Comparing our +temporally adapted models with pretrained VideoCLIP is +not fair since adapted models see data beyond the pretrain- +ing phase. In addition to the zero-shot comparison, we com- +pare against a baseline model that is trained for standard +video-text retrieval on the same datasets as temporal adap- +tation. Instead of using stitched clips, we use simple canon- +ical pairs, i.e., (vi, ζi) instead of (uij, tij). +Results on synthetic data. As our first downstream eval- +uation, we check if TACT performs better on our synthetic +data (Sec. 3). On the video-to-text variant, the TEMPO- +adapted model attains 78.1% accuracy, ActivityNet 59.4%, +Charades 88.3% and Charades-Ego 86.7%. This is signif- +icantly higher than random performance that non-adapted +models achieve in Tab. 1. This highlights that TACT mod- +els indeed learn useful signals to match the time-order in +video and language. +I. Text-to-video retrieval. We consider two widely used +benchmarks: MSR-VTT [118] and YouCookII [132] and +adopt standard retrieval metrics for these tasks. Several re- +cent works have identified a bias for spatial-understanding +in these datasets, particularly MSR-VTT [8, 13, 42, 55, 58, +65]. +Thus, we consider this class of tasks as requiring +low time awareness. As shown in Tab. 5 set I, on MSR- +VTT [118], we observe that TACT is worse (marked in red) +or at-par with the baselines for all adaptation datasets. This +aligns well with findings in [13,55] that these benchmarks +do not need time awareness. On YouCookII [132], TACT +models based on Charades and Charades-Ego outperform +the baseline (marked in green). We believe this may be as +a consequence of both YouCookII and Charades being cap- +tured indoors, which lowers the domain-shift. +II. Temporal video QA. Next, we use subsets of recently +released multiple-choice video question answering bench- +marks: Next-QA [113] and AGQA [33]. +The idea is to +check if we can probe models for temporal understand- +ing by asking questions with temporal language. Buch et +al. [13] a identify subset of Next-QA, dubbed as ATP-hard2, +with a higher concentration of temporally challenging data. +For AGQA, we pick a subset of ∼6k questions that explic- +itly have a question with before/after relation. We consider +these benchmarks as requiring moderate-high level of time +awareness and AGQA in particular is also close to our adap- +tation task. We use accuracy as the standard metric. +We observe (see Tab. 5 set II) that indeed TACT al- +most always outperforms (marked in green) baselines on +both Next-QA and AGQA. TEMPO-adapted TACT seems +to generalize particularly well on both benchmarks. Like- +wise, Charades-adapted TACT does well on AGQA since +AGQA is also based on the Charades videos accounting for +reduced domain-shift. We affirm that temporal adaptation +is useful, especially when the downstream tasks require it. +III. Action-to-video retrieval. Finally, we consider action +recognition benchmarks such as Something-Something +(SSv2) [32] and Temporal [88]. SSv2 was designed to cap- +ture richer temporal information [32,55] . We follow Lie et +al. [55], who propose the template-retrieval task that en- +courages temporal modelling and use their evaluation split3 +containing C=174 actions and K=12 videos per class. In- +terestingly, different actions in SSv2 require differing lev- +els of time awareness. We create a subset SSv2 (events) +2Available here: github.com/StanfordVL/atp-video-language +3Available here: github.com/jayleicn/singularity +7 + +Low +Time awareness +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ High +Adaptation +I. Text-to-Video Retrieval +II. Temporal VQA +III. Action-to-Video Retrieval +Dataset +Method +MSR-VTT +YouCookII +Next-QA (ATP) +AGQA +SSv2 +SSv2 (events) +Temporal +R@1 +R@5 +R@10 +R@1 +R@5 +R@10 +Accuracy +Accuracy +mAP +mAP +mAP +- +Chance +0.1 +0.5 +1.0 +0.1 +0.5 +1.0 +20.0 +50.0 +0.6 +2.0 +2.0 +- +None +10.6 +23.4 +29.9 +18.2 +45.5 +59.9 +23.4 +49.9 +3.4 +6.4 +13.0 +TEMPO +Baseline +12.0 +29.3 +37.3 +21.5 +48.2 +61.8 +25.0 +50.8 +3.9 +6.8 +15.9 +TACT +12.8 +26.5 +35.7 +20.4 +45.1 +58.7 +27.6 +57.1 +4.2 +7.7 +16.1 +ActivityNet +Baseline +15.7 +34.4 +44.9 +15.6 +38.8 +51.4 +23.7 +50.7 +3.7 +7.0 +16.0 +TACT +13.8 +29.6 +39.6 +16.0 +36.9 +49.8 +25.4 +50.3 +3.5 +7.2 +16.2 +Charades +Baseline +12.3 +25.8 +33.6 +21.5 +48.6 +61.7 +26.0 +50.5 +4.1 +7.1 +13.7 +TACT +11.7 +25.2 +32.4 +22.4 +49.1 +62.4 +25.2 +54.8 +4.3 +7.8 +14.1 +Charades-Ego +Baseline +13.1 +27.5 +34.5 +19.4 +47.1 +60.8 +24.3 +49.7 +4.0 +6.9 +14.7 +TACT +11.1 +24.6 +30.6 +21.9 +48.2 +61.9 +25.6 +58.4 +3.9 +6.9 +13.5 +Table 5. Results on downstream zero-shot evaluation with tasks requiring increasing time awareness from I to III. None corresponds +to direct evaluation of the VideoCLIP model on the downstream dataset. Green denotes an improvement for the TACT adapted model +w.r.t. the baseline, red denotes a deterioration. As we move from tasks that need low to high time awareness, the effectiveness of TACT +increases. See Sec. 6 for a more detailed discussion. The table is best viewed on screen in color. +with Cevents=49 actions that have at least two verbs in the +label as occurrence of multiple verbs is indicative of mul- +tiple events occurring in sequence. Finally, we also eval- +uate against the Temporal benchmark [88], a combination +of 50 action classes from SSv2 [32] and Kinetics-400 [51] +for which temporal information is deemed to be essential +for recognition. Similar to text-to-video retrieval, we use +the action class as a text query and obtain a ranking over +all videos. Different from the retrieval setup, since a single +query has multiple correct answers (upto K=12 videos), +we report mAP as the metric for these benchmarks. This +task set needs high time awareness. Furthermore, unlike QA +tasks in II, there is a shift in several (uncontrolled) factors +as we move from temporal adaptation task to these tasks. +From Tab. 5, we observe that TEMPO- and Charades- +adapted models generalize well across set III benchmarks. +ActivityNet-adapted TACT underperforms on SSv2 but +outperforms the baseline on strongly temporal actions in +SSv2 (events) and Temporal. Finally, TACT adapted on +Charades-Ego is at-par or slightly worse than the base- +line on SSv2 variants, and also on Temporal, perhaps due +to the shift from egocentric to third-person videos. Over- +all, despite SSv2 and Temporal requiring high time aware- +ness, TACT models shows promising zero-shot generaliza- +tion with the right choice of the adaptation dataset. +7. Discussion and Conclusion +Spatial vs. temporal understanding. An interesting facet +of TACT is αsame which controls how well a model adapts +to temporal tasks. We highlight this on the TEMPO dataset +in Tab. 6, where, αsame=0 results in Atime ∼50% while +Hyperparameters +Adaptation +Downstream +αsame +αcross +β +TEMPO +MSR-VTT +AGQA +Atime ↑ +R@1 ↑ +MedR ↓ +Accuracy↑ +0 +0 +0 +49.4 +15.0 +20.0 +50.5 +0 +0 +1 +49.5 +14.2 +20.0 +49.9 +0 +1 +0 +49.3 +14.4 +19.0 +50.2 +0 +1 +1 +49.5 +15.1 +19.0 +50.2 +1 +0 +0 +60.6 +11.7 +27.0 +56.6 +1 +0 +1 +62.9 +9.4 +36.0 +58.3 +1 +1 +0 +59.7 +9.1 +37.0 +56.9 +1 +1 +1 +62.5 +12.8 +27.0 +57.1 +Table 6. Impact of αsame on spatial- vs. temporal understanding. +Gray denotes better performance between αsame=0 or 1. While +αsame=1 drives temporal understanding, it comes at a cost of re- +trieval performance on MSR-VTT [118]. This hints at αsame con- +trolling the trade-off between spatial- and temporal-understanding. +αsame=1 improves performance. Further investigation on +downstream tasks shows that adaptation with αsame=1 does +not perform well on MSR-VTT (a non-temporal bench- +mark) but shows consistent improvements on AGQA (a +temporal benchmark). This hints at αsame controlling the +trade-off between spatial and temporal understanding. +Generalization to other temporal prompts. The time or- +der of events in language can be described using a variety +of sentence structures. Although we train video-language +models using temporal relations such as before/after, it +is natural to ask if the model still correctly associates +time order for a different prompt such as First, +.., +then, ... To systematically evaluate this, we gather event +pairs E1, E2 (E1 occurs before E2 in the video) for each +sample in the validation set and stitch them using three +8 + +Temporal accuracy +0 +25 +50 +75 +100 +TEMPO +ActivityNet +Charades +Charades-Ego +E1 before E2 +E2 after E1 +First, E1, then E2 +Effect of different prompts on inferring time-order +Chance +Figure 5. Models trained by TACT with before/after relations gen- +eralize to a new kind of prompt such as First, ..., then .... +This indicates learning of the underlying time order of events +rather than the mere order of words. +prompts as follows: (i) E1 before E2, (ii) E2 after E1, +(iii) First E1, then E2. As shown in Fig. 5, TACT-adapted +models generalize well to a new prompt (iii). This substan- +tiates the learning of time order of events rather than merely +learning the order of words in the sentence. +Limitations. +While we present a promising way of in- +stilling time in video-language models, our work is limited +to the VideoCLIP [117] pretrained model. Our initial ex- +periments with Frozen in Time [6] were not as promising, +perhaps because it uses a much shorter temporal context +(4 frames). Furthermore, we consider a specific definition +of time awareness derived from temporal relations like be- +fore/after. It is natural to ask if this can be extended to more +general notions of temporality, e.g., as defined by Allen [4]. +Finally, there can always be more downstream tasks that one +could consider such as (spatio-)temporal localization. +Conclusion. Given the essence of time in video-language +models, we present a simple experiment based on synthetic +data to test for time awareness in existing models. 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Datasets and Pre-processing +We sketch out the procedure we use for stitching two +clips within a video. +Clip stitching. +Consider a video containing two events +(clips) vi , vj with associated captions ζi, ζj as shown in +Fig. 6. +We assume these are non-overlapping (in time). +We stitch the text descriptions to construct a new caption +tij := [ζi; τ; ζj]. Since τ can be either before or after, we +end up with two newly constructed sentences. Correspond- +ing to each of these new sentences, we also stitch the video +events to construct a stitched video. Note that the order of +stitching video events depends on the value of τ. For exam- +ple, if τ is before, then uij := [vi; vj] as shown in first of +the two stitched clips. If τ is after, then uij := [vj; vi] as +shown in the second of the two stitched clips. +From each stitched clip in Fig. 6, we construct negatives +for the contrastive loss by reversing the time order in ei- +ther video or text. This step happens on-the-fly during loss +computation, and hence, we do not show it here. For a +given dataset, we can either use all possible tuples of non- +overlapping events to create such stitched clips or sample +from all possible tuples. Since the TEMPO dataset already +comes with stitched event descriptions (based on DiDeMo), +we directly use its subset which has before/after relations +in the text. For all the other datasets, we apply the stitching +process as described. Recall, ∆time is the time distance be- +tween the two events, and plays a key role in deciding the +difficulty of temporal adaptation, as observed empirically. +Next, we describe dataset properties and show some +qualitative examples after the clip stitching step. +Adaptation datasets. To gain a sense of the diversity in +the datasets we consider for adaptation, we present exam- +ples of stitched clips from these datasets in Fig. 8. Please +refer to the attached HTML page for corresponding videos. +Since TEMPO has short adjacent clips, the context remains +almost the same, we think this is important to instill a sense +of time in models. In contrast, for ActivityNet, since the +stitched events are far apart, the context changes make it +easy to infer which event description goes with which part +of the video, or the time order of events. In this regard, +Charades and Charades-Ego are similar to TEMPO. Quan- +Video Stream +Event X +Event Y +Stitched clips +Description(X) before Description(Y). +Video +Text +Description(X) after Description(Y). +Video +Text +Figure 6. +Illustration of clip stitching. +We consider two non- +overlapping events in a video and stitch them with temporal re- +lations - before and after. ∆time denotes the time difference be- +tween midpoints of the two events. +0 +50 +100 +150 +200 +0 +200 +400 +600 +800 +1000 +1200 +1400 +Number of videos +TEMPO +0 +50 +100 +150 +200 +0 +100 +200 +300 +400 +500 +600 +700 +Charades +0 +50 +100 +150 +200 +Number of clips in a video +0 +20 +40 +60 +80 +100 +120 +Number of videos +CharadesEgo +0 +50 +100 +150 +200 +Number of clips in a video +0 +500 +1000 +1500 +2000 +ActivityNet +Figure 7. Number of clips in a video. The number of clips per +video is lower in TEMPO and ActivityNet as compared to Cha- +rades and Charades-Ego. +titatively, this change in context is captured by ∆time which +is lowest for TEMPO (mean 6.8s), followed by Charades- +Ego (13.3s), Charades (14.5s) and ActivityNet (58.8s). +Distribution of number of clips in a video. A single video +with 10 non-overlapping individual event clips can make +upto 10C2=45 stitched clips. We plot the number of clips +per video against the number of videos in a given dataset +in Fig. 7. A single video with >30 stitched clips is rare +in TEMPO and ActivityNet while much more frequent in +Charades and Charades-Ego. Overall, the number of clips +per video is lower in TEMPO and ActivityNet as compared +to Charades and Charades-Ego. +Downstream datasets. In Fig. 9, we also show some ex- +amples from some downstream datasets (tasks) that need +higher time awareness since they typically involve multiple +temporally linked events (e.g., walk and eat in Fig. 9(b)). +15 + +timeA rabbit lays down on its stomach before bunny lying on it’s side +Little girl eats from cup after the child walks downhill +(a) TEMPO +A woman is standing in a room holding a hula hoop before she begins to use the hula hoop +The team shakes hands with the opposing team after a team groups together holding a trophy +(b) ActivityNet +Putting on shoe/shoes before holding a mirror +(c) Charades +Taking a broom from somewhere before holding a dish +(d) Charades-Ego +Figure 8. Examples from datasets used for temporal adaptation. +The first two frames are linearly spaced from the first event while +the last two from the second event. Notice how there is a sig- +nificant change in visual context between the two events in Activi- +tyNet in contrast to other datasets. Best viewed on a screen. Please +refer to the attached HTML page for corresponding videos. +On these datasets, we perform zero-shot evaluation of tem- +porally adapted models in Sec. 6 of the main paper. +B. Experiments +What makes temporal adaptation difficult? To recall, +we define ∆time as the time-distance (in seconds) between +the midpoints of the two clips in a stitched pair. We hy- +pothesize that ∆time is inversely related to the difficulty of +temporal adaptation, i.e., the larger ∆time, the easier it is +to distinguish between two stitched clips that have opposite +time order. For example, consider ActivityNet examples +in Fig. 8(b) where the visual context changes significantly +making inference of the time order of events relatively eas- +ier. +We further test our hypothesis by sampling individual +clips from the Charades-Ego dataset to match the ∆time dis- +tribution of TEMPO. Concretely, assuming ∆time for both +these datasets follows a multinomial distribution, we con- +struct a new distribution using a convex combination of +the individual distributions , where the mixing parameter +λ ∈ [0, 1] controls the extent to which we modify the dis- +tribution from TEMPO (λ=0) to Charades-Ego (λ=1). The +resulting distributions are presented in Fig. 10 (left). With +λ=1, we sample from the original Charages-Ego distribu- +tion and gradually move towards TEMPO as λ → 0. +We then sample stitched clips according to this new dis- +tribution and post-pretrain temporal adaptation for varying +values of λ. Note that for this experiment, we keep fixed +NC=10, 000 for each λ. From Fig. 10 (right), we indeed +find that as we move towards a more TEMPO-like distri- +bution (shorter ∆time), temporal accuracy deteriorates. The +best fit also confirms that the distribution of ∆time is strongly +correlated (ρ = −0.92) with the difficulty of inferring time- +order consistency. +C. Qualitative Analysis +To get an intuitive sense of whether a TACT model un- +derstands time order of events, we perform a qualitative +analysis on the model trained on TEMPO. Our demo in- +terface looks like the one shown in Fig. 11. First, a user +uploads a video and adds text descriptions for two events +within the video. These descriptions are then connected +via a temporal relation such as before or after. We also +experiment with a new temporal connector First, ..., +then, .... to check if our model generalizes beyond be- +fore/after. We will release the demo code on our project +website. +First, we consider samples from the TEMPO validation +set and show their results in Fig. 12. Notably, for some +examples, it connects time order for before relations but +not the other two. We suspect this is because a majority +(∼ 60%) of the TEMPO dataset has descriptions involv- +ing before. Note that TEMPO already comes with tem- +poral captions of which we pick subset of before/after rela- +tions. Second, we also consider samples from other datasets +which the model has never seen. To our surprise, albeit +qualitatively, the model does generalize well to such exam- +ples as shown in Fig. 13. +These results reinforce the promise of our method and +also raise the possibility of extending this work to consider +more general temporal relations. Having said that, we re- +iterate that these are qualitative examples and should be +treated as such. +16 + +Nathalie Veilleux +watchwm +Ownerof StudiosVertPranaQuestion: How did the boy react when he entered the room at the start? +Answer: Smile. +Question: Why does the baby turn around near the end of the video? +Answer: Sits down to play. +(a) Next-QA: Video question answering +Question: Did they reach for and grab a picture before or after putting a bag +somewhere? +Answer: Before +Question: Did they walk through a doorway before or after they +eating the last thing they touched? +Answer: After +(b) AGQA: Video question answering +Template: Spinning [something] that quickly stops +spinning +(c) Something-Something: Template-based video retrieval +Figure 9. Examples from datasets used for downstream evaluation. +These tasks demand time awareness since it is often not possible +to infer the action from a single frame. Please refer to the attached +HTML page for corresponding videos. +17 + +(0.0, 5.0] +(5.0, 7.5] +(7.5, 10.0] +(10.0, 12.5] +(12.5, inf] +0.0 +0.2 +0.4 +0.6 +Density +λ = 1.0 (CharadesEgo) +(0.0, 5.0] +(5.0, 7.5] +(7.5, 10.0] +(10.0, 12.5] +(12.5, inf] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +λ = 0.75 +(0.0, 5.0] +(5.0, 7.5] +(7.5, 10.0] +(10.0, 12.5] +(12.5, inf] +0.0 +0.1 +0.2 +0.3 +λ = 0.50 +(0.0, 5.0] +(5.0, 7.5] +(7.5, 10.0] +(10.0, 12.5] +(12.5, inf] +0.0 +0.1 +0.2 +0.3 +0.4 +λ = 0.25 +(0.0, 5.0] +(5.0, 7.5] +(7.5, 10.0] +(10.0, 12.5] +(12.5, inf] +0.0 +0.2 +0.4 +0.6λ = 0.0 (TEMPO) +1.0 +0.75 +0.5 +0.25 +0.0 +λ −→ 0 +60 +62 +64 +66 +68 +Temporal Accuracy (Atime) +Atime vs Distribution of ∆time +ρ = -0.92 +CharadesEgo +TEMPO +Figure 10. Impact of changing distribution of ∆time, the time gap between two stitched clips. Left: We vary the distribution of ∆time for +Charades-Ego and make it similar to that of TEMPO as λ → 0. Thus, crudely, as λ decreases, so does ∆time. Right: Atime on Charades-Ego +where the time difference between sampled clips is according to the distributions on the left. We observe that the accuracy deteriorates +as the time-distance between a pair of clips decreases indicating a strong correlation between the distribution of ∆time and difficulty of +temporal adaptation. +Figure 11. Interface of our demo for qualitative analysis. The user uploads a video and is asked to describe two events in the video. These +event descriptions are then connected via one of the three temporal relations shown at the bottom left. We construct one sentence that is +consistent with the time order of events in the video and another that is not. The output on the right shows the ranking of the constructed +sentences in terms of cosine similarity with the video representation. Higher score for correct matching indicated by a longer orange bar. +18 + +Test of Time: Instilling Video-Language Models with a Sense of Time +Rank sentences based on their relevance to a video + Video (stitched with two events) +Constructed sentence 1 +The child runs into the room before he sits near the gifts +Constructed sentence 2 +he sits near the gifts before the child runs into the room +The child runs into the room before he sits near the gifts +54% +he sits near the gifts before the child runs into the room +46% +Refresh video. Check this if you load a new video. +Write a description for event X (any event within the video) +The child runs into the room +Write a description for event Y (any event within the video) +he sits near the gifts +Choose a relation between the two events +O before +after +First,.,.then.. +Clear +SubmitBefore +After +First, … +then, …. +Before +After +First, … +then, …. +Figure 12. Qualitative examples from TEMPO validation set. We +evaluate similarity of a given video with sentences with different +temporal order with the usual temporal connectors (before/after). +Green bordered boxes indicate correct predictions (consistent +time order between video and language) while red denote mis- +predictions. For some examples, e.g., in the bottom example, the +model gets predictions incorrect particularly for relations other +than before. Furthermore, we also try a new temporal connector +First, ..., then, ... and observe that the model qualitatively +generalizes to that as well. +Before +After +First, … +then, …. +(a) Example from Charades-Ego +Before +After +First, … +then, …. +(b) Example from Next-QA +Figure 13. Qualitative results on samples not from TEMPO. We +see that despite not having seen these examples, the model still +connects the time order across video and language correctly. +19 + +First, The stuffed panda is visible on zooming in occurs, +68% +then the bus drives by occurs +First, The bus drives by occurs, then the stuffed panda is +32% +visible on zooming in occursThe stuffed panda is visible on zooming in before the bus +69% +drives by +The bus drives by before the stuffed panda is visible on +31% +zooming inThe bus drives by after the stuffed panda is visible on +72% +zooming in +The stuffed panda is visible on zooming in after the bus +28% +drives byFirst, she eats an ice-cream occurs, then the child walks +59% +down the hill occurs +First, The child walks down the hill occurs, then she eats +41% +an ice-cream occursThe child walks down the hill before she eats an ice-cream +98% +she eats an ice-cream before the child walks down the hill +2%The child walks down the hill after she eats an ice-cream +72% +she eats an ice-cream after the child walks down the hill +28%Ranking over sentences +First The man picks up a broom occurs, then he looks +at the television occurs +First The man picks up a broom occurs, then he looks at the +88% +television occurs +First He looks at the television occurs, then the man picks +12% +up a broom occurs Ranking oversentences +The man picks up a broom before he looks at the +television +The man picks up a broom before he looks at the television +100% +He looks at the television before the man picks up a broom +0%Ranking oversentences +He looks at the television after the man picks up a +broom +He looks at the television after the man picks up a broom +100% +The man picks up a broom after he looks at the television +0%First, The child runs into the room occurs, then he sits +74% +near the gifts occurs +First, he sits near the gifts occurs, then the child runs +26% +into the room occursThe child runs into the room before he sits near the gifts +54% +he sits near the gifts before the child runs into the room +46%he sits near the gifts after the child runs into the room +91% +The child runs into the room after he sits near the gifts +%6 \ No newline at end of file diff --git a/iNA0T4oBgHgl3EQfIP_x/content/tmp_files/load_file.txt b/iNA0T4oBgHgl3EQfIP_x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ca16522f7862bd1071a60d1ce151c20fd8c4cab2 --- /dev/null +++ b/iNA0T4oBgHgl3EQfIP_x/content/tmp_files/load_file.txt @@ -0,0 +1,1482 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf,len=1481 +page_content='Test of Time: Instilling Video-Language Models with a Sense of Time Piyush Bagad University of Amsterdam piyush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='bagad@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='nl Makarand Tapaswi IIIT Hyderabad makarand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='tapaswi@iiit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='in Cees G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Snoek University of Amsterdam c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='snoek@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='nl Abstract Modeling and understanding time remains a challenge in contemporary video understanding models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' With language emerging as a key driver towards powerful generalization, it is imperative for foundational video-language models to have a sense of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In this paper, we consider a specific as- pect of temporal understanding: consistency of time order as elicited by before/after relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We establish that six existing video-language models struggle to understand even such simple temporal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We then question whether it is feasible to equip these foundational models with tem- poral awareness without re-training them from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' To- wards this, we propose a temporal adaptation recipe on top of one such model, VideoCLIP, based on post-pretraining on a small amount of video-text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We conduct a zero-shot evaluation of the adapted models on six datasets for three downstream tasks which require a varying degree of time awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We observe encouraging performance gains es- pecially when the task needs higher time awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Our work serves as a first step towards probing and instilling a sense of time in existing video-language models without the need for data and compute-intense training from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Introduction Self-supervised pretraining at scale on multimodal web corpora tied with powerful architectures [105] has led to foundational models [12] for images [2, 48, 58, 83, 84] and videos [2, 6, 25, 107, 117, 125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' These models have en- abled remarkable improvements on a plethora of down- stream tasks, video-language tasks such as video-text re- trieval, video question-answering, and action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Given the cost and difficulty of video annotations, even for a small amount of downstream data, such foundational models are emerging as the de-facto backbone for zero- shot [117, 121, 127] and few-shot generalization [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' How- ever, it remains unclear if these video-language models cap- ture essential properties of a video beyond what can be learned from static images, most notably: time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Many before us have shown that existing video-language models [6,56,65,117] can achieve impressive performance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Dog runs away before it brings a ball to the man The baby eats food after it looks into the camera The dog brings a a ball to the man before it runs away The baby looks into the camera after it eats food Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Can you match the correct video-text pairs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Under- standing the time order of events in both video and language is necessary to be able to solve this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' See footnote on next page for answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' on several video benchmarks [21, 40, 118] without reli- ably encoding time [13, 55, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For example, Buch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' [13] show that a model that uses a single (carefully selected) frame often outperforms recent video-language models [56, 117] on standard video benchmarks such as MSR-VTT [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Lie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' [55] report similar findings with a single-frame pretraining approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' These findings hint at a lack of time awareness in video models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' However, it re- mains unclear if these findings are caused, indeed, by the lack of time in video models or whether the benchmarks themselves do not mandate time awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Furthermore, there is no clear definition of what it means for a model to be time aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In this paper we strive to shed light on all these factors of time awareness in video-language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' As a first step, we consider a simple notion of under- standing time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', understanding temporal relations such as before and after [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Consider the task presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' A time invariant model shall be able to associate (A) with (1) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='02074v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='CV] 5 Jan 2023 or (2) and (B) with (3) or (4) based on static frames alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' But to distinguish between (1) and (2), one needs to be able to understand time order and connect it across video and language1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Thus, the first question we ask in Section 3: do the representations learnt by foundational video-language models encode this sense of time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' To reliably attribute lack of time awareness to models and not existing benchmarks, we design our own synthetic dataset to probe models for this sense of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We create video-language pairs that show a sequence of two events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Then, we alter the order of events either in the text or the video and check if models can con- nect the order in video and language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We find that existing video-language models indeed struggle to associate the time order across video and language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In light of these findings, the second question we ask in Section 4 is: can we adapt a video-language model, with- out expensive re-training from scratch, to instill this sense of time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Towards this, we take inspiration from literature on understanding time in natural language, where there has been much work on developing time aware language mod- els [19, 35, 36, 130, 131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Our objective is to instill time awareness in a video-language model without having to pre- train from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' To do that, we propose TACT: Temporal Adaptation by Consistent Time-ordering based on two key components: (i) we artificially create samples that provide temporal signal, for example, by flipping the order of events in the video or the text, (ii) we introduce a modified con- trastive loss to learn time order consistency based on these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Instead of training from scratch, we adapt an ex- isting video-language model, VideoCLIP [60], using the paradigm of post-pretraining on a small amount of video- text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We demonstrate the effectiveness of TACT in con- necting the time order in video and language on four diverse datasets in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Finally, in line with the original motive of video- language models for zero-shot generalization, we evalu- ate in Section 6 our TACT adapted model for three sets of tasks on six downstream datasets which require vary- ing degree of time awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' On tasks that need higher time awareness, with the appropriate choice of adaptation dataset, TACT outperforms a strong baseline that is based on post-pretraining on canonical clip-text pairs without con- sideration of time-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In summary, our contributions are as follows: (i) We show that existing video-language models struggle to as- sociate time order in video and language through a con- trolled experiment on synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' (ii) Based on Video- CLIP [117], we propose TACT, a method for temporal adap- tation using this time order consistency without having to pretrain from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' (iii) We demonstrate improved zero- shot generalizability of our temporally adapted models on tasks that require higher time awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 1Answers: (A)-(2), (B)-(1), (C)-(4), (D)-(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Background and Related Work We briefly discuss recent advances in video-language models followed by their consideration of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Foundational video-language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Large-scale datasets, self-supervision, and the advent of Transform- ers [105] have led to the emergence of powerful encoders for images [20,38,101], videos [5,11,23,102,115], language models [18,63,69,86] and even universal encoders [31,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' These encoders form the basis for several vision-language foundational models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Popular image-language models such as CLIP [83] and ALIGN [48] are trained on massive datasets by using web images and alt-text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Similarly, video- language models are catching up and can be categorised into two broad directions: (i) adapting image-language mod- els for videos [8, 22, 49, 50, 62, 65, 71, 108, 110, 119], and (ii) pure video-based models that are learned using large video-text datasets [3,7,26–28,30,57,61,64,67,68,95,117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Recently, a new paradigm of post-pretraining has emerged where an existing image- or video-language model goes through another stage of self-supervised pretraining on a small amount of video data before it is evaluated on down- stream tasks [65, 119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This is promising as it circumvents the prohibitive cost of pretraining on large datasets from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In [65] the post-pretraining uses time-invariant mean-pooling, while [119] strives to bridge the domain-gap between image captions and video subtitles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In contrast, our proposed temporal adaptation involves post-pretraining of VideoCLIP [117] with a small amount of data that instills the model to learn the time-order of events in a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Time in vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Time separates videos from static images or an unordered set of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' While modeling time re- mains a challenge, it also presents a natural source of su- pervision that has been exploited for self-supervised learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For example, as a proxy signal by posing pretext tasks involving spatio-temporal jigsaw [1, 43, 52], video speed [10,16,47,94,109,123], arrow of time [78,80,112], frame/clip ordering [24, 70, 90, 97, 116], video continu- ity [60], or tracking [44,106,111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Several works have also used contrastive learning to obtain spatio-temporal repre- sentations by (i) contrasting temporally augmented versions of a clip [46, 77, 81], or (ii) encouraging consistency be- tween local and global temporal contexts [9, 17, 85, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Nevertheless, it remains unclear if the learnt representations actually encode time reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' There has been some very recent work on evaluating self-supervised video representa- tions [87, 98] on their temporal recognition ability instead of only relying on time as a guidance for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In the same spirit, a related direction pursues evaluation and benchmarking of time awareness in video datasets [88], models [13, 14, 29, 55, 89, 124] or both [42, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' [42] measure the effect of motion on temporal action recognition to find that only a subset of classes in UCF-101 2 and Kinetics-400 require motion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Ghodrati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' [29] propose new tasks to evaluate temporal asymme- try, continuity and causality in video models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Our work derives inspiration from these but applies more generally to video-language models as language provides a basis for open-world generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Time in language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Time has also been extensively stud- ied in the natural language literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Early works identified temporal structures in language such as temporal preposi- tions and quantifiers [4,79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' More recent literature focuses on tasks such as extracting temporal relations [34, 72–74], as well as temporal reasoning [35,36,82,130,131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For ex- ample, Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' [35,36] and Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' [131] pretrain lan- guage models specifically to focus on understanding tem- poral relations such as before, after, during, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Emergence of large language models has also spurred an increased in- terest in developing benchmarks to test for time awareness in these models [19, 75, 76, 100, 104, 129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For example, Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' [75] propose a new benchmark of reading com- prehension with questions involving before/after relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Since temporal relations in language are grounded in the video, we draw inspiration from [35, 36, 131] and aim to instill time awareness in video-language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Visual-linguistic compositionality has been explored for image-language models [66, 99, 120, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The composi- tional nature of language allows the evaluation of various aspects: meaning change due to change in word order [99], relationship between objects [126], systematicity and pro- ductivity [66], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Similar to the Winograd scheme pre- sented in [99], we change the word order keeping the tem- poral prepositions constant which changes the order of time in language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This enables us to evaluate the temporal under- standing of video-language models beyond static images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Time in video-language models appears implicitly through tasks like video-text alignment [37] and temporal ground- ing [41,59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In this work, we consider large self-supervised video-language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We do not consider supervised models designed for specific downstream tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', tem- poral grounding, question-answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Some recent works have shown the under-utilisation of time in classic video- text benchmarks such as MSR-VTT [118], YouCook [132], ActivityNet [21], and DiDeMo [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For example, [13, 55, 56] discover that on several benchmarks, using only one or few frames or clips achieves competitive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Adaptations of the popular CLIP architecture for videos (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', CLIP4Clip [65]) show that weighted mean pooling of a set of frames already achieves impressive performance on retrieval benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' These raise some key questions: do existing video- language models truly understand time in the sense of cor- rectly associating order of events in language and video?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' If not, can we adapt them to instill time awareness?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Our A red circle appears before a yellow circle A yellow circle appears before a red circle A red circle appears A yellow circle appears Attractor Distractor Time-order Consistency Control Task Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Overview of the proposed task to evaluate time- order consistency across synthetic video-language pairs having be- fore/after relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We also define a control task to check if the synthetic videos are considered out-of-distribution by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' work addresses these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' There has been some work in using time-order across video and language as a source of self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Specifically, concurrent to our work, both Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' [96] and Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' [15] propose fine-grained temporal alignment between video and text as the pretrain- ing objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Different from these works, we consider the notion of time order and we aim to adapt a given video- language model using post-pretraining, which circumvents the need for a new round of compute-intense pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Do Video-Language Models Sense Time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Probing video-language models for temporal under- standing is an open direction of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In this work, we consider a specific sense of temporal understanding: con- sistency in the order of events in a video with the associated textual description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For example, consider a text descrip- tion: A red circle appears before a yellow circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This imposes an order constraint on the video stream to have the event red circle appears happen before the event yellow circle appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Can existing video-language models connect time-order in text with that in video?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' To answer this, we design an experiment with synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We construct simple videos that com- prise of a pair of events such as the ones mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We generate N=180 video-language pairs as a combination of C=6 colors, S=3 shapes, and |τ|=2 temporal relations: before and after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The corresponding caption describes the order of events connected with a before/after temporal rela- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We call this caption as an attractor since it is consis- tent with the time-ordering in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Likewise, we con- struct a distractor where we flip the order of event descrip- tions while retaining the temporal relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' An example pair is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 2 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Ideally, a time aware video- language model should be able to associate the video with the temporally consistent text, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We refer to this task as time-order consistency check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In order to rule out the possibility that synthetic videos are out-of-distribution, we also perform the same experiment with canonical clips 3 Paradigm Method Video-to-Text Text-to-Video Chance 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 Image-Language adapted to video CLIP4Clip [65] 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 CLIP2Video [22] 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='8 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='8 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3 CenterCLIP [128] 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 Video-Language Contrastive VideoCLIP [117] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3 Frozen in Time [6] 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='6 Video-Language Masking BridgeFormer [28] 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='2 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Results on synthetic control ( ) and time-order consis- tency ( ) task as described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Existing video-language models show random performance on our time-order task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' where a video displays a single event and the text describes that same event as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We refer to this as the control task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Choice of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We consider recent video-language models, broadly categorized into three groups: (i) image- language models like CLIP [83] that are adapted to videos [22,65,128], (ii) pure video-language models trained on a contrastive learning objective [6, 117], and (iii) pure video-language models trained on a masking objective [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We evaluate video-to-text and text-to-video re- trieval on both time-order consistency and control tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' From Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 1, we observe that while most video-language models perform well on the control task, all of them strug- gle and perform on par with random chance on the temporal task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This gap in performance deserves attention given the importance of time in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Adaptation by Consistent Time-Ordering We describe a post-pretraining recipe for instilling a sense of time into a video-language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We pro- pose TACT: Temporal Adaptation by Consistency of Time- order, that is based on two key components: (i) we artifi- cially create samples that provide temporal signals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', by flipping the order of events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' and (ii) we introduce a modi- fied contrastive loss to learn temporal consistency based on these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We start by defining the notation and then describe the key components of our adaptation recipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Let V be the space of video clips and T be the space of text clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Consider two non-overlapping video clips vi, vj ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Let ζi, ζj ∈ T be their respective captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Let τ be a temporal relation, τ ∈ {before, after}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Then, we denote a stitched and time-order consistent clip as (uij, tij), where uij := [vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' vj], tij := [ζi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' ζi], and [·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' ·] denotes concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Note that depending on τ, the order of vi and vj may need to change in uij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For brevity, we drop the subscripts and refer to the stitched pair as (u, t) unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Time-order reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The classical contrastive learning paradigm for video-language models aligns components of a video clip vi with it’s text counterpart ζi and contrasts against other texts ζj that usually describe a completely dif- ferent clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This makes such models ignore the finer de- tails of temporal understanding as it is easier to contrast the negatives by simply focusing on the objects or the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This is evident from simple bag-of-word like methods that are shown to work well for contrastive learning, both on the visual (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', CLIP4Clip [65]) and textual (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', MIL- NCE [67]) modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We hypothesize that unless there are negatives in a contrastive setup that contain the same scenes and objects, models do not need to learn a sense of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Thus, we present a simple strategy to generate nega- tives that force the learning process to focus on the temporal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We define a time-order reversal function T that operates on the stitched video clip or text description and temporally swaps its constituents : T(u) = T([vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' vj]) := [vj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' vi], and (1) T(t) = T([ζi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' ζj]) := [ζj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' ζi] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' (2) An illustration of T is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Note that T does not reverse the actual video, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', time does not flow back- wards, but only changes the order in which events happen in the stitched clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Our objective is to train a model that is able to distinguish between the original pair (u, t) and time-reversed versions (u, T(t)), and (T(u), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We assume access to an existing pre-trained video-language model with a visual encoder fθ and text en- coder gφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We obtain the video encoding zu := fθ(u) ∈ Rd and the text encoding zt := gφ(t) ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Our goal is to adapt Θ = {θ, φ} via post pre-training such that the re- sulting model is time aware while maintaining its original performance on tasks such as retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' As we aim to use a small amount of data, we only update some parameters of the model (Θ), such as the last few layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We now introduce our recipe for temporal adaptation based on the InfoNCE loss [103] to learn time-order sen- sitive video-text correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For a positive (or time- order consistent) video-text pair (u, t), we first define a for- ward loss where the stitched pair is in its original time-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Lf = � (u,t)∈B (TNCE(zu, zt) + TNCE(zt, zu)) , (3) where TNCE is the Noise Contrastive Estimation (NCE) loss for temporal adaptation, defined as: TNCE(zu, zt) := − log exp(zu · zt) � t′∈Bt exp(zu · zt′) + Ctime , (4) where B is the batch of (u, t) pairs and Bt specifically refers to other stitched text captions in the batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Ctime accumu- 4 Usual Positives Usual Negatives Time-order reversed Negatives (Cross sample) Time-order reversed Negatives (Same sample) A red circle appears before a yellow circle A yellow circle appears before a red circle 𝕋 𝕋 Time-order Reversal function 𝕋 ℒ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' ℒ" Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Overview of TACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Along with the usual contrastive loss, where negatives come from other samples in the batch, we make use of time-order reversal within the same sample and cross samples to generate additional negatives for both video and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We also extend the contrastive loss to time-order reversed video and text corresponding to reverse consistency Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' lates negatives defined using time-order reversal as: Ctime = αsame exp(zu·zT(t))+αcross � t′∈Bt\\{t} exp(zu·zT(t′)), (5) where αsame controls the effect of contrasting text from the same sample but with reversed text time-order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', T(t), and αcross encourages the model to contrast between re- versed versions of other text captions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', T(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Note that when both αsame and αcross are 0, we revert back to the standard NCE formulation, albeit on stitched pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' While Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' (4) corresponds to the video-text loss TNCE(zu, zt), the text-video loss TNCE(zt, zu) is defined symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Furthermore, we also apply a reverse loss Lr to bring time-order reversed versions of both the video and the text together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Note that as we consider (u, t) as a positive pair, (T(u), T(t)) also form a positive pair, Lr = � (T(u),T(t))∈B � TNCE(zT(u), zT(t)) + TNCE(zT(t), zT(u)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' (6) Here, the TNCE term operates on time-reversed clips and Ctime contrasts (T(u), T(t)) with un-reversed text clips in the batch (T(u), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The overall loss function is defined as a combination, L = Lf + βLr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' (7) Depending on the choice of loss coefficients αsame, αcross, β ∈ {0, 1}, we can vary properties of the adapted model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For example, setting αsame=1 encour- ages high sensitivity to time-order reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' As we will see empirically, the loss coefficients also provide the flexibility to adapt the model to various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We illustrate this temporal extension of the contrastive loss in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 3 (best seen in color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' T illustrates the time or- der reversal function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The top half corresponds to Lf while the bottom half visualizes Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In particular, the top-left quadrant alone corresponds to the standard contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' While the green diagonal terms are positive pairs, the red di- agonal terms are the strongest drivers for instilling temporal understanding in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Experiments: TACT Ablations Base model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We demonstrate the effectiveness of TACT as an adaptation recipe on top of VideoCLIP [117] ow- ing to its simple architecture, contrastive objective, and use of pre-computed S3D [114] features that enable faster ex- perimentation and allow encoding a long temporal context (∼32 secs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We initialize Θ from the model pretrained on HowTo100M [68] and post-pretrain on multiple datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' One of our key objectives is to post-pretrain on a small amount of data in contrast to massive pretrain- ing datasets such as WebVid2M [7] or HowTo100M [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We consider dense video captioning datasets that offer suf- ficient diversity in terms of size, backgrounds, clip dura- tions, viewpoints and activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Specifically, we experi- ment with: (i) TEMPO [39]: the subset of stitched di- verse third-person videos from DiDeMo [40] with text de- scriptions for fixed 5s segments that contain before/after relations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' (ii) ActivityNet Captions [54]: a dense video captioning dataset with human-centric actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' (iii) Cha- rades [93]: a scripted indoor daily human activities video dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' and (iv) Charades-Ego [91]: similar to Charades, scripted human activities from the egocentric viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' To construct stitched clips, we randomly sample any two non- overlapping clip-text pairs in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Since we require stitched clips instead of raw videos, we create new splits for each dataset (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We will release all the splits publicly on our project page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We evaluate the post-pretrained model using two sets of metrics: (i) standard retrieval metrics, recall R@1, R@5, R@10 and median-rank evaluated on stitched video-text clips;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' and (ii) time-order consistency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', the fraction of videos for which the model correctly associates text that is time order consistent with the video: Atime := 1 |D| � (u,t)∈D I[d(zu, zt) < d(zu, zT(t))], (8) where (u, t) are time-order consistent pairs, D is the dataset, and d(·, ·) is a distance metric based on cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 5 Dataset Train Validation Test Ego Length NV NC NV NC NV NC (s) TEMPO 3,904 28,427 411 1,000 396 1,000 \x17 30 ActivityNet 7,440 95,474 453 906 456 912 \x17 120 Charades 5,262 99,928 500 1,000 500 1,000 \x17 30 Charades-Ego 2,679 155,306 500 1,000 210 420 \x13 31 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Statistics of datasets we consider for temporal adapta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' NV is the number of unique videos and NC is the number of stitched clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Based on NV, TEMPO and Charades-Ego are smaller as compared to ActivityNet and Charades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Post-pretraining details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We freeze the word embeddings and layers 1 to 5 for both the video and text encoders in VideoCLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For adaptation, we use the Adam optimizer [53] with learning rate 5e−6, batch size 32 trained on a single node with 4 GeForce GTX 1080 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' On TEMPO, we train for 60 epochs while on the other datasets, we train for 10 epochs and pick the checkpoint that maximizes the geo- metric mean of R@1 and Atime on the respective validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' A typical training run takes about 1-3 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Evaluation on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 3 show that TACT⋆ with optimal loss coefficients outperforms TACT† (all 0 loss coefficients) and the zero-shot baseline (no post- pretraining), both on the retrieval and time-order consis- tency tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This indicates the robustness of the adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Impact of loss coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Choosing appropriate Dataset Method Retrieval Time-order R@1↑ MedR ↓ Atime↑ Zero-shot 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 TEMPO TACT† 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 TACT⋆ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 Zero-shot 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='6 ActivityNet TACT† 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='8 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 TACT⋆ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='8 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 Zero-shot 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 Charades TACT† 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 TACT⋆ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 Zero-shot 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='6 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 Charades-Ego TACT† 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 TACT⋆ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='2 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Results for the best TACT model on test sets of various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' TACT⋆ is the model with optimal loss coefficients and TACT† is a baseline with all coefficients 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' On time order, TACT generalizes well with TACT⋆ outperforming the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' On re- trieval, for TEMPO and Charades-Ego, TACT⋆ outperforms the baseline as their optimal models have β=1 which helps retrieval with small amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 ∆time between clips (sec) 0 100 200 300 400 500 600 Frequency TEMPO Mean: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 0 100 200 ∆time between clips (sec) 0 50 100 150 200 250 300 ActivityNet Mean: 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='8 10 20 30 40 ∆time between clips (sec) 0 50 100 150 200 Charades Mean: 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 0 10 20 30 ∆time between clips (sec) 0 50 100 150 200 250 CharadesEgo Mean: 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Time-distance between stitched clips in datasets for tem- poral adaptation (∆time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' TEMPO has stitched clips close to each other while those in Charades-Ego are farthest apart suggesting a correlation between ∆time and the difficulty of temporal adapta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' values for loss coefficients Θl:={αsame, αcross, β} al- lows the model to learn various aspects and adapt us- ing different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' On each dataset, we vary Θl∈{0, 1}3 and find the best configuration based on the GeometricMean(R@1, max(Atime − 50, 0)) on the valida- tion sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The above metric ensures the geometric mean is not overpowered by Atime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' As αsame is directly responsible for discriminating be- tween original and time-reversed orders, unsurprisingly, setting it to 1 is necessary to achieve the best results on Atime for all the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For TEMPO and Charades-Ego, using all loss components (all 1) provides the best results, whereas αcross=1 and β=0 achieves a better trade-off for Activi- tyNet and Charades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Choosing β=1 leads to an improve- ment in retrieval performance for TEMPO and Charades- Ego but leads to a decline for ActivityNet and Charades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We attribute this to the number of unique videos in the train set for these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' As ActivityNet and Charades have more videos than TEMPO or Charades-Ego (see train NV Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 2) additional positives introduced by setting β=1 are not necessary and in fact hurt performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' What makes temporal adaptation hard?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We observe a large gap in Atime between TEMPO and ActivityNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We hypothesize that the distance (in seconds) between the two clips (∆time) in a stitched clip is strongly correlated with the difficulty of adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Intuitively, it is easier to in- fer the time-order consistency for a stitched clip u with the text t that has distant constituent clips vi, vj since the ob- jects and scene could be vastly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In contrast, it is harder to discern the correct time-order when the con- stituent clips are closer in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 4 shows the distribu- tion of ∆time for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Indeed, the mean distance between clips in ActivityNet (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='8s) is much higher than that in TEMPO (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4s) making the task harder on TEMPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' To further strengthen our hypothesis, we conduct a con- trolled experiment where we gradually vary the distribution of ∆time for Charades-Ego to match it to that of TEMPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We find a strong correlation (ρ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='92) between ∆time and hardness of adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' More details can be found in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 6 Loss coefficients TEMPO ActivityNet Charades Charades-Ego αsame αcross β R@1 ↑ MedR ↓ Atime ↑ R@1 ↑ MedR ↓ Atime ↑ R@1 ↑ MedR ↓ Atime ↑ R@1 ↑ MedR ↓ Atime ↑ Chance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 50.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Impact of loss coefficients for TACT post-pretraining on validation sets of various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For each dataset, the corresponding color-marked row denotes the best configuration based on the geometric mean of R@1 and Atime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' TACT is able to connect time-order in video and language while maintaining its retrieval capabilities across several datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Experiments: Downstream Evaluation The goal of foundational video-language models such as VideoCLIP is to pretrain them on massive video-text datasets and generalize in a zero- or few-shot manner to a diverse range of downstream video understanding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We evaluate TACT adapted models on three sets of downstream tasks that need a low-to-high time awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Baseline: Standard post-pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Comparing our temporally adapted models with pretrained VideoCLIP is not fair since adapted models see data beyond the pretrain- ing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In addition to the zero-shot comparison, we com- pare against a baseline model that is trained for standard video-text retrieval on the same datasets as temporal adap- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Instead of using stitched clips, we use simple canon- ical pairs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', (vi, ζi) instead of (uij, tij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Results on synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' As our first downstream eval- uation, we check if TACT performs better on our synthetic data (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' On the video-to-text variant, the TEMPO- adapted model attains 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1% accuracy, ActivityNet 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4%, Charades 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3% and Charades-Ego 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This is signif- icantly higher than random performance that non-adapted models achieve in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This highlights that TACT mod- els indeed learn useful signals to match the time-order in video and language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Text-to-video retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We consider two widely used benchmarks: MSR-VTT [118] and YouCookII [132] and adopt standard retrieval metrics for these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Several re- cent works have identified a bias for spatial-understanding in these datasets, particularly MSR-VTT [8, 13, 42, 55, 58, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Thus, we consider this class of tasks as requiring low time awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' As shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 5 set I, on MSR- VTT [118], we observe that TACT is worse (marked in red) or at-par with the baselines for all adaptation datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This aligns well with findings in [13,55] that these benchmarks do not need time awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' On YouCookII [132], TACT models based on Charades and Charades-Ego outperform the baseline (marked in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We believe this may be as a consequence of both YouCookII and Charades being cap- tured indoors, which lowers the domain-shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Temporal video QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Next, we use subsets of recently released multiple-choice video question answering bench- marks: Next-QA [113] and AGQA [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The idea is to check if we can probe models for temporal understand- ing by asking questions with temporal language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Buch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' [13] a identify subset of Next-QA, dubbed as ATP-hard2, with a higher concentration of temporally challenging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For AGQA, we pick a subset of ∼6k questions that explic- itly have a question with before/after relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We consider these benchmarks as requiring moderate-high level of time awareness and AGQA in particular is also close to our adap- tation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We use accuracy as the standard metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We observe (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 5 set II) that indeed TACT al- most always outperforms (marked in green) baselines on both Next-QA and AGQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' TEMPO-adapted TACT seems to generalize particularly well on both benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Like- wise, Charades-adapted TACT does well on AGQA since AGQA is also based on the Charades videos accounting for reduced domain-shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We affirm that temporal adaptation is useful, especially when the downstream tasks require it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Action-to-video retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Finally, we consider action recognition benchmarks such as Something-Something (SSv2) [32] and Temporal [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' SSv2 was designed to cap- ture richer temporal information [32,55] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We follow Lie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' [55], who propose the template-retrieval task that en- courages temporal modelling and use their evaluation split3 containing C=174 actions and K=12 videos per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In- terestingly, different actions in SSv2 require differing lev- els of time awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We create a subset SSv2 (events) 2Available here: github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='com/StanfordVL/atp-video-language 3Available here: github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='com/jayleicn/singularity 7 Low Time awareness −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ High Adaptation I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Text-to-Video Retrieval II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Temporal VQA III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Action-to-Video Retrieval Dataset Method MSR-VTT YouCookII Next-QA (ATP) AGQA SSv2 SSv2 (events) Temporal R@1 R@5 R@10 R@1 R@5 R@10 Accuracy Accuracy mAP mAP mAP Chance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 TACT 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='8 29.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 TACT 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='6 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='9 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='9 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='6 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Results on downstream zero-shot evaluation with tasks requiring increasing time awareness from I to III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' None corresponds to direct evaluation of the VideoCLIP model on the downstream dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Green denotes an improvement for the TACT adapted model w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' the baseline, red denotes a deterioration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' As we move from tasks that need low to high time awareness, the effectiveness of TACT increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 6 for a more detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The table is best viewed on screen in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' with Cevents=49 actions that have at least two verbs in the label as occurrence of multiple verbs is indicative of mul- tiple events occurring in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Finally, we also eval- uate against the Temporal benchmark [88], a combination of 50 action classes from SSv2 [32] and Kinetics-400 [51] for which temporal information is deemed to be essential for recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Similar to text-to-video retrieval, we use the action class as a text query and obtain a ranking over all videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Different from the retrieval setup, since a single query has multiple correct answers (upto K=12 videos), we report mAP as the metric for these benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This task set needs high time awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Furthermore, unlike QA tasks in II, there is a shift in several (uncontrolled) factors as we move from temporal adaptation task to these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' From Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 5, we observe that TEMPO- and Charades- adapted models generalize well across set III benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' ActivityNet-adapted TACT underperforms on SSv2 but outperforms the baseline on strongly temporal actions in SSv2 (events) and Temporal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Finally, TACT adapted on Charades-Ego is at-par or slightly worse than the base- line on SSv2 variants, and also on Temporal, perhaps due to the shift from egocentric to third-person videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Over- all, despite SSv2 and Temporal requiring high time aware- ness, TACT models shows promising zero-shot generaliza- tion with the right choice of the adaptation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Discussion and Conclusion Spatial vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' temporal understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' An interesting facet of TACT is αsame which controls how well a model adapts to temporal tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We highlight this on the TEMPO dataset in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 6, where, αsame=0 results in Atime ∼50% while Hyperparameters Adaptation Downstream αsame αcross β TEMPO MSR-VTT AGQA Atime ↑ R@1 ↑ MedR ↓ Accuracy↑ 0 0 0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 0 0 1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='9 0 1 0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='2 0 1 1 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='2 1 0 0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='6 1 0 1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3 1 1 0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='9 1 1 1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='8 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Impact of αsame on spatial- vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' temporal understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Gray denotes better performance between αsame=0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' While αsame=1 drives temporal understanding, it comes at a cost of re- trieval performance on MSR-VTT [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This hints at αsame con- trolling the trade-off between spatial- and temporal-understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' αsame=1 improves performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Further investigation on downstream tasks shows that adaptation with αsame=1 does not perform well on MSR-VTT (a non-temporal bench- mark) but shows consistent improvements on AGQA (a temporal benchmark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This hints at αsame controlling the trade-off between spatial and temporal understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Generalization to other temporal prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The time or- der of events in language can be described using a variety of sentence structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Although we train video-language models using temporal relations such as before/after, it is natural to ask if the model still correctly associates time order for a different prompt such as First, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='., then, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' To systematically evaluate this, we gather event pairs E1, E2 (E1 occurs before E2 in the video) for each sample in the validation set and stitch them using three 8 Temporal accuracy 0 25 50 75 100 TEMPO ActivityNet Charades Charades-Ego E1 before E2 E2 after E1 First, E1, then E2 Effect of different prompts on inferring time-order Chance Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Models trained by TACT with before/after relations gen- eralize to a new kind of prompt such as First, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', then .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='. This indicates learning of the underlying time order of events rather than the mere order of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' prompts as follows: (i) E1 before E2, (ii) E2 after E1, (iii) First E1, then E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 5, TACT-adapted models generalize well to a new prompt (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This substan- tiates the learning of time order of events rather than merely learning the order of words in the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' While we present a promising way of in- stilling time in video-language models, our work is limited to the VideoCLIP [117] pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Our initial ex- periments with Frozen in Time [6] were not as promising, perhaps because it uses a much shorter temporal context (4 frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Furthermore, we consider a specific definition of time awareness derived from temporal relations like be- fore/after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' It is natural to ask if this can be extended to more general notions of temporality, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', as defined by Allen [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Finally, there can always be more downstream tasks that one could consider such as (spatio-)temporal localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Given the essence of time in video-language models, we present a simple experiment based on synthetic data to test for time awareness in existing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We find that existing models lack a sense of time defined in terms of consistency of order of events in video and language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' To fill this gap, building upon VideoCLIP [117], we present TACT, a recipe to instill this sense of time in video-language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Finally, we analyze the zero-shot generalizabil- ity of TACT-adapted models to a diverse set of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We hope that this work provokes further probing and instilling time awareness in video-language models, and also inspires other adaptations of foundational models to solve various challenging tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' References [1] Unaiza Ahsan, Rishi Madhok, and Irfan Essa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Video jig- saw: Unsupervised learning of spatiotemporal context for video action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In Winter Conference on Applica- tions of Computer Vision (WACV), pages 179–189.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 3, 7 14 Supplementary Material As part of the supplementary material, we describe pre- processing steps as well as some qualitative examples from the datasets in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In Appendix B, we present ad- ditional ablations on what makes temporal adaptation hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This expands on the last paragraph of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 5 of the main pa- per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Finally, in Appendix C, we conduct a qualitative anal- ysis to verify if the model has indeed learnt to connect the time order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Datasets and Pre-processing We sketch out the procedure we use for stitching two clips within a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Clip stitching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Consider a video containing two events (clips) vi , vj with associated captions ζi, ζj as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We assume these are non-overlapping (in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We stitch the text descriptions to construct a new caption tij := [ζi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' ζj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Since τ can be either before or after, we end up with two newly constructed sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Correspond- ing to each of these new sentences, we also stitch the video events to construct a stitched video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Note that the order of stitching video events depends on the value of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For exam- ple, if τ is before, then uij := [vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' vj] as shown in first of the two stitched clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' If τ is after, then uij := [vj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' vi] as shown in the second of the two stitched clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' From each stitched clip in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 6, we construct negatives for the contrastive loss by reversing the time order in ei- ther video or text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' This step happens on-the-fly during loss computation, and hence, we do not show it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For a given dataset, we can either use all possible tuples of non- overlapping events to create such stitched clips or sample from all possible tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Since the TEMPO dataset already comes with stitched event descriptions (based on DiDeMo), we directly use its subset which has before/after relations in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For all the other datasets, we apply the stitching process as described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Recall, ∆time is the time distance be- tween the two events, and plays a key role in deciding the difficulty of temporal adaptation, as observed empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Next, we describe dataset properties and show some qualitative examples after the clip stitching step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Adaptation datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' To gain a sense of the diversity in the datasets we consider for adaptation, we present exam- ples of stitched clips from these datasets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Please refer to the attached HTML page for corresponding videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Since TEMPO has short adjacent clips, the context remains almost the same, we think this is important to instill a sense of time in models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In contrast, for ActivityNet, since the stitched events are far apart, the context changes make it easy to infer which event description goes with which part of the video, or the time order of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In this regard, Charades and Charades-Ego are similar to TEMPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Quan- Video Stream Event X Event Y Stitched clips Description(X) before Description(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Video Text Description(X) after Description(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Video Text Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Illustration of clip stitching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We consider two non- overlapping events in a video and stitch them with temporal re- lations - before and after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' ∆time denotes the time difference be- tween midpoints of the two events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 0 50 100 150 200 0 200 400 600 800 1000 1200 1400 Number of videos TEMPO 0 50 100 150 200 0 100 200 300 400 500 600 700 Charades 0 50 100 150 200 Number of clips in a video 0 20 40 60 80 100 120 Number of videos CharadesEgo 0 50 100 150 200 Number of clips in a video 0 500 1000 1500 2000 ActivityNet Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Number of clips in a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The number of clips per video is lower in TEMPO and ActivityNet as compared to Cha- rades and Charades-Ego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' titatively, this change in context is captured by ∆time which is lowest for TEMPO (mean 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='8s), followed by Charades- Ego (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3s), Charades (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5s) and ActivityNet (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='8s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Distribution of number of clips in a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' A single video with 10 non-overlapping individual event clips can make upto 10C2=45 stitched clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We plot the number of clips per video against the number of videos in a given dataset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' A single video with >30 stitched clips is rare in TEMPO and ActivityNet while much more frequent in Charades and Charades-Ego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Overall, the number of clips per video is lower in TEMPO and ActivityNet as compared to Charades and Charades-Ego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Downstream datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 9, we also show some ex- amples from some downstream datasets (tasks) that need higher time awareness since they typically involve multiple temporally linked events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', walk and eat in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 9(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 15 timeA rabbit lays down on its stomach before bunny lying on it’s side Little girl eats from cup after the child walks downhill (a) TEMPO A woman is standing in a room holding a hula hoop before she begins to use the hula hoop The team shakes hands with the opposing team after a team groups together holding a trophy (b) ActivityNet Putting on shoe/shoes before holding a mirror (c) Charades Taking a broom from somewhere before holding a dish (d) Charades-Ego Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Examples from datasets used for temporal adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The first two frames are linearly spaced from the first event while the last two from the second event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Notice how there is a sig- nificant change in visual context between the two events in Activi- tyNet in contrast to other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Best viewed on a screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Please refer to the attached HTML page for corresponding videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' On these datasets, we perform zero-shot evaluation of tem- porally adapted models in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 6 of the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Experiments What makes temporal adaptation difficult?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' To recall, we define ∆time as the time-distance (in seconds) between the midpoints of the two clips in a stitched pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We hy- pothesize that ∆time is inversely related to the difficulty of temporal adaptation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', the larger ∆time, the easier it is to distinguish between two stitched clips that have opposite time order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For example, consider ActivityNet examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 8(b) where the visual context changes significantly making inference of the time order of events relatively eas- ier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We further test our hypothesis by sampling individual clips from the Charades-Ego dataset to match the ∆time dis- tribution of TEMPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Concretely, assuming ∆time for both these datasets follows a multinomial distribution, we con- struct a new distribution using a convex combination of the individual distributions , where the mixing parameter λ ∈ [0, 1] controls the extent to which we modify the dis- tribution from TEMPO (λ=0) to Charades-Ego (λ=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The resulting distributions are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 10 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' With λ=1, we sample from the original Charages-Ego distribu- tion and gradually move towards TEMPO as λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We then sample stitched clips according to this new dis- tribution and post-pretrain temporal adaptation for varying values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Note that for this experiment, we keep fixed NC=10, 000 for each λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 10 (right), we indeed find that as we move towards a more TEMPO-like distri- bution (shorter ∆time), temporal accuracy deteriorates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The best fit also confirms that the distribution of ∆time is strongly correlated (ρ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='92) with the difficulty of inferring time- order consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Qualitative Analysis To get an intuitive sense of whether a TACT model un- derstands time order of events, we perform a qualitative analysis on the model trained on TEMPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Our demo in- terface looks like the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' First, a user uploads a video and adds text descriptions for two events within the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' These descriptions are then connected via a temporal relation such as before or after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We also experiment with a new temporal connector First, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', then, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='. to check if our model generalizes beyond be- fore/after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We will release the demo code on our project website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' First, we consider samples from the TEMPO validation set and show their results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Notably, for some examples, it connects time order for before relations but not the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We suspect this is because a majority (∼ 60%) of the TEMPO dataset has descriptions involv- ing before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Note that TEMPO already comes with tem- poral captions of which we pick subset of before/after rela- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Second, we also consider samples from other datasets which the model has never seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' To our surprise, albeit qualitatively, the model does generalize well to such exam- ples as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' These results reinforce the promise of our method and also raise the possibility of extending this work to consider more general temporal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Having said that, we re- iterate that these are qualitative examples and should be treated as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 16 Nathalie Veilleux watchwm Ownerof StudiosVertPranaQuestion: How did the boy react when he entered the room at the start?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Answer: Smile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Question: Why does the baby turn around near the end of the video?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Answer: Sits down to play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' (a) Next-QA: Video question answering Question: Did they reach for and grab a picture before or after putting a bag somewhere?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Answer: Before Question: Did they walk through a doorway before or after they eating the last thing they touched?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Answer: After (b) AGQA: Video question answering Template: Spinning [something] that quickly stops spinning (c) Something-Something: Template-based video retrieval Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Examples from datasets used for downstream evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' These tasks demand time awareness since it is often not possible to infer the action from a single frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Please refer to the attached HTML page for corresponding videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5] (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='6 Density λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 (CharadesEgo) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5] (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0] (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5] (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5, inf] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='75 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5] (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0] (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5] (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5, inf] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='50 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5] (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0] (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5] (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5, inf] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='25 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5] (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0] (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5] (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5, inf] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='6λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 (TEMPO) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='0 λ −→ 0 60 62 64 66 68 Temporal Accuracy (Atime) Atime vs Distribution of ∆time ρ = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='92 CharadesEgo TEMPO Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Impact of changing distribution of ∆time, the time gap between two stitched clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Left: We vary the distribution of ∆time for Charades-Ego and make it similar to that of TEMPO as λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Thus, crudely, as λ decreases, so does ∆time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Right: Atime on Charades-Ego where the time difference between sampled clips is according to the distributions on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We observe that the accuracy deteriorates as the time-distance between a pair of clips decreases indicating a strong correlation between the distribution of ∆time and difficulty of temporal adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Interface of our demo for qualitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The user uploads a video and is asked to describe two events in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' These event descriptions are then connected via one of the three temporal relations shown at the bottom left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We construct one sentence that is consistent with the time order of events in the video and another that is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The output on the right shows the ranking of the constructed sentences in terms of cosine similarity with the video representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Higher score for correct matching indicated by a longer orange bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 18 Test of Time: Instilling Video-Language Models with a Sense of Time Rank sentences based on their relevance to a video Video (stitched with two events) Constructed sentence 1 The child runs into the room before he sits near the gifts Constructed sentence 2 he sits near the gifts before the child runs into the room The child runs into the room before he sits near the gifts 54% he sits near the gifts before the child runs into the room 46% Refresh video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Check this if you load a new video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Write a description for event X (any event within the video) The child runs into the room Write a description for event Y (any event within the video) he sits near the gifts Choose a relation between the two events O before after First,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=',.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='. Clear SubmitBefore After First, … then, ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Before After First, … then, ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Qualitative examples from TEMPO validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We evaluate similarity of a given video with sentences with different temporal order with the usual temporal connectors (before/after).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Green bordered boxes indicate correct predictions (consistent time order between video and language) while red denote mis- predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' For some examples, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', in the bottom example, the model gets predictions incorrect particularly for relations other than before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Furthermore, we also try a new temporal connector First, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=', then, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' and observe that the model qualitatively generalizes to that as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Before After First, … then, ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' (a) Example from Charades-Ego Before After First, … then, ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' (b) Example from Next-QA Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' Qualitative results on samples not from TEMPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' We see that despite not having seen these examples, the model still connects the time order across video and language correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 19 First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The stuffed panda is visible on zooming in occurs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' 68% then the bus drives by occurs First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The bus drives by occurs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' then the stuffed panda is 32% visible on zooming in occursThe stuffed panda is visible on zooming in before the bus 69% drives by The bus drives by before the stuffed panda is visible on 31% zooming inThe bus drives by after the stuffed panda is visible on 72% zooming in The stuffed panda is visible on zooming in after the bus 28% drives byFirst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' she eats an ice-cream occurs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' then the child walks 59% down the hill occurs First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The child walks down the hill occurs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' then she eats 41% an ice-cream occursThe child walks down the hill before she eats an ice-cream 98% she eats an ice-cream before the child walks down the hill 2%The child walks down the hill after she eats an ice-cream 72% she eats an ice-cream after the child walks down the hill 28%Ranking over sentences First The man picks up a broom occurs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' then he looks at the television occurs First The man picks up a broom occurs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' then he looks at the 88% television occurs First He looks at the television occurs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' then the man picks 12% up a broom occurs Ranking oversentences The man picks up a broom before he looks at the television The man picks up a broom before he looks at the television 100% He looks at the television before the man picks up a broom 0%Ranking oversentences He looks at the television after the man picks up a broom He looks at the television after the man picks up a broom 100% The man picks up a broom after he looks at the television 0%First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' The child runs into the room occurs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' then he sits 74% near the gifts occurs First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' he sits near the gifts occurs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} +page_content=' then the child runs 26% into the room occursThe child runs into the room before he sits near the gifts 54% he sits near the gifts before the child runs into the room 46%he sits near the gifts after the child runs into the room 91% The child runs into the room after he sits near the gifts %6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNA0T4oBgHgl3EQfIP_x/content/2301.02074v1.pdf'} diff --git a/itAzT4oBgHgl3EQfM_uk/content/tmp_files/2301.01142v1.pdf.txt b/itAzT4oBgHgl3EQfM_uk/content/tmp_files/2301.01142v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4657267e52e9e4d6fe5556a0158ae5fc22e4bd45 --- /dev/null +++ b/itAzT4oBgHgl3EQfM_uk/content/tmp_files/2301.01142v1.pdf.txt @@ -0,0 +1,2118 @@ +Mutual Information Regularization for Vertical Federated Learning +Tianyuan Zou +Yang Liu +Ya-Qin Zhang +Institute for AI Industry Research, Tsinghua University +Beijing, China +zty22@mails.tsinghua.edu.cn, liuy03@air.tsinghua.edu.cn, zhangyaqin@tsinghua.edu.cn +Abstract +Vertical Federated Learning (VFL) is widely utilized in +real-world applications to enable collaborative learning +while protecting data privacy and safety. However, previous +works show that parties without labels (passive parties) in +VFL can infer the sensitive label information owned by the +party with labels (active party), or execute backdoor attacks +to VFL. Meanwhile, active party can also infer sensitive +feature information from passive party. All these pose new +privacy and security challenges to VFL systems. We pro- +pose a new general defense method which limits the mutual +information between private raw data, including both fea- +tures and labels, and intermediate outputs to achieve a better +trade-off between model utility and privacy. We term this +defense Mutual Information Regularization Defense (MID). +We theoretically and experimentally testify the effectiveness +of our MID method in defending existing attacks in VFL, in- +cluding label inference attacks, backdoor attacks and feature +reconstruction attacks. +1. Introduction +Federated Learning (FL) [29] was first proposed to train +cross-device machine learning models and protect data pri- +vacy simultaneously which can be also regarded as horizon- +tal FL (HFL) [39] as data are partitioned horizontally in the +database by sample. Another kind of FL framework is verti- +cal FL (VFL) [6,14,16,24,25,39] where data are partitioned +by feature, which means each participant owns a portion of +the data features of each sample. This framework is consis- +tent with several real-world situations. For example, a bank +and an E-commerce company each obtains some features +of the same group of users and they collaboratively train +a model for preference prediction. Similar to HFL, partic- +ipants in VFL aim to collaboratively train a shared model +on the premise of keeping their local private data safe by +communicating privacy-preserving intermediate results. As +shown in Fig. 1, in basic VFL framework with 2 parties, lo- +cal data and local model of each party are kept locally while +(a) Label Inference Attacks [11,21,46] and Feature Reconstruction Attacks +[19] +(b) Targeted [46] and Non-targeted Backdoor [23] Attacks +Figure 1. Demonstration of different attacks in 2-party VFL. +local intermediate results and gradient information are trans- +mitted between an active and a passive party. To attack this +basic framework, recent studies [43–46] have explored data +reconstruction attacks by exploiting the intermediate results +exchanged, as well as backdoor attacks by manipulating the +input data. As for data reconstruction attacks, both label +inference attacks [11,21,46] and feature reconstruction at- +tacks [19,28] have been proposed. As for backdoor attacks, +malicious passive parties can modify the shared model for +their own purpose by adding a trigger to a few of the at- +tacker’s local samples in a targeted backdoor attack [46], +or hurt the overall model utility by adding noise or failing +1 +arXiv:2301.01142v1 [cs.LG] 1 Jan 2023 + +Feature Reconstruction +Model +Inversion +"horse" +Gradient +Intermediate +Result +Model +Gradient +Completion +Inversion +Auxiliary +"horse" +Labeled Data +"horse" +Label Inference +Label Inference"horse" +Intermediate +Gradient +Result +Missing +Non-targeted +Backdoor +Noisy +sample +argetec +Triggered +Backdoor +Sample +riggerto transmit intermediate results in non-targeted backdoor +attacks [23]. We summarize these attacks in Fig. 1. +To mitigate these threats, various defense methods can be +applied, including Adding Noise [3,9,38], Gradient Sparsi- +fication (GS) [22] and Discrete Gradients (DG) [11]. How- +ever, these defense methods suffer from accuracy drop for +the main task. There are also specific defense methods for +targeted scenarios, such as MARVELL [21] to defend label +leakage in binary classification , Confusional AutoEncoder +(CAE) [46] to defend label leakage by model inversion at- +tacks, RVFR [23] to defend robustness-related attacks such +as missing features and adversarial input attacks. However +these defense scenarios are task-specific. +In this work, we observe that the root cause for data at- +tacks by either active or passive party lies in the fundamental +dependency between the local model at a passive party and +the label or local features. Therefore we proposed a new +general defense method that aims to defend existing attacks +from the perspective of information theory. Specifically, +we design a Mutual Information Regularization Defense +(MID) for restricting the level of information about local +data contained in exchanged intermediate outputs. We per- +form extensive experiments which demonstrate that MID is +very effective in defending all kinds of data reconstruction +attacks and backdoor attacks compared with existing defense +methods. Moreover, we provide theoretical guarantee for +model robustness with MID under VFL scenario. +In summary, our contributions are: +• We propose a new general defense method for VFL, +Mutual Information Regularization Defense (MID), +which regularizes the information dependency between +parties’ local sensitive data and exchanged intermediate +outputs. We show theoretically that MID is effective +in preventing information leakage from exposed inter- +mediate outputs and improving model robustness to +defend against backdoor attacks. +• We perform comprehensive experimental evaluations +and show that with proper design of information bottle- +neck, MID is a promising universal defense method that +achieves better utility-privacy trade-off than other gen- +eral defense methods for various feature reconstruction +attacks, label inference attacks and backdoor attacks. +2. Related Work +Federated Learning (FL) [30,39,40] is a novel machine +learning paradigm in which participants collaboratively train +a machine learning model without centralizing each parties’ +local data. FL can be further categorized into horizontal +federated learning (HFL) where data are partitioned by sam- +ples, and vertical federated learning (VFL) where data are +partitioned by features [39]. VFL [6, 18, 26] is commonly +used in real-world cross-silo applications in finance and ad- +vertising [7,10]. +Existing attacks to VFL protocols are either to recon- +struct private data [11, 17, 21, 46] or to hurt model robust- +ness [23, 23, 27, 31, 46]. For data reconstruction attacks, +the target of these attacks is either private labels or private +features. Label inference attacks can be performed using +sample-level gradients (SLI) [11,21], or batch-level gradi- +ents (BLI) [46], or trained local models [11]. Reconstruction +of private features also pose great threat to data safety of VFL +system. Most related works focus on simple models includ- +ing logistic regression [15,28,37] and tree [28]. While for +neural networks (NN), recovering image data [19] or tabular +data [28] can be done by model inversion under white-box +setting, and for black-box setting, prior information about +data is required [17] or the targeted features are limited to +binary values [42]. In addition, passive parties can launch +backdoor attacks by assigning specific label to triggered sam- +ples [46](targeted backdoor), or by adding noise to some +randomly selected samples or by adding missing features to +harm the model utility [13,23](non-targeted backdoor). +For defense, cryptographic techniques like Homomor- +phic Encryption (HE) or Secure Multi-Party Computation +(MPC) [39] have been proposed to protect in-transit mes- +sages. However, since they do not protect learned results, +VFL with such protections still opens doors to attacks that +only exploit trained model results or malicious backdoor [46]. +Some other general defense strategies focus on reducing in- +formation by adding noise [9,11,21], Gradient Discretiza- +tion [8,11], Gradient Sparsification [1] and Gradient Com- +pression [22], or combined [11,32]. These methods suffer +from utility losses. Other emerging defense methods tar- +gets to specific attacks or scenarios, such as data augmenta- +tion [12] or disguising labels [19,46] to defend against gra- +dient inversion attacks, MARVELL [21] to defend against +label inference in binary classification tasks, RVFR [23] +to defend against backdoor attacks. Mutual Information +has been explored as an effective regularization to machine +learning models to improve the robustness of model against +malicious attacks in the past [2,35,36] but has never been +explored in VFL setting before. +3. Problem Definition +3.1. Vertical Federated Learning Setting +Under a typical VFL system, K data owners together ob- +tain a dataset of N samples D = {xi, yi}N +i=1 with each par- +ticipant k holding a portion of the features Xk = {xk +i }N +i=1 +and only one party controls the label information Y += +{yi}N +i=1. We refer this party as the active party and other +parties as the passive parties. Without loss of generality, +we assume party K is the active party, and other parties +are passive parties. In VFL, each party k adopts a local +2 + +model Gk with model parameters θk. Note that Gk can +adopt various kinds of model, like logistic regression, tree, +support vector machine, neural network, etc. With the local +model and data, each participant k calculates its local output +Hk = {Hk +i }N +i=1 = {Gk(xk +i , θk)}N +i=1 = Gk(Xk, θk) and +sends them to the active party for loss calculation. Therefore, +the overall objective for VFL is formulated as: +min +Θ L(Θ; D) ≜ 1 +N +N +� +i=1 +ℓ(S(H1 +i , . . . , HK +i ), yi) +(1) +where Θ = [θ1; . . . ; θK] are training parameters, S denotes +a global model which can be either a prediction function +or a model with trainable parameters, and ℓ denotes a loss +function, such as a cross entropy loss. To perform training +with back propagation, active party performs gradient com- +putation with received Hk and transmits back { ∂ℓ +∂Hi }N +i=1 to +each party. See Algorithm 1 for a complete algorithm. +To further protect transmitted sample-level information, +cryptographic techniques such as Homomorphic Encryption +(HE) can be applied [39] and gradient is calculated under en- +cryption while a coordinator is introduced to the VFL system +for distributing encryption keys and decryption. Under HE- +protected VFL, sample-level gradient information is protect +while batch-level gradient information is revealed. +Algorithm 1 A VFL framework with and without MID (at +active party) +Input: Learning rate η; MID hyper-parameter λ +Output: Model parameters θ1, θ2, . . . , θK +1: Party 1,2,. . . ,K, initialize θ1, θ2, ... θK; +2: for each iteration j=1,2, ... do +3: +Randomly sample S ⊂ [N]; +4: +for each party k in parallel do +5: +Computes {Hk +i }i∈S; +6: +Sends {Hk +i }i∈S to party K; +7: +end for +8: +if MID is applied then +9: +Active party computes Zk +i = MVIB(Hk +i ) and loss +ℓ using Eq. (6); +10: +Active party computes { ∂ℓ +∂Zk +i }i∈S and updates +MVIB; +11: +else +12: +Active party computes loss ℓ using Eq. (1); +13: +end if +14: +Active party sends { ∂ℓ +∂Hi }i∈S to all other parties; +15: +for each party k=1,2,...,K in parallel do +16: +Computes ∇kℓ = +∂ℓ +∂Hi +∂Hk +i +∂θk ; +17: +Updates θj+1 +k += θj +k − η∇kℓ; +18: +end for +19: end for +To simplify our discussion, we first consider a VFL sys- +tem with 1 active party and 1 passive party only, whose input +spaces are Xa and Xp respectively. The training objective +under this setting can be written as: +L = ℓ( ˆY , Y ) = ℓ(S(Ha, Hp), Y ) +(2) +where Ha, Hp are the intermediate local outputs of active +party and passive party respectively and ˆY denotes the pre- +dicted labels. Multi-party scenario can be easily extended +and will be studied in the following sections. +3.2. Attacks +Label Inference Attacks. In label inference attacks, pas- +sive parties try to steal the private labels from the active +party. Multiple routes can be taken to complete these attacks: +Model Completion attack (MC) [11] infers label by complet- +ing the local model with an additional layer and fine-tuning +the whole model using auxiliary labeled data. Depending +on whether the attacker updates its local model actively to +infer more information, MC attack can be separated into +active MC attack (AMC) and passive MC attack (PMC). +Sample-level Label Inference attack (SLI) [11,21] assumes +sample-level gradient information is exposed to the attacker. +Direct Label Inference attack (DLI) [11,21] exploits the fact +that sample-level gradient +∂ℓ +∂Hp +i exhibits a different sign value +on the label position when a global softmax function S is ap- +plied. Assuming the gradient of one random positive sample +is known, Direction Scoring attack (DS) [21] exploits the +cosine similarity between each gradient pairs to cluster each +sample into positive or negative class. Batch-level Label In- +ference attack (BLI) [46] assumes only the local batch-level +gradient is locally available, such as in the case of VFL with +HE-protection, and trains a neural network (NN) model to +invert label information from batch-level gradients. +Backdoor attacks. Depending on whether a separate +training target exhibits, backdoor attacks can be catego- +rized into targeted and non-targeted backdoor attacks. Tar- +geted Backdoor attack. Gradient Replacement Backdoor at- +tack [46] is a targeted backdoor where the attacker attempts +to assign a previously chosen target label τ to triggered sam- +ples. Non-targeted backdoor attacks include Noisy-sample +Backdoor attack which aims to harm the model utility by +adding random noise δxp +(n) to randomly chosen samples to +get noisy sample xp +i +′ and Missing Backdoor attack [23] in +which some Hp +i are randomly lost (set to 0), equivalent to +setting xp +i +′ = xp +(m) that satisfies HP ′ = Gp(xp +(m)) = 0, +through out training process. We summarize the backdoor +dataset Xp′ = {xp +i +′} +N +i=1 with: +xp +i +′ ≜ +� +� +� +� +� +� +� +� +� +xp +i + δxp +(t) +triggered sample i +xp +i + δxp +(n) +noisy sample i +xp +(m) +missing sample i +xp +i +others +3 + +(a) Active Party with MID +(b) Passive Party with MID +Figure 2. Demonstration of MID implementation in a 2-party VFL +system. Xp, Xa denotes local data sample set at passive and active +party separately. +and the false label set Y f = {yf +i } +N +i=1 with: +yf +i ≜ +� +� +� +τ +triggered sample i +˜yi ̸= yi +noisy/missing sample i +yi +others +Then, the training goal of a backdoor attacker is: +min +Θ Lb(Θ; D′) ≜ 1 +N +N +� +i=1 +ℓ(S(Ha +i , Hp +i +′), yf +i ) += 1 +N +N +� +i=1 +ℓ(S(Ga(xa +i ), Gp(xp +i +′)), yf +i ) +(3) +Feature Reconstruction Attacks. Parties in VFL can +also utilize its local data and knowledge to reconstruct private +local features belonging to other parties. CAFE [19] provides +a possible feature reconstruction method by inverting the +parties’ local models Gk using neural network under a white- +box VFL setting, which means that the active party has +knowledge of passive parties’ local models {Gk}K−1 +k=1 . +4. Mutual Information Regularization +4.1. Defense Against Label Inference Attacks +In order to prevent all the passive party’s attacks in +Sec. 3.2 , one possible way is for the active party to re- +duce the dependency of their local models on the label and +predicted label. Following works on Information Bottleneck +(IB) [2,33,34], we regard the neural network considering Xp +as a Markov chain Y −Xp −Hp −T −Z − ˆY , where Hp is +the original model output, T is a stochastic encoding layer, Z +is the new model output which aims to decode Y from T and +ˆY is the VFL model prediction. Following the Data Process- +ing Inequality (DPI) theory [4], I(Hp, ˆY ) ≤ I(Hp, T). To +minimize the mutual information (MI) between ˆY and Hp, +I( ˆY , Hp), following [36], we can replace I( ˆY , Hp) with its +upper bound I(Hp, T) and the training objective with: +min +T {−I(Y, T) + λI(Hp, T)} +(4) +Since I(Y, T) is maximized simultaneously as Eq. (2) is +minimized [5], we then combine Eqs. (2) and (4) to rewrite +the loss function for VFL training as the following: +L = ℓ( ˆY , Y ) + λ · I(Hp, T) += ℓ(S(Ha, Z), Y ) + λ · I(Hp, T), λ ≥ 0 +(5) +When minimizing L, I( ˆY , Y ) is maximized to guaranty the +model performance while I(Hp, T) is minimized to prevent +the passive party from inferring active party’s private label +information Y . If there exits more than one passive party, +the loss function can be generalized as: +min +Θ L(Θ; D) ≜ 1 +N +N +� +i=1 +ℓ(S(Z1 +i , . . . , ZK−1 +i +, HK +i ), yi) ++ +K−1 +� +k=1 +λkI(Hk, T k), λk ≥ 0 +(6) +Although the idea is straight forward, in reality, it is hard +to precisely calculate the mutual information I(Y, T) and +I(Hp, T). To overcome this difficulty, we follow the imple- +mentation of Variational Information Bottleneck (VIB) [2]. +The idea is to use parametric modeling to approximate the +calculation of those two mutual information value, with an +encoder to approximate I(Hp, T) and a decoder to approx- +imate I(Y, T). Reparameterization trick is also applied to +make the decoder derivable thus making the backward prop- +agation process possible. The process can be denoted as: +Z = MVIB(Hp) +(7) +where MVIB is the "encoder-decoder" structure with repa- +rameterization trick for derivable guarantee. MVIB first +transmits Hp to the bottleneck layer T which ignores as +much detail of Hp as possible but keeps sufficient infor- +mation about Y , and then decodes Y related information +from T and outputs Z as the decoded representation of Y . +Specifically, the encoder Me is to estimate the µ, σ for T +to achieve p(t|hp) = N(t|µ, σ2) which is needed in the cal- +culation of I(Hp, T) = +�� +p(hp, t) log +p(hp,t) +p(hp)p(t) dhpdt = +�� +p(hp, t) log p(t|hp) +p(t) +dhpdt. The stochastic attribute of T +lies in the random generation of T according to µ, σ, that +is T = µ + ϵ · σ, ϵ ∼ N(0, 1). And the decoder Md is +a variational approximation to p(y|t) which is needed in +the calculation of I(Y, T) = +�� +p(y, t) log +p(y,t) +p(y)p(t) dydt = +4 + +Passive +Gp +Party +Active +Party +Md +GaGp +M +e +Md +Passive +Party +Active +Party�� +p(y, t) log p(y|t) +p(y) dydt. See Fig. 2a for detailed demon- +stration. In Fig. 2, we use Me, Md to denote the encoder +and decoder inside MVIB, T is the output of reparameteri- +zation and Z is the output of MVIB, also is the local model +prediction under MID. +We provide a detailed training algorithm with MID pro- +tection in Algorithm 1. As this defense method is designed +from MI perception, we term it Mutual Information Regular- +ization Defense (MID). For MID, λ is the hyper-parameter +that controls the balance between information compression +of Hp in T and the representation ability of T according +to Y . A large λ indicates a high compression rate which +should result in a better defense ability but may harm the +VFL utility at the same time. When λ = 0.0, no information +bottleneck regularization is applied but only Me and Md +are added as additional model layers to the VFL system since +their existence or absence is regardless of the value of λ. +4.2. Defense Against Backdoor Attacks +Lemma 4.1. When MID is applied, the goal of defend- +ing against backdoor attacks is to min |I(Y, T) − I(Y, T ′)| +where T, T ′ is the MID bottleneck representation for the +original and the modified local data sample. +Proof. As describe in Sec. 3.2, in targeted and non-targeted +backdoor attacks, the passive attacker aims to achieve Eq. (3), +making the prediction ˆy′ closer to yf +i rather than the sam- +ple’s original label yi. Let ˆY ′ = {ˆy′ +i}N +i=1 = S(Ha, Hp′), +then I( ˆY ′, Y f) ≥ I( ˆY ′, Y ) while I( ˆY , Y f) ≤ I( ˆY , Y ) +holds for true for ˆY = S(Ha, Hp). Therefore, to defend +against all these backdoor attacks, the goal is to minimize +the change in I( ˆY ′, Y ) compared to I( ˆY , Y ), that is to +min |I( ˆY ′, Y ) − I( ˆY , Y )|. With Hp′ converted to T ′ in +MID, this is equivalent to +min |I(Y, T) − I(Y, T ′)| +(8) +Theorem 4.1. The performance gap |I(Y, T) − I(Y, T ′)| is +bounded by the following: +|I(Y, T) − I(Y, T ′)| ≤ B1|T |1/2(I(Hp, T))1/2 ++ B2|T |3/4(I(Hp, T))1/4 ++ B3|T |1/2(I(Hp′, T ′))1/2 ++ B4|T |3/4(I(Hp′, T ′))1/4 + B0 +(9) +where B1 = B2 log +1 +B2 , B2 = +4√2 log 2 +minhp∈Hp{p(hp)}, B3 = +B4 log +1 +B4 , B4 = +4√2 log 2 +minhp′∈Hp′{p(hp′)}, B0 = log M and +M = supt∈T {M(t)} with M(t) being the number of adver- +sarial representation t′ ∈ T ′ = T that satisfies ||t−t′||2 ≤ ϵ +given any ϵ > 0. +Thus, according to Theorem 4.1 and Eq. (5), when the +active party applies MID, by improving model robustness, +backdoor attacks launched by the passive party is prevented. +4.3. Defense Against Feature Reconstruction At- +tacks +When the attacker’s target is to recover features, the +defending party can also utilize MID to protect its data +by adding MVIB behinds its original local model output +Hp, generating Zp = MVIB(Hp) to further decrease +I(Xp, Zp). With MID, the defender (passive party) is able +to defend against feature reconstruction attacks, even for at- +tacks that directly exploits local models such as CAFE [19]. +See Fig. 2b for its implementation. Note that, from the omni- +scient perspective, the whole model architecture is the same +whether the MID defense is implemented in the passive or +active party. A detailed training algorithm is provided in +Algorithm 2 in the appendix. +Theorem 4.2. When applying MID, passive party is able to +protect local private data Xp by minimizing I(Xp, Z). +Proof. In the case of MID implemented in the passive party, +the Markov chain Y − Xp − Hp − T − Z − ˆY can still +apply. As the reverse sequence of a Markov chain also +forms a Markov chain, according to DPI theory [4], we have +I(Xp, Z) ≤ I(Xp, T) ≤ I(Hp, T). Since I(Hp, T) is an +upper bound of I(Xp, Z), I(Xp, Z) is simultaneously min- +imized as Eq. (4) is achieved. So, the passive party can +still apply this objective function for its MID and obtains +a stochastic layer T containing all the available knowledge +about Y but only the minimum sufficient statistical knowl- +edge about Hp and Xp. +5. Experiments +5.1. Models and Datasets +We conduct our experiments on 3 different datasets: +MNIST, CIFAR10 and CIFAR100. In MNIST dataset [41], +each image sample is evenly split and assigned to each party +respectively. A 2-layer MLP model with a 32-neuron layer +as the hidden middle layer is used for each party’s local +model except in CAFE attack which adopts a Convolution- +MaxPool-Convolution-MaxPool model structure followed +by a 3-layer-FC model as each party’s local model follow- +ing the original work [19]. In CIFAR10 and CIFAR100 +dataset [20], each image sample is evenly split and assigned +to each party respectively. Resnet20 is used for each party’s +local model in model completion attacks (PMC and AMC) +to be consistent with the original work [11] and the same +model structure for MNIST dataset is applied for these 2 +datasets in CAFE. While for other attacks, Resnet18 is used. +Through out our experiments, all data from the 3 datasets +are used for multi-class classification tasks. We use the +5 + +training and testing dataset provided therein. For binary +classification tasks, we randomly select 2 classes and use the +belonging data to compose a balanced dataset. +As for the global prediction model S, a global soft- +max function is used at the active party, except for MC +attacks [11], DS attack and CAFE attack [19], which adopts +a 4-layer FC model, 1-layer FC model and 1-layer FC model +respectively with trainable parameters. Global trainable +model is not used for other attacks, namely DLI attack, BLI +attack, targeted and non-targeted backdoor attacks, in order +to guarantee a stronger attack performance. +5.2. Attacks +We test the effectiveness of MID on 9 kinds of attacks +designed for VFL systems, namely, Passive Model Comple- +tion attack (PMC) [11], Active Model Completion attack +(AMC) [11], Direct Label Inference attack (DLI) [11, 21], +Direction Scoring attack (DS) [21], Batch-level Label In- +ference attack (BLI) [46], Label Replacement Backdoor +attack [46], Noisy-sample attack [23], Missing attack [23] +and CAFE [19]. The first 5 attacks are label inference at- +tacks, the last attack is feature reconstruction attack, and the +rest are backdoor attacks. +For MC attacks, we use CIFAR10 dataset with 40 and +10 auxiliary labeled data and CIFAR100 dataset with 400 +and 100 auxiliary labeled data, which means each class of +CIFAR10 or CIFAR100 owns 4 or 1 auxiliary labeled data +belonging to that class. In BLI attack, we follow the im- +plementation detail in [46] which means batch size is set to +2048. For label replacement backdoor attack, 1% of data +samples are randomly selected and marked with trigger while +target label τ is also randomly chosen [46]. 1% of data sam- +ples are added with noise δxp +(n) ∼ N(0, 2) for noisy-sample +attack, while 25% of passive model outputs failed to get to +the active party, i.e. Hp +i +′ = 0, for missing attack. For CAFE, +we follow the CAFE implementation [19] with default hyper- +parameters and use a batch size of 40 with the number of +iterations for feature reconstruction set to 10000 for MNIST +and 20000 for CIFAR10 and CIFAR100. Notice that the first +FC layer, of which CAFE first recovers its output and input +before recovering the input data sample features, is selected +differently depending on whether MID is applied. If MID is +applied, the first FC layer is the one-layer MID decoder, also +the last layer. Otherwise, same as the original paper [19], +there are 2 more FC layers after the first FC layer. +5.3. Baseline Defense Methods +In our experiments, we evaluate MID with 3 general de- +fense method: Adding Noise with Gaussian distribution +(DP-G) or Laplace distribution (DP-L) and Gradient Spar- +sification (GS). We also evaluate DiscreteSGD (DG) [11] +against MC attacks and DLI attack, MARVELL [21] against +DS attack which is conducted under binary classification +(a) CIFAR10 PMC-40 +(b) CIFAR10 AMC-40 +(c) CIFAR10 PMC-10 +(d) CIFAR10 AMC-10 +(e) CIFAR100 PMC-100 +(f) CIFAR100 AMC-100 +Figure 3. Comparison of various kinds of defense methods on pas- +sive model completion attack (PMC) and active model completion +attack (AMC) using CIFAR10 and CIFAR100 datasets. The num- +ber after PMC and AMC is the number of total auxiliary labeled +data used in the experiment. +task, Confusional AutoEncoder (CAE) [46] against BLI at- +tack and RVFR [23] against backdoor attacks. +Adding Noise. A Gaussian or Laplacian noise with stan- +dard deviation ranging from 5e−5 to 1.0 is added to the +gradients after they are 2-norm clipped with 0.2. Gaussian +noise is also added to defend against data reconstruction +attack in which gradients are 2-norm clipped with 3 with +noise of standard deviation ranging from 0.1 to 10 added. +GS. [1] Various drop rate ranging from 50.0% to 99.9% is +evaluated in the experiment. DG. [11] Number of bins for +gradient quantification ranging from 3 to 24 is evaluated in +the experiments. MARVELL. [21] The power constraint +hyper-parameter ranging from 0.1 to 10 times the norm of +gradients is evaluated. CAE. [46] Following the original pa- +per, both encoder and decoder of CAE have the architecture +of 2-layer-FC. Hyper-parameter λ2 that controls the confu- +sion level ranging from 0.0 to 2.0 is evaluated. RVFR. [23] +We evaluate this defense method in backdoor attacks follow- +ing the default parameter setting of the original paper. Note +that, the forth server training stage in RVFR is inapplicable +under our VFL setting as no trainable global model exits. +6 + +0.14 +1e-4 +DP-L +3e-4 +GS +18 +0.12 +12 +accuracy +DG +0.5 +MID +0.6 +0.10 +w/o defense +0.75 +0.0 +0.9 +Label recovery +0.08 +1e-7 +0.06 +1e-3 +0.04 +0.02 +D +1e-5 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Main task accuracy0.7 +0.5 +DP-L +2e-4 1e-4 +GS +-0.6 +0.6 +DG +24 +MID +0.5 +w/o defense +18 +0.4 +0.0 +0.3 +1e-3 +2 +0.2 +0.S +0. +-1e-7 +le-2 +0.1 +6 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Main task accuracyDP-L +0.7 +0.75 +GS +0.5l +accuracy +0.6 +1e-4 +DG +0.6 +0.70 - +0.0 +MID +0.78 +0.80 +0.82 +18 +0.5 +w/o defense +Label recovery +0.4 +-1e-5 +0.3 +0.2 +1e-2 +0.75 +1e-? +0.1 +0.9 +0.1 +1e-4 +6 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Main task accuracy0.5 +DP-L +0.6 +GS +accuracy +DG +e +0.5 +24 +MID +18 +w/o defense +0.6 +Label recovery +0.4 +2e-4 +0.3 +0.75 +le-3 +0.0 +0.9 +0.2 +2 +-1e-7 +e. +1e-3 +0.1 +0.1)+-1e-4 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Main task accuracyDP-L +0.6 +2e- +GS +accuracy +24 +DG +0.6 +MID +18 +0.5 +0.0 +w/o defense +Label recovery +0.4 +1e-3 +0.3 +0.75 +1e-5 +0.2 +0.9 +12 +le-2 +0.1 +0.11e-4 +6 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Main task accuracyDP-L +0.10 +1e-4 +GS +18 +Label recovery accuracy +DG +0.5 +MID +0.08 +3e-4 +12 +w/o defense +0.6 +0.75 +0.06 +0.04 +6 +0.0 +1e-3 +0.9 +X1e-7 +0.02 +Te: +1e-4J +1e-3 e-5 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Main task accuracy(a) CIFAR10 DLI +(b) CIFAR100 DLI +(c) MNIST BLI +(d) CIFAR10 BLI +Figure 4. Comparison of various kinds of defense methods against +direct label inference attack (DLI) and batch-level label inference +attack (BLI) on 3 different datasets. +For our MID defense, we evaluate different hyper- +parameters λ ranging from 0.0 to 1e4. Note when λ = 0.0, +the MID defense is degraded to an encoder-decoder neural +network, which is still effective to defend certain gradient- +based attacks due to the modification of the local model +structure. Comparing it with experimental results of MID +with λ > 0, we can see the effectiveness of generating a vari- +ational information bottleneck rather than adding additional +model layers to the VFL system. +5.4. Evaluation Metrics +To evaluate different defense methods, we mainly put two +metrics in the same figure: attack success rate (y-axis) and +main task utility (x-axis). A defense method is considered +superior if the attack success rate is lower at the same level +of main task utility, thus appearing on the bottom right of the +figure. The definition for attack success rate varies slightly +for different tasks. For label inference attacks, we use the +ratio of the correctly recovered labels; for targeted backdoor +attack, we use backdoor accuracy, the ratio of triggered +backdoor samples that are predicted as target class; for non- +targeted backdoor attacks, we use the drop of main task +accuracy on attacked samples; for feature reconstruction +attack, we use Peak Signal-to-Noise Ratio (PSNR) that is +widely utilized for assessing the quality of images [19,45], +where a low PSNR value indicates a high ratio of noise and +a low success rate. +5.5. Defending Against Label Inference Attacks +Model Completion Attacks. We compare MID with 3 +other baseline methods following previous work [11, 46]: +(a) MNIST Targeted +(b) CIFAR10 Targeted +(c) MNIST Noisy-sample +(d) CIFAR100 Noisy-sample +(e) MNIST Missing +(f) CIFAR100 Missing +Figure 5. Comparison of various kinds of defense methods against +targeted backdoor attack, namely label replacement backdoor at- +tack, and non-targeted backdoor attacks including noisy-sample +backdoor attack and missing backdoor attack on 3 different datasets. +DP-L, GS and DG. The results are shown in Fig. 3 and Fig. 7 +in the appendix. From Figs. 3 and 7, we can see that all +methods exhibit a trade-off between attack accuracy (y-axis) +and main task accuracy (x-axis). Increasing defense strength +by increasing noise level, sparsification rate or regularization +hyper-parameter λ in MID will lead to lower attack accuracy +and main task accuracy. However, our MID defense outper- +forms all the other baseline methods with much lower attack +accuracy while maintaining a high main task accuracy over +a wide range of λ values. The experiments demonstrates the +effectiveness of MID defense in suppressing the information +of true label distribution Y contained in the local model Gp +and local model output Hp at passive party. Other defense +methods fail to limit the attack accuracy to the same level +when maintaining a similar main task accuracy. +Sample-level Label Inference Attacks. Direct label in- +ference attack (DLI) and direction scoring attack (DS) are +2 typical types of sample-level label inference attack. We +first evaluate MID with 3 other baseline methods, DP-L, GS +and DG, against DLI attack. Defense results are shown in +Figs. 4a and 4b. We can see that MID outperforms most +of the baseline methods. Since DS attack can only recover +7 + +1.0 +12.18 +0.6 +Label recovery accuracy +Te- +4 +0.8 +DP-L +0.6 +GS +DG +0.01e-3 +MID +0.4 +5e-2 +w/o defense +1e-2 +-1e-6 +0.2 +1e- +0.9 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Main task accuracy1.0 +1.05 +0.75 18 +0.5 +1.00 +Label recovery accuracy +0.9 +0.8 +0.6 +1e +0.95 +0.50 +0.52 +DP-L +1e-5 +0.6 +GS +DG +MID +0.4 +w/o defense +0.2 +1e-70.0 +1e-2 +Te-6 +0.0 +3e-3 +11e-4 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Main task accuracy100 +DP-G +1e-5 +DP-L +1e-4 +1e-5 +accuracy +80 +GS +CAE +1e-4 +MID +60 +Label recovery +w/o defense +1e-3 +40 +1e-3 +95 +97 +98 +96 +20 +1e-2 +99 +0.0 +0.1 +1e-2 1e-3 +e-6 +0.1 +1e-2 +0.1 +0.05 +0 +0.0 +84 +87 +90 +93 +96 +Main task accuracy100 +1e-5 +1e-5 +90 +accuracy +80 +DP-G +DP-L +60 +Label recovery +95 +GS +96 +CAE +40 +MID +97 +w/o defense +20 +1e-3 +1e-3 +0.1 +0.1 +1e-2 +0.5 +2.0 +0. +0.0 +68 +70 +72 +74 +76 +78 +80 +82 +Main task accuracy95.0 +99.0 +1e-3 +99.5 +99.9 +1e-3 +80 +Backdoor task accuracy +1e-2 +60 +w/o defense +40 +DP-G +D5e-2 +DP-L +5e-2 +20 +GS +0.1 +RVFR +0.1 +1e-6 +1e- +MID +0.0 +0 +0.1 +55 +60 +65 +70 +75 +80 +85 +90 +Main task accuracy1e-3 +le-4 +97.0 +95.0 +Te-4 +99.0 +1e-3 +80 +Backdoor task accuracy +1e-2 +1e-2 +60 +w/o defense +40 +DP-G +7.5 +1.0 +0.1 +DP-L +0.0 +5.0 +20 +1é-3 +0.1 +GS +1e-6 +2.5 +67 +68 +69 +RVFR +0.1 +MID +0 +20 +30 +40 +50 +60 +70 +Main task accuracy1e-2 +Noisy-sample main task difference +50 +DP-G +95.0 +e2 +DP-L +GS +99.0 +40 +RVFR +99.5 +MID +w/o defense +99.9 +5et2 +30 +0.Φ +0. +e- +20 +0.1 +10 +1.0 +1.0 +20 +40 +60 +80 +Main task accuracy35 +1e-4 +main task difference +DP-G +1e-3 +21e-4 +32.5 +DP-L +1e-3 +30 +30.0 +95.0 +GS +99.0 +41 +42 +1e-2 +RVFR +25 +MID +10.0 +20 +99.9 +w/o defense +11e-4 +15 +Noisy-sample r +1e-2 +10 +↑e-3 +5 +11.0 +0.1 +0 +0.1 +0 +10 +20 +30 +40 +Main task accuracy sample main task difference +17.5 +99.5 +DP-G +DP-L +99.0 +15.0 +GS +1e-3 +M1e-3 +RVFR +99.9 +12.5 +MID +1e-2 +1e-2 +0.1 +10.0 +w/o defense +95.0 +0.1 +7.5 +0.0 +1e-3 +5.0 +2 +1e-4 +11.0 +Missing +2.5 +0 +88 +90 +1.0 +0.0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Main task accuracy95.0 +DP-G +10 +DP-L +99.5 +1e-B +1e-2 +GS +1e4 +990 +RVFR +1e-2 +Te-3 +8 +99.9 +MID +w/o defense +6 +4 +Te +0.5 +0 +2 +0.0 +1e-3 +0.1 +35 +36 +0 +0.1 +0 +5 +10 +15 +20 +25 +30 +35 +40 +Main task accuracy(a) MNIST +(b) CIFAR10 +(c) CIFAR100 +Figure 6. Effectiveness of MID against CAFE at various λ. +binary label, aside from DP-G, DP-L and GS, we also com- +pare MID with MARVELL, which is specifically designed +for defending DS [21] attack. Results are shown in Fig. 8 in +the appendix. We can see that MID results in the same level +of attack accuracy compared with DP-G, DP-L and GS with +a slightly lower main task accuracy than MARVELL. +Batch-level Label Inference Attacks. We evaluate MID +and other defending methods against BLI attack with MNIST +and CIFAR10 dataset, and results are shown in Figs. 4c +and 4d. It’s clear that, MID performs better than DP-G, DP- +L and GS, the 3 commonly used defending methods under +VFL scenario, with a much lower attack accuracy while +maintaining the same level main task accuracy. Notice that +CAE, a specific defense method designed for BLI attack in +which real labels are disguised with soft fake labels, achieves +the same level of main task accuracy with a slightly lower +attack accuracy compared to MID. +5.6. Defending Against Backdoor Attacks +The results for backdoor attacks are shown in Fig. 5. +From Figs. 5a and 5b, we can see that MID is the most +effective defense method among all the 5 defending meth- +ods we evaluated (MID, DP-G, DP-L, GS and RVFR), as +it achieves a much lower backdoor success rate at a high +main task accuracy for targeted backdoor attack. For non- +targeted backdoor attacks, MID is also the most effective +defending method, especially for missing attack (see Figs. 5e +and 5f). For noisy sample attack, RVFR is slightly better +than or comparable to MID (Figs. 5c and 5d). Notice that the +point with λ = 0.0 appears closer to the bottom of Figs. 5a, +5b, 5e and 5f, due to the fact that targeted backdoor attack +and missing attack are more vulnerable to the changes in the +model settings, i.e., when MVIB is added, resulting in a low +attack accuracy even without information regularization. +5.7. Defending Against Feature Reconstruction At- +tack +As the attacker is the active party under this setting, MID +is applied at the passive party like shown in Fig. 2b. +Defense Method +CIFAR10 +CIFAR100 +PSNR +Value +Main +ACC +PSNR +Value +Main +ACC +No defense +21.4417 +0.6015 +20.5476 +0.3296 +MID, λ = 0.0 +20.2628 +0.5956 +20.4584 +0.3281 +MID, λ = 1.0 +18.6929 +0.5920 +18.9796 +0.3235 +MID, λ = 100.0 +8.6667 +0.5881 +11.4972 +0.3213 +MID, λ = 10000.0 +6.1831 +0.5844 +6.1711 +0.3209 +DP-G, ϵ = 0.1 +7.1257 +0.2754 +6.6617 +0.0525 +Table 1. PSNR value for recovered data and main task accuracy of +CAFE for CIFAR10 and CIFAR100 datasets. +Results of reconstruction images are shown in Fig. 6 and +Tab. 1. More results are listed in Tab. 2 in the appendix. We +can see that CAFE successfully recovers original data using +gradients and local models. However MID and DP-G can +both successfully prevent the attacker from successfully re- +cover data features while MID can maintain a high main task +accuracy at the same time. With the increase of λ in MID, +the model becomes more robust against feature reconstruc- +tion attack since less information can be recovered within the +same number of iterations, both visually and quantitatively +indicated by a lower PSNR value, while the model utility is +just slightly harmed (see Tab. 1). Compared with DP-G, MID +can simultaneously achieve a lower PSNR value and a much +higher main task accuracy as shown in Tab. 1, indicating a +better defense ability against reconstruction attacks. +6. Conclusion +In this paper, we introduce a novel general defense +method MID which is able to defend against various kinds of +label inference attacks, backdoor attacks and feature recon- +struction attacks under VFL scenario. We provide theoretical +analysis and comprehensive experimental evaluations to tes- +tify the effectiveness of MID compared to existing defense +methods. We believe this work will shed light on future re- +search directions towards improving privacy and robustness +of VFL systems. +8 + +12345678gReferences +[1] Alham Fikri Aji and Kenneth Heafield. +Sparse commu- +nication for distributed gradient descent. +arXiv preprint +arXiv:1704.05021, 2017. +[2] Alexander A Alemi, Ian Fischer, Joshua V Dillon, and Kevin +Murphy. Deep variational information bottleneck. arXiv +preprint arXiv:1612.00410, 2016. +[3] Eugene Bagdasaryan, Andreas Veit, Yiqing Hua, Deborah +Estrin, and Vitaly Shmatikov. How to backdoor federated +learning. 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Algorithm of MID adopted by passive party +We describe how MID is applied at passive party in detail +in Algorithm 2. +Algorithm 2 A VFL framework with MID (at passive party) +Input: Learning rate η; MID hyper-parameter λ +Output: Model parameters θ1, θ2, . . . , θK +1: Party 1,2,. . . ,K, initialize θ1, θ2, ... θK; +2: for each iteration j=1,2, ... do +3: +Randomly sample S ⊂ [N]; +4: +for each passive party k (̸= K) in parallel do +5: +Computes {Hk +i }i∈S; +6: +Applies MID to generate Zk +i = MMID +k(Hk +i ); +7: +Sends {Zk +i }i∈S to party K; +8: +end for +9: +Active party K computes {HK +i }i∈S; +10: +Active party computes loss ℓ using Eq. (6); +11: +Active party sends { ∂ℓ +∂Zi }i∈S to all other parties; +12: +for each party k=1,2,...,K in parallel do +13: +Passive party k(̸= K) computes { ∂ℓ +∂Zi ← +∂ℓ +∂Zi + +∂I(Hk,Zk) +∂Zk +i +}i∈S and ∇kℓ = +∂ℓ +∂Zi +∂Zk +i +∂θk ; +14: +Active party K computes ∇Kℓ = +∂ℓ +∂Hi +∂HK +i +∂θK ; +15: +Each party updates θj+1 +k += θj +k − η∇kℓ; +16: +end for +17: end for +The main difference between this algorithm and Algo- +rithm 1 is that in this algorithm, MMID +k is now kept at +each passive party instead of the active party in Algorithm 1. +B. Proof for Theorem 4.1 +Proof. From the relation of mutual information to entropy +and conditional entropy, i.e. I(X, Y ) = H(X) − H(X|Y ), +we have: +|I(Y, T) − I(Y, T ′)| += |H(T) − H(T|Y ) − H(T ′) + H(T ′|Y )| += |[H(T) − H(T ′)] − [H(T|Y ) − H(T ′|Y )]| +≤ |H(T|Y ) − H(T ′|Y )| + |H(T) − H(T ′)| +In the following, we will show that |H(T|Y ) − H(T ′|Y )| +and |H(T) − H(T ′)| each has an upper bound. +Following Theorem 3.2 in [35], |H(T|Y ) − H(T ′|Y )| +has an upper bound: +|H(T|Y ) − H(T ′|Y )| ≤ B2 log 1 +B2 +|T |1/2(I(Hp, T))1/2 ++ B2|T |3/4(I(Hp, T))1/4 ++ B4 log 1 +B4 +|T |1/2(I(Hp′, T ′))1/2 ++ B4|T |3/4(I(Hp′, T ′))1/4 +(10) +This upper bound is symmetric to T and T ′ and is posi- +tively correlated to I(Hp, T) and I(Hp′, T ′) respectively. +If we define B1 ≜ B2 log +1 +B2 and B3 ≜ B4 log +1 +B4 , then +Eq. (10) has the form that is the same to the first 4 items of +the right side of Eq. (9). Notice that, the four coefficients +B1, B2, B3, B4 and |T |, the size of the finite set of possible +values of T, are all independent of Hp and T. +Moreover, |H(T) − H(T ′)| can be bounded with a con- +stant value. If t ∈ T , t′ ∈ T ′ satisfy ||t − t′||2 ≤ ϵ, then we +refer to t′ as an ϵ-bounded modified representation of t. If we +denote the number of the ϵ-bounded modified representation +t′ around t as M(t), then following Equation (82) in [35], +we have: +|H(T) − H(T ′)| ≤ | +� +t∈T +p(t) log M(t)| +≤ | +� +t∈T +p(t) log M| += | log M| +(11) +where M = supt∈T M(t). This means, |H(T) − H(T ′)| +can be bounded by a value independent to Hp, T and ϵ. +Summing up Eqs. (10) and (11), we can get Eq. (9). And +Eq. (9) shows that Eq. (8) can be achieved by achieving +min I(Hp, T). +C. Additional Experimental Results +C.1. Defending Against Label Inference Attacks +The results of various defending methods against model +completion attacks, including passive model completion at- +tack (PCM) and active model completion attack (ACM), on +CIFAR100 dataset with 400 auxiliary labeled data are pre- +sented in Fig. 7. It’s clear to see from the figure that MID +performs better than the 3 other baseline methods, including +DP-G, DP-L and GS, since a lower recovery accuracy is +achieved at the same level of main task utility. +For direction scoring attack (DS), the defense results are +shown in Fig. 8. We can see from Figs. 8a and 8b that all +the defense methods can reduce the attack accuracy to a low +1 + +(a) CIFAR100 PMC-400 +(b) CIFAR100 AMC-400 +Figure 7. Comparison of various kinds of defense methods on +passive and active model completion attack (PMC, AMC) using +CIFAR100 dataset. The number after PMC and AMC is the number +of total auxiliary labeled data used in the experiment. +(a) CIFAR10 DS +(b) CIFAR100 DS +Figure 8. Comparison of various kinds of defense methods against +direction scoring attack (DS) on CIFAR10 and CIFAR100 datasets. +level with comparable main task accuracy while MARVELL +achieves a slightly higher main task accuracy. +C.2. Defending Against Backdoor Attacks +We also conduct targeted backdoor attack under 4-party +VFL setting. In this setting, the 3 passive parties cooperate +with each other by sharing the same target label τ and adding +local triggers to the same set of triggered samples to launch +a gradient replacement backdoor attack. We evaluate MID +with the same 4 other baseline defense mechanisms we use +in Sec. 5.6 and the results are presented in Fig. 9. From +this figure, we can see that MID can limit the backdoor +accuracy to a much lower level compared to other methods +(DP-G, DP-L and GS). Moreover, RVFR, a defense designed +for defending against backdoor attacks, achieves a similar +defense ability compared with MID. +C.3. Defending Against Feature Reconstruction At- +tack +We present more experimental results of MID and DP- +G against CAFE attack in Tab. 2. We evaluate MID with +hyper-parameter λ ranging from 0.0 to 10000.0 and DP-G +with noise of standard deviation ranging from 0.1 to 10.0 +following the original work [19]. Results for DP-G exhibit +very similar trend, consistent with the original work [19]. +For MID, we observe that as λ increases, the feature recon- +struction quality is worsened, but the main task accuracy is +(a) 4-party Backdoor MNIST +(b) 4-party Backdoor CIFAR10 +Figure 9. Comparison of various kinds of defense methods against +4-party targeted backdoor attack on MNIST dataset and CIFAR10 +dataset. +Defense Method +CIFAR10 +CIFAR100 +PSNR +Value +Main +ACC +PSNR +Value +Main +ACC +No defense +21.4417 +0.6015 +20.5476 +0.3296 +MID, λ = 0.0 +20.2628 +0.5956 +20.4584 +0.3281 +MID, λ = 0.1 +20.0116 +0.5944 +19.2968 +0.3249 +MID, λ = 1.0 +18.6929 +0.5920 +18.9796 +0.3235 +MID, λ = 10.0 +15.4265 +0.5908 +16.3231 +0.3230 +MID, λ = 100.0 +8.6667 +0.5881 +11.4972 +0.3213 +MID, λ = 1000.0 +7.1028 +0.5873 +6.9704 +0.3219 +MID, λ = 10000.0 +6.1831 +0.5844 +6.1711 +0.3209 +DP-G, ϵ = 0.1 +7.1257 +0.2754 +6.6617 +0.0525 +DP-G, ϵ = 1.0 +7.1126 +0.2742 +6.6578 +0.0497 +DP-G, ϵ = 10.0 +7.1174 +0.2722 +6.1279 +0.0489 +Table 2. PSNR value for recovered data and main task accuracy of +CAFE for CIFAR10 and CIFAR100 datasets. +not affected as much as defense with DP-G. +2 + +0.200 +e-4 +DP-L +18 +GS +12 +0.175 +accuracy +DG +MID +0.5 +0.150 +0.6 +w/o defense +3e-4 +0.125 +Label recovery +0.75 +0.100 +0.075 +0.0 +0.050 +Te-3 +0.9 +Y1e-7 +0.025 +1e-4 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Main task accuracy0.6 +DP-L +0.75 +0.25 +GS +e-4 +accuracy +DG +MID +12 +0.20 +w/o defense +0.9 +Label recovery +0.15 +0.0 +1e +1e-3 +0.10 +0.27 +0.5 +e-6 +0.26 +0.05 +0.25 +0.46 +0.48 +e- +0.00 1le-43 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Main task accuracy100 +DP-G +95 +DP-L +Label recovery accuracy +90 +GS +85 +MARVELL +MID +80 +w/o defense +75 +1e-4 +70 +65 +10 +60 +X +55 +0.0X +0.5 +5e- +Te: +0.5 +50 +99 +45 +93.6 +94.2 +94.8 +95.4 +96.0 +Main task accuracy100 +DP-G +95 +DP-L +Label recovery accuracy +90 +GS +85 +MARVELL +MID +80 +w/o defense +1e-4 +75 +70 +65 +1e-47 +60 +90 +55 +0.1 +50 +1e-3 95 +99 +98 +5e-21e-3 +45 +80 +82 +84 +86 +88 +90 +92 +Main task accuracy95.0 +100 +99.9 +99.5 99.0 +1.1e-2. 1e-3 +0.1 +0.11e-2 +Backdoor task accuracy +80 +w/o defense +DP-G +60 +DP-L +GS +40 +3.5 +RVFR +Te-! +MID +3.0 - +1e-3 +20 +2.5 +30.X5 31.00 31.2 +1.0 +0.1 +0 +0.5 +10 +15 +20 +25 +30 +35 +Main task accuracyw/o defense +1e-2 +DP-G +40 +Backdoor task accuracy +DP-L +GS +30 +RVFR +MID +99.5 +20 +95.0 +99.9 +99.0 +10 +1.02 +1é-6 +0.1 +0.1 +25 +30 +35 +40 +45 +50 +55 +60 +Main task accuracy \ No newline at end of file diff --git a/itAzT4oBgHgl3EQfM_uk/content/tmp_files/load_file.txt b/itAzT4oBgHgl3EQfM_uk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..51ba2793ead4fa57c5426e288a672dc34dada1d5 --- /dev/null +++ b/itAzT4oBgHgl3EQfM_uk/content/tmp_files/load_file.txt @@ -0,0 +1,1013 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf,len=1012 +page_content='Mutual Information Regularization for Vertical Federated Learning Tianyuan Zou Yang Liu Ya-Qin Zhang Institute for AI Industry Research, Tsinghua University Beijing, China zty22@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='cn, liuy03@air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='cn, zhangyaqin@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='cn Abstract Vertical Federated Learning (VFL) is widely utilized in real-world applications to enable collaborative learning while protecting data privacy and safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' However, previous works show that parties without labels (passive parties) in VFL can infer the sensitive label information owned by the party with labels (active party), or execute backdoor attacks to VFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Meanwhile, active party can also infer sensitive feature information from passive party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' All these pose new privacy and security challenges to VFL systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We pro- pose a new general defense method which limits the mutual information between private raw data, including both fea- tures and labels, and intermediate outputs to achieve a better trade-off between model utility and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We term this defense Mutual Information Regularization Defense (MID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We theoretically and experimentally testify the effectiveness of our MID method in defending existing attacks in VFL, in- cluding label inference attacks, backdoor attacks and feature reconstruction attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Introduction Federated Learning (FL) [29] was first proposed to train cross-device machine learning models and protect data pri- vacy simultaneously which can be also regarded as horizon- tal FL (HFL) [39] as data are partitioned horizontally in the database by sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Another kind of FL framework is verti- cal FL (VFL) [6,14,16,24,25,39] where data are partitioned by feature, which means each participant owns a portion of the data features of each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' This framework is consis- tent with several real-world situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For example, a bank and an E-commerce company each obtains some features of the same group of users and they collaboratively train a model for preference prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Similar to HFL, partic- ipants in VFL aim to collaboratively train a shared model on the premise of keeping their local private data safe by communicating privacy-preserving intermediate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 1, in basic VFL framework with 2 parties, lo- cal data and local model of each party are kept locally while (a) Label Inference Attacks [11,21,46] and Feature Reconstruction Attacks [19] (b) Targeted [46] and Non-targeted Backdoor [23] Attacks Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Demonstration of different attacks in 2-party VFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' local intermediate results and gradient information are trans- mitted between an active and a passive party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' To attack this basic framework, recent studies [43–46] have explored data reconstruction attacks by exploiting the intermediate results exchanged, as well as backdoor attacks by manipulating the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' As for data reconstruction attacks, both label inference attacks [11,21,46] and feature reconstruction at- tacks [19,28] have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' As for backdoor attacks, malicious passive parties can modify the shared model for their own purpose by adding a trigger to a few of the at- tacker’s local samples in a targeted backdoor attack [46], or hurt the overall model utility by adding noise or failing 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='01142v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='LG] 1 Jan 2023 Feature Reconstruction Model Inversion "horse" Gradient Intermediate Result Model Gradient Completion Inversion Auxiliary "horse" Labeled Data "horse" Label Inference Label Inference"horse" Intermediate Gradient Result Missing Non-targeted Backdoor Noisy sample argetec Triggered Backdoor Sample riggerto transmit intermediate results in non-targeted backdoor attacks [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We summarize these attacks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' To mitigate these threats, various defense methods can be applied, including Adding Noise [3,9,38], Gradient Sparsi- fication (GS) [22] and Discrete Gradients (DG) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' How- ever, these defense methods suffer from accuracy drop for the main task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' There are also specific defense methods for targeted scenarios, such as MARVELL [21] to defend label leakage in binary classification , Confusional AutoEncoder (CAE) [46] to defend label leakage by model inversion at- tacks, RVFR [23] to defend robustness-related attacks such as missing features and adversarial input attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' However these defense scenarios are task-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' In this work, we observe that the root cause for data at- tacks by either active or passive party lies in the fundamental dependency between the local model at a passive party and the label or local features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Therefore we proposed a new general defense method that aims to defend existing attacks from the perspective of information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Specifically, we design a Mutual Information Regularization Defense (MID) for restricting the level of information about local data contained in exchanged intermediate outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We per- form extensive experiments which demonstrate that MID is very effective in defending all kinds of data reconstruction attacks and backdoor attacks compared with existing defense methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Moreover, we provide theoretical guarantee for model robustness with MID under VFL scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' In summary, our contributions are: We propose a new general defense method for VFL, Mutual Information Regularization Defense (MID), which regularizes the information dependency between parties’ local sensitive data and exchanged intermediate outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We show theoretically that MID is effective in preventing information leakage from exposed inter- mediate outputs and improving model robustness to defend against backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We perform comprehensive experimental evaluations and show that with proper design of information bottle- neck, MID is a promising universal defense method that achieves better utility-privacy trade-off than other gen- eral defense methods for various feature reconstruction attacks, label inference attacks and backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Related Work Federated Learning (FL) [30,39,40] is a novel machine learning paradigm in which participants collaboratively train a machine learning model without centralizing each parties’ local data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' FL can be further categorized into horizontal federated learning (HFL) where data are partitioned by sam- ples, and vertical federated learning (VFL) where data are partitioned by features [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' VFL [6, 18, 26] is commonly used in real-world cross-silo applications in finance and ad- vertising [7,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Existing attacks to VFL protocols are either to recon- struct private data [11, 17, 21, 46] or to hurt model robust- ness [23, 23, 27, 31, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For data reconstruction attacks, the target of these attacks is either private labels or private features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Label inference attacks can be performed using sample-level gradients (SLI) [11,21], or batch-level gradi- ents (BLI) [46], or trained local models [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Reconstruction of private features also pose great threat to data safety of VFL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Most related works focus on simple models includ- ing logistic regression [15,28,37] and tree [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' While for neural networks (NN), recovering image data [19] or tabular data [28] can be done by model inversion under white-box setting, and for black-box setting, prior information about data is required [17] or the targeted features are limited to binary values [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' In addition, passive parties can launch backdoor attacks by assigning specific label to triggered sam- ples [46](targeted backdoor), or by adding noise to some randomly selected samples or by adding missing features to harm the model utility [13,23](non-targeted backdoor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For defense, cryptographic techniques like Homomor- phic Encryption (HE) or Secure Multi-Party Computation (MPC) [39] have been proposed to protect in-transit mes- sages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' However, since they do not protect learned results, VFL with such protections still opens doors to attacks that only exploit trained model results or malicious backdoor [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Some other general defense strategies focus on reducing in- formation by adding noise [9,11,21], Gradient Discretiza- tion [8,11], Gradient Sparsification [1] and Gradient Com- pression [22], or combined [11,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' These methods suffer from utility losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Other emerging defense methods tar- gets to specific attacks or scenarios, such as data augmenta- tion [12] or disguising labels [19,46] to defend against gra- dient inversion attacks, MARVELL [21] to defend against label inference in binary classification tasks, RVFR [23] to defend against backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Mutual Information has been explored as an effective regularization to machine learning models to improve the robustness of model against malicious attacks in the past [2,35,36] but has never been explored in VFL setting before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Problem Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Vertical Federated Learning Setting Under a typical VFL system, K data owners together ob- tain a dataset of N samples D = {xi, yi}N i=1 with each par- ticipant k holding a portion of the features Xk = {xk i }N i=1 and only one party controls the label information Y = {yi}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We refer this party as the active party and other parties as the passive parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Without loss of generality, we assume party K is the active party, and other parties are passive parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' In VFL, each party k adopts a local 2 model Gk with model parameters θk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Note that Gk can adopt various kinds of model, like logistic regression, tree, support vector machine, neural network, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' With the local model and data, each participant k calculates its local output Hk = {Hk i }N i=1 = {Gk(xk i , θk)}N i=1 = Gk(Xk, θk) and sends them to the active party for loss calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Therefore, the overall objective for VFL is formulated as: min Θ L(Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' D) ≜ 1 N N � i=1 ℓ(S(H1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' , HK i ), yi) (1) where Θ = [θ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' θK] are training parameters, S denotes a global model which can be either a prediction function or a model with trainable parameters, and ℓ denotes a loss function, such as a cross entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' To perform training with back propagation, active party performs gradient com- putation with received Hk and transmits back { ∂ℓ ∂Hi }N i=1 to each party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' See Algorithm 1 for a complete algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' To further protect transmitted sample-level information, cryptographic techniques such as Homomorphic Encryption (HE) can be applied [39] and gradient is calculated under en- cryption while a coordinator is introduced to the VFL system for distributing encryption keys and decryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Under HE- protected VFL, sample-level gradient information is protect while batch-level gradient information is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Algorithm 1 A VFL framework with and without MID (at active party) Input: Learning rate η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' MID hyper-parameter λ Output: Model parameters θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' , θK 1: Party 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' ,K, initialize θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' θK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 2: for each iteration j=1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' do 3: Randomly sample S ⊂ [N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 4: for each party k in parallel do 5: Computes {Hk i }i∈S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 6: Sends {Hk i }i∈S to party K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 7: end for 8: if MID is applied then 9: Active party computes Zk i = MVIB(Hk i ) and loss ℓ using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 10: Active party computes { ∂ℓ ∂Zk i }i∈S and updates MVIB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 11: else 12: Active party computes loss ℓ using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 13: end if 14: Active party sends { ∂ℓ ∂Hi }i∈S to all other parties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 15: for each party k=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=',K in parallel do 16: Computes ∇kℓ = ∂ℓ ∂Hi ∂Hk i ∂θk ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 17: Updates θj+1 k = θj k − η∇kℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 18: end for 19: end for To simplify our discussion, we first consider a VFL sys- tem with 1 active party and 1 passive party only, whose input spaces are Xa and Xp respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' The training objective under this setting can be written as: L = ℓ( ˆY , Y ) = ℓ(S(Ha, Hp), Y ) (2) where Ha, Hp are the intermediate local outputs of active party and passive party respectively and ˆY denotes the pre- dicted labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Multi-party scenario can be easily extended and will be studied in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Attacks Label Inference Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' In label inference attacks, pas- sive parties try to steal the private labels from the active party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Multiple routes can be taken to complete these attacks: Model Completion attack (MC) [11] infers label by complet- ing the local model with an additional layer and fine-tuning the whole model using auxiliary labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Depending on whether the attacker updates its local model actively to infer more information, MC attack can be separated into active MC attack (AMC) and passive MC attack (PMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Sample-level Label Inference attack (SLI) [11,21] assumes sample-level gradient information is exposed to the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Direct Label Inference attack (DLI) [11,21] exploits the fact that sample-level gradient ∂ℓ ∂Hp i exhibits a different sign value on the label position when a global softmax function S is ap- plied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Assuming the gradient of one random positive sample is known, Direction Scoring attack (DS) [21] exploits the cosine similarity between each gradient pairs to cluster each sample into positive or negative class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Batch-level Label In- ference attack (BLI) [46] assumes only the local batch-level gradient is locally available, such as in the case of VFL with HE-protection, and trains a neural network (NN) model to invert label information from batch-level gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Depending on whether a separate training target exhibits, backdoor attacks can be catego- rized into targeted and non-targeted backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Tar- geted Backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Gradient Replacement Backdoor at- tack [46] is a targeted backdoor where the attacker attempts to assign a previously chosen target label τ to triggered sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Non-targeted backdoor attacks include Noisy-sample Backdoor attack which aims to harm the model utility by adding random noise δxp (n) to randomly chosen samples to get noisy sample xp i ′ and Missing Backdoor attack [23] in which some Hp i are randomly lost (set to 0), equivalent to setting xp i ′ = xp (m) that satisfies HP ′ = Gp(xp (m)) = 0, through out training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We summarize the backdoor dataset Xp′ = {xp i ′} N i=1 with: xp i ′ ≜ � � � � � � � � � xp i + δxp (t) triggered sample i xp i + δxp (n) noisy sample i xp (m) missing sample i xp i others 3 (a) Active Party with MID (b) Passive Party with MID Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Demonstration of MID implementation in a 2-party VFL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Xp, Xa denotes local data sample set at passive and active party separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' and the false label set Y f = {yf i } N i=1 with: yf i ≜ � � � τ triggered sample i ˜yi ̸= yi noisy/missing sample i yi others Then, the training goal of a backdoor attacker is: min Θ Lb(Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' D′) ≜ 1 N N � i=1 ℓ(S(Ha i , Hp i ′), yf i ) = 1 N N � i=1 ℓ(S(Ga(xa i ), Gp(xp i ′)), yf i ) (3) Feature Reconstruction Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Parties in VFL can also utilize its local data and knowledge to reconstruct private local features belonging to other parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' CAFE [19] provides a possible feature reconstruction method by inverting the parties’ local models Gk using neural network under a white- box VFL setting, which means that the active party has knowledge of passive parties’ local models {Gk}K−1 k=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Mutual Information Regularization 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defense Against Label Inference Attacks In order to prevent all the passive party’s attacks in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 , one possible way is for the active party to re- duce the dependency of their local models on the label and predicted label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Following works on Information Bottleneck (IB) [2,33,34], we regard the neural network considering Xp as a Markov chain Y −Xp −Hp −T −Z − ˆY , where Hp is the original model output, T is a stochastic encoding layer, Z is the new model output which aims to decode Y from T and ˆY is the VFL model prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Following the Data Process- ing Inequality (DPI) theory [4], I(Hp, ˆY ) ≤ I(Hp, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' To minimize the mutual information (MI) between ˆY and Hp, I( ˆY , Hp), following [36], we can replace I( ˆY , Hp) with its upper bound I(Hp, T) and the training objective with: min T {−I(Y, T) + λI(Hp, T)} (4) Since I(Y, T) is maximized simultaneously as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (2) is minimized [5], we then combine Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (2) and (4) to rewrite the loss function for VFL training as the following: L = ℓ( ˆY , Y ) + λ · I(Hp, T) = ℓ(S(Ha, Z), Y ) + λ · I(Hp, T), λ ≥ 0 (5) When minimizing L, I( ˆY , Y ) is maximized to guaranty the model performance while I(Hp, T) is minimized to prevent the passive party from inferring active party’s private label information Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' If there exits more than one passive party, the loss function can be generalized as: min Θ L(Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' D) ≜ 1 N N � i=1 ℓ(S(Z1 i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' , ZK−1 i , HK i ), yi) + K−1 � k=1 λkI(Hk, T k), λk ≥ 0 (6) Although the idea is straight forward, in reality, it is hard to precisely calculate the mutual information I(Y, T) and I(Hp, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' To overcome this difficulty, we follow the imple- mentation of Variational Information Bottleneck (VIB) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' The idea is to use parametric modeling to approximate the calculation of those two mutual information value, with an encoder to approximate I(Hp, T) and a decoder to approx- imate I(Y, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Reparameterization trick is also applied to make the decoder derivable thus making the backward prop- agation process possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' The process can be denoted as: Z = MVIB(Hp) (7) where MVIB is the "encoder-decoder" structure with repa- rameterization trick for derivable guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' MVIB first transmits Hp to the bottleneck layer T which ignores as much detail of Hp as possible but keeps sufficient infor- mation about Y , and then decodes Y related information from T and outputs Z as the decoded representation of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Specifically, the encoder Me is to estimate the µ, σ for T to achieve p(t|hp) = N(t|µ, σ2) which is needed in the cal- culation of I(Hp, T) = �� p(hp, t) log p(hp,t) p(hp)p(t) dhpdt = �� p(hp, t) log p(t|hp) p(t) dhpdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' The stochastic attribute of T lies in the random generation of T according to µ, σ, that is T = µ + ϵ · σ, ϵ ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' And the decoder Md is a variational approximation to p(y|t) which is needed in the calculation of I(Y, T) = �� p(y, t) log p(y,t) p(y)p(t) dydt = 4 Passive Gp Party Active Party Md GaGp M e Md Passive Party Active Party�� p(y, t) log p(y|t) p(y) dydt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 2a for detailed demon- stration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 2, we use Me, Md to denote the encoder and decoder inside MVIB, T is the output of reparameteri- zation and Z is the output of MVIB, also is the local model prediction under MID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We provide a detailed training algorithm with MID pro- tection in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' As this defense method is designed from MI perception, we term it Mutual Information Regular- ization Defense (MID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For MID, λ is the hyper-parameter that controls the balance between information compression of Hp in T and the representation ability of T according to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' A large λ indicates a high compression rate which should result in a better defense ability but may harm the VFL utility at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' When λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0, no information bottleneck regularization is applied but only Me and Md are added as additional model layers to the VFL system since their existence or absence is regardless of the value of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defense Against Backdoor Attacks Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' When MID is applied, the goal of defend- ing against backdoor attacks is to min |I(Y, T) − I(Y, T ′)| where T, T ′ is the MID bottleneck representation for the original and the modified local data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' As describe in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2, in targeted and non-targeted backdoor attacks, the passive attacker aims to achieve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (3), making the prediction ˆy′ closer to yf i rather than the sam- ple’s original label yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Let ˆY ′ = {ˆy′ i}N i=1 = S(Ha, Hp′), then I( ˆY ′, Y f) ≥ I( ˆY ′, Y ) while I( ˆY , Y f) ≤ I( ˆY , Y ) holds for true for ˆY = S(Ha, Hp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Therefore, to defend against all these backdoor attacks, the goal is to minimize the change in I( ˆY ′, Y ) compared to I( ˆY , Y ), that is to min |I( ˆY ′, Y ) − I( ˆY , Y )|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' With Hp′ converted to T ′ in MID, this is equivalent to min |I(Y, T) − I(Y, T ′)| (8) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' The performance gap |I(Y, T) − I(Y, T ′)| is bounded by the following: |I(Y, T) − I(Y, T ′)| ≤ B1|T |1/2(I(Hp, T))1/2 + B2|T |3/4(I(Hp, T))1/4 + B3|T |1/2(I(Hp′, T ′))1/2 + B4|T |3/4(I(Hp′, T ′))1/4 + B0 (9) where B1 = B2 log 1 B2 , B2 = 4√2 log 2 minhp∈Hp{p(hp)}, B3 = B4 log 1 B4 , B4 = 4√2 log 2 minhp′∈Hp′{p(hp′)}, B0 = log M and M = supt∈T {M(t)} with M(t) being the number of adver- sarial representation t′ ∈ T ′ = T that satisfies ||t−t′||2 ≤ ϵ given any ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Thus, according to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (5), when the active party applies MID, by improving model robustness, backdoor attacks launched by the passive party is prevented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defense Against Feature Reconstruction At- tacks When the attacker’s target is to recover features, the defending party can also utilize MID to protect its data by adding MVIB behinds its original local model output Hp, generating Zp = MVIB(Hp) to further decrease I(Xp, Zp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' With MID, the defender (passive party) is able to defend against feature reconstruction attacks, even for at- tacks that directly exploits local models such as CAFE [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 2b for its implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Note that, from the omni- scient perspective, the whole model architecture is the same whether the MID defense is implemented in the passive or active party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' A detailed training algorithm is provided in Algorithm 2 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' When applying MID, passive party is able to protect local private data Xp by minimizing I(Xp, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' In the case of MID implemented in the passive party, the Markov chain Y − Xp − Hp − T − Z − ˆY can still apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' As the reverse sequence of a Markov chain also forms a Markov chain, according to DPI theory [4], we have I(Xp, Z) ≤ I(Xp, T) ≤ I(Hp, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Since I(Hp, T) is an upper bound of I(Xp, Z), I(Xp, Z) is simultaneously min- imized as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (4) is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' So, the passive party can still apply this objective function for its MID and obtains a stochastic layer T containing all the available knowledge about Y but only the minimum sufficient statistical knowl- edge about Hp and Xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Experiments 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Models and Datasets We conduct our experiments on 3 different datasets: MNIST, CIFAR10 and CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' In MNIST dataset [41], each image sample is evenly split and assigned to each party respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' A 2-layer MLP model with a 32-neuron layer as the hidden middle layer is used for each party’s local model except in CAFE attack which adopts a Convolution- MaxPool-Convolution-MaxPool model structure followed by a 3-layer-FC model as each party’s local model follow- ing the original work [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' In CIFAR10 and CIFAR100 dataset [20], each image sample is evenly split and assigned to each party respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Resnet20 is used for each party’s local model in model completion attacks (PMC and AMC) to be consistent with the original work [11] and the same model structure for MNIST dataset is applied for these 2 datasets in CAFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' While for other attacks, Resnet18 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Through out our experiments, all data from the 3 datasets are used for multi-class classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We use the 5 training and testing dataset provided therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For binary classification tasks, we randomly select 2 classes and use the belonging data to compose a balanced dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' As for the global prediction model S, a global soft- max function is used at the active party, except for MC attacks [11], DS attack and CAFE attack [19], which adopts a 4-layer FC model, 1-layer FC model and 1-layer FC model respectively with trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Global trainable model is not used for other attacks, namely DLI attack, BLI attack, targeted and non-targeted backdoor attacks, in order to guarantee a stronger attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Attacks We test the effectiveness of MID on 9 kinds of attacks designed for VFL systems, namely, Passive Model Comple- tion attack (PMC) [11], Active Model Completion attack (AMC) [11], Direct Label Inference attack (DLI) [11, 21], Direction Scoring attack (DS) [21], Batch-level Label In- ference attack (BLI) [46], Label Replacement Backdoor attack [46], Noisy-sample attack [23], Missing attack [23] and CAFE [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' The first 5 attacks are label inference at- tacks, the last attack is feature reconstruction attack, and the rest are backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For MC attacks, we use CIFAR10 dataset with 40 and 10 auxiliary labeled data and CIFAR100 dataset with 400 and 100 auxiliary labeled data, which means each class of CIFAR10 or CIFAR100 owns 4 or 1 auxiliary labeled data belonging to that class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' In BLI attack, we follow the im- plementation detail in [46] which means batch size is set to 2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For label replacement backdoor attack, 1% of data samples are randomly selected and marked with trigger while target label τ is also randomly chosen [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 1% of data sam- ples are added with noise δxp (n) ∼ N(0, 2) for noisy-sample attack, while 25% of passive model outputs failed to get to the active party, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Hp i ′ = 0, for missing attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For CAFE, we follow the CAFE implementation [19] with default hyper- parameters and use a batch size of 40 with the number of iterations for feature reconstruction set to 10000 for MNIST and 20000 for CIFAR10 and CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Notice that the first FC layer, of which CAFE first recovers its output and input before recovering the input data sample features, is selected differently depending on whether MID is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' If MID is applied, the first FC layer is the one-layer MID decoder, also the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Otherwise, same as the original paper [19], there are 2 more FC layers after the first FC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Baseline Defense Methods In our experiments, we evaluate MID with 3 general de- fense method: Adding Noise with Gaussian distribution (DP-G) or Laplace distribution (DP-L) and Gradient Spar- sification (GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We also evaluate DiscreteSGD (DG) [11] against MC attacks and DLI attack, MARVELL [21] against DS attack which is conducted under binary classification (a) CIFAR10 PMC-40 (b) CIFAR10 AMC-40 (c) CIFAR10 PMC-10 (d) CIFAR10 AMC-10 (e) CIFAR100 PMC-100 (f) CIFAR100 AMC-100 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Comparison of various kinds of defense methods on pas- sive model completion attack (PMC) and active model completion attack (AMC) using CIFAR10 and CIFAR100 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' The num- ber after PMC and AMC is the number of total auxiliary labeled data used in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' task, Confusional AutoEncoder (CAE) [46] against BLI at- tack and RVFR [23] against backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Adding Noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' A Gaussian or Laplacian noise with stan- dard deviation ranging from 5e−5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 is added to the gradients after they are 2-norm clipped with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Gaussian noise is also added to defend against data reconstruction attack in which gradients are 2-norm clipped with 3 with noise of standard deviation ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 to 10 added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' [1] Various drop rate ranging from 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0% to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9% is evaluated in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' [11] Number of bins for gradient quantification ranging from 3 to 24 is evaluated in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' MARVELL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' [21] The power constraint hyper-parameter ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 to 10 times the norm of gradients is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' CAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' [46] Following the original pa- per, both encoder and decoder of CAE have the architecture of 2-layer-FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Hyper-parameter λ2 that controls the confu- sion level ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' RVFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' [23] We evaluate this defense method in backdoor attacks follow- ing the default parameter setting of the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Note that, the forth server training stage in RVFR is inapplicable under our VFL setting as no trainable global model exits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='14 1e-4 DP-L 3e-4 GS 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='12 12 accuracy DG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 MID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='10 w/o defense 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 Label recovery 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='08 1e-7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='06 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='02 D 1e-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 Main task accuracy0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 DP-L 2e-4 1e-4 GS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 DG 24 MID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 w/o defense 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='8 Main task accuracyDP-L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='75 GS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5l accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 1e-4 DG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='70 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 MID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='82 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 w/o defense Label recovery 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 1e-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 1e-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='75 1e-?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 1e-4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='8 Main task accuracy0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 DP-L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 GS accuracy DG e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 24 MID 18 w/o defense 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 Label recovery 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 2e-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='75 le-3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 2e- GS accuracy 24 DG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 MID 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 w/o defense Label recovery 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='75 1e-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 12 le-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='11e-4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='8 Main task accuracyDP-L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='10 1e-4 GS 18 Label recovery accuracy DG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 MID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='08 3e-4 12 w/o defense 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='04 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 X1e-7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='02 Te: 1e-4J 1e-3 e-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 Main task accuracy(a) CIFAR10 DLI (b) CIFAR100 DLI (c) MNIST BLI (d) CIFAR10 BLI Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Comparison of various kinds of defense methods against direct label inference attack (DLI) and batch-level label inference attack (BLI) on 3 different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For our MID defense, we evaluate different hyper- parameters λ ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 to 1e4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Note when λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0, the MID defense is degraded to an encoder-decoder neural network, which is still effective to defend certain gradient- based attacks due to the modification of the local model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Comparing it with experimental results of MID with λ > 0, we can see the effectiveness of generating a vari- ational information bottleneck rather than adding additional model layers to the VFL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Evaluation Metrics To evaluate different defense methods, we mainly put two metrics in the same figure: attack success rate (y-axis) and main task utility (x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' A defense method is considered superior if the attack success rate is lower at the same level of main task utility, thus appearing on the bottom right of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' The definition for attack success rate varies slightly for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For label inference attacks, we use the ratio of the correctly recovered labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' for targeted backdoor attack, we use backdoor accuracy, the ratio of triggered backdoor samples that are predicted as target class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' for non- targeted backdoor attacks, we use the drop of main task accuracy on attacked samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' for feature reconstruction attack, we use Peak Signal-to-Noise Ratio (PSNR) that is widely utilized for assessing the quality of images [19,45], where a low PSNR value indicates a high ratio of noise and a low success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defending Against Label Inference Attacks Model Completion Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We compare MID with 3 other baseline methods following previous work [11, 46]: (a) MNIST Targeted (b) CIFAR10 Targeted (c) MNIST Noisy-sample (d) CIFAR100 Noisy-sample (e) MNIST Missing (f) CIFAR100 Missing Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Comparison of various kinds of defense methods against targeted backdoor attack, namely label replacement backdoor at- tack, and non-targeted backdoor attacks including noisy-sample backdoor attack and missing backdoor attack on 3 different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' DP-L, GS and DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 7 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' From Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 3 and 7, we can see that all methods exhibit a trade-off between attack accuracy (y-axis) and main task accuracy (x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Increasing defense strength by increasing noise level, sparsification rate or regularization hyper-parameter λ in MID will lead to lower attack accuracy and main task accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' However, our MID defense outper- forms all the other baseline methods with much lower attack accuracy while maintaining a high main task accuracy over a wide range of λ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' The experiments demonstrates the effectiveness of MID defense in suppressing the information of true label distribution Y contained in the local model Gp and local model output Hp at passive party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Other defense methods fail to limit the attack accuracy to the same level when maintaining a similar main task accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Sample-level Label Inference Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Direct label in- ference attack (DLI) and direction scoring attack (DS) are 2 typical types of sample-level label inference attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We first evaluate MID with 3 other baseline methods, DP-L, GS and DG, against DLI attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defense results are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 4a and 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We can see that MID outperforms most of the baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Since DS attack can only recover 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 Label recovery accuracy Te- 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='8 DP-L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 GS DG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='01e-3 MID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 5e-2 w/o defense 1e-2 1e-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 1e- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='8 Main task accuracy1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='75 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='00 Label recovery accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 1e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='52 DP-L 1e-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 GS DG MID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 w/o defense 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 1e-70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 1e-2 Te-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 3e-3 11e-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 Main task accuracy100 DP-G 1e-5 DP-L 1e-4 1e-5 accuracy 80 GS CAE 1e-4 MID 60 Label recovery w/o defense 1e-3 40 1e-3 95 97 98 96 20 1e-2 99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 1e-2 1e-3 e-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 1e-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 84 87 90 93 96 Main task accuracy100 1e-5 1e-5 90 accuracy 80 DP-G DP-L 60 Label recovery 95 GS 96 CAE 40 MID 97 w/o defense 20 1e-3 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 1e-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 68 70 72 74 76 78 80 82 Main task accuracy95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 1e-3 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 1e-3 80 Backdoor task accuracy 1e-2 60 w/o defense 40 DP-G D5e-2 DP-L 5e-2 20 GS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 RVFR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 1e-6 1e- MID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 55 60 65 70 75 80 85 90 Main task accuracy1e-3 le-4 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 Te-4 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 1e-3 80 Backdoor task accuracy 1e-2 1e-2 60 w/o defense 40 DP-G 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 DP-L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 20 1é-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 GS 1e-6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 67 68 69 RVFR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 MID 0 20 30 40 50 60 70 Main task accuracy1e-2 Noisy-sample main task difference 50 DP-G 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 e2 DP-L GS 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 40 RVFR 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 MID w/o defense 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 5et2 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='Φ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' e- 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 20 40 60 80 Main task accuracy35 1e-4 main task difference DP-G 1e-3 21e-4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 DP-L 1e-3 30 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 GS 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 41 42 1e-2 RVFR 25 MID 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 20 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 w/o defense 11e-4 15 Noisy-sample r 1e-2 10 ↑e-3 5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0 10 20 30 40 Main task accuracy sample main task difference 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 DP-G DP-L 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 GS 1e-3 M1e-3 RVFR 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 MID 1e-2 1e-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 w/o defense 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 1e-3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 2 1e-4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 Missing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 0 88 90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 10 20 30 40 50 60 70 80 90 Main task accuracy95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 DP-G 10 DP-L 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 1e-B 1e-2 GS 1e4 990 RVFR 1e-2 Te-3 8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 MID w/o defense 6 4 Te 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 35 36 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0 5 10 15 20 25 30 35 40 Main task accuracy(a) MNIST (b) CIFAR10 (c) CIFAR100 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Effectiveness of MID against CAFE at various λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' binary label, aside from DP-G, DP-L and GS, we also com- pare MID with MARVELL, which is specifically designed for defending DS [21] attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 8 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We can see that MID results in the same level of attack accuracy compared with DP-G, DP-L and GS with a slightly lower main task accuracy than MARVELL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Batch-level Label Inference Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We evaluate MID and other defending methods against BLI attack with MNIST and CIFAR10 dataset, and results are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 4c and 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' It’s clear that, MID performs better than DP-G, DP- L and GS, the 3 commonly used defending methods under VFL scenario, with a much lower attack accuracy while maintaining the same level main task accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Notice that CAE, a specific defense method designed for BLI attack in which real labels are disguised with soft fake labels, achieves the same level of main task accuracy with a slightly lower attack accuracy compared to MID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defending Against Backdoor Attacks The results for backdoor attacks are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' From Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5a and 5b, we can see that MID is the most effective defense method among all the 5 defending meth- ods we evaluated (MID, DP-G, DP-L, GS and RVFR), as it achieves a much lower backdoor success rate at a high main task accuracy for targeted backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For non- targeted backdoor attacks, MID is also the most effective defending method, especially for missing attack (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5e and 5f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For noisy sample attack, RVFR is slightly better than or comparable to MID (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5c and 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Notice that the point with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 appears closer to the bottom of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5a, 5b, 5e and 5f, due to the fact that targeted backdoor attack and missing attack are more vulnerable to the changes in the model settings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=', when MVIB is added, resulting in a low attack accuracy even without information regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defending Against Feature Reconstruction At- tack As the attacker is the active party under this setting, MID is applied at the passive party like shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defense Method CIFAR10 CIFAR100 PSNR Value Main ACC PSNR Value Main ACC No defense 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6015 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3296 MID, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5956 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4584 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3281 MID, λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6929 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5920 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3235 MID, λ = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5881 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3213 MID, λ = 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5844 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1711 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3209 DP-G, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2754 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6617 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0525 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' PSNR value for recovered data and main task accuracy of CAFE for CIFAR10 and CIFAR100 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Results of reconstruction images are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 6 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' More results are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 2 in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We can see that CAFE successfully recovers original data using gradients and local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' However MID and DP-G can both successfully prevent the attacker from successfully re- cover data features while MID can maintain a high main task accuracy at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' With the increase of λ in MID, the model becomes more robust against feature reconstruc- tion attack since less information can be recovered within the same number of iterations, both visually and quantitatively indicated by a lower PSNR value, while the model utility is just slightly harmed (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Compared with DP-G, MID can simultaneously achieve a lower PSNR value and a much higher main task accuracy as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 1, indicating a better defense ability against reconstruction attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Conclusion In this paper, we introduce a novel general defense method MID which is able to defend against various kinds of label inference attacks, backdoor attacks and feature recon- struction attacks under VFL scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We provide theoretical analysis and comprehensive experimental evaluations to tes- tify the effectiveness of MID compared to existing defense methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We believe this work will shed light on future re- search directions towards improving privacy and robustness of VFL systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 8 12345678gReferences [1] Alham Fikri Aji and Kenneth Heafield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Sparse commu- nication for distributed gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' arXiv preprint arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='05021, 2017.' metadata={'source': 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+page_content=' idlg: Im- proved deep leakage from gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' CoRR, abs/2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='02610, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' [45] Ligeng Zhu, Zhijian Liu, and Song Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Deep leakage from gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Advances in neural information processing systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' [46] Tianyuan Zou, Yang Liu, Yan Kang, Wenhan Liu, Yuanqin He, Zhihao Yi, Qiang Yang, and Ya-Qin Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defending batch-level label inference and replacement attacks in vertical federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' IEEE Transactions on Big Data, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 10 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Algorithm of MID adopted by passive party We describe how MID is applied at passive party in detail in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Algorithm 2 A VFL framework with MID (at passive party) Input: Learning rate η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' MID hyper-parameter λ Output: Model parameters θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' , θK 1: Party 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' ,K, initialize θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' θK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 2: for each iteration j=1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' do 3: Randomly sample S ⊂ [N];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 4: for each passive party k (̸= K) in parallel do 5: Computes {Hk i }i∈S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 6: Applies MID to generate Zk i = MMID k(Hk i );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 7: Sends {Zk i }i∈S to party K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 8: end for 9: Active party K computes {HK i }i∈S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 10: Active party computes loss ℓ using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 11: Active party sends { ∂ℓ ∂Zi }i∈S to all other parties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 12: for each party k=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=',K in parallel do 13: Passive party k(̸= K) computes { ∂ℓ ∂Zi ← ∂ℓ ∂Zi + ∂I(Hk,Zk) ∂Zk i }i∈S and ∇kℓ = ∂ℓ ∂Zi ∂Zk i ∂θk ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 14: Active party K computes ∇Kℓ = ∂ℓ ∂Hi ∂HK i ∂θK ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 15: Each party updates θj+1 k = θj k − η∇kℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 16: end for 17: end for The main difference between this algorithm and Algo- rithm 1 is that in this algorithm, MMID k is now kept at each passive party instead of the active party in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Proof for Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' From the relation of mutual information to entropy and conditional entropy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' I(X, Y ) = H(X) − H(X|Y ), we have: |I(Y, T) − I(Y, T ′)| = |H(T) − H(T|Y ) − H(T ′) + H(T ′|Y )| = |[H(T) − H(T ′)] − [H(T|Y ) − H(T ′|Y )]| ≤ |H(T|Y ) − H(T ′|Y )| + |H(T) − H(T ′)| In the following, we will show that |H(T|Y ) − H(T ′|Y )| and |H(T) − H(T ′)| each has an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Following Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 in [35], |H(T|Y ) − H(T ′|Y )| has an upper bound: |H(T|Y ) − H(T ′|Y )| ≤ B2 log 1 B2 |T |1/2(I(Hp, T))1/2 + B2|T |3/4(I(Hp, T))1/4 + B4 log 1 B4 |T |1/2(I(Hp′, T ′))1/2 + B4|T |3/4(I(Hp′, T ′))1/4 (10) This upper bound is symmetric to T and T ′ and is posi- tively correlated to I(Hp, T) and I(Hp′, T ′) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' If we define B1 ≜ B2 log 1 B2 and B3 ≜ B4 log 1 B4 , then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (10) has the form that is the same to the first 4 items of the right side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Notice that, the four coefficients B1, B2, B3, B4 and |T |, the size of the finite set of possible values of T, are all independent of Hp and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Moreover, |H(T) − H(T ′)| can be bounded with a con- stant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' If t ∈ T , t′ ∈ T ′ satisfy ||t − t′||2 ≤ ϵ, then we refer to t′ as an ϵ-bounded modified representation of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' If we denote the number of the ϵ-bounded modified representation t′ around t as M(t), then following Equation (82) in [35], we have: |H(T) − H(T ′)| ≤ | � t∈T p(t) log M(t)| ≤ | � t∈T p(t) log M| = | log M| (11) where M = supt∈T M(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' This means, |H(T) − H(T ′)| can be bounded by a value independent to Hp, T and ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Summing up Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (10) and (11), we can get Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' And Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (9) shows that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (8) can be achieved by achieving min I(Hp, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Additional Experimental Results C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defending Against Label Inference Attacks The results of various defending methods against model completion attacks, including passive model completion at- tack (PCM) and active model completion attack (ACM), on CIFAR100 dataset with 400 auxiliary labeled data are pre- sented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' It’s clear to see from the figure that MID performs better than the 3 other baseline methods, including DP-G, DP-L and GS, since a lower recovery accuracy is achieved at the same level of main task utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For direction scoring attack (DS), the defense results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We can see from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 8a and 8b that all the defense methods can reduce the attack accuracy to a low 1 (a) CIFAR100 PMC-400 (b) CIFAR100 AMC-400 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Comparison of various kinds of defense methods on passive and active model completion attack (PMC, AMC) using CIFAR100 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' The number after PMC and AMC is the number of total auxiliary labeled data used in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' (a) CIFAR10 DS (b) CIFAR100 DS Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Comparison of various kinds of defense methods against direction scoring attack (DS) on CIFAR10 and CIFAR100 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' level with comparable main task accuracy while MARVELL achieves a slightly higher main task accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defending Against Backdoor Attacks We also conduct targeted backdoor attack under 4-party VFL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' In this setting, the 3 passive parties cooperate with each other by sharing the same target label τ and adding local triggers to the same set of triggered samples to launch a gradient replacement backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We evaluate MID with the same 4 other baseline defense mechanisms we use in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 and the results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' From this figure, we can see that MID can limit the backdoor accuracy to a much lower level compared to other methods (DP-G, DP-L and GS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Moreover, RVFR, a defense designed for defending against backdoor attacks, achieves a similar defense ability compared with MID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defending Against Feature Reconstruction At- tack We present more experimental results of MID and DP- G against CAFE attack in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' We evaluate MID with hyper-parameter λ ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 to 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 and DP-G with noise of standard deviation ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 following the original work [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Results for DP-G exhibit very similar trend, consistent with the original work [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' For MID, we observe that as λ increases, the feature recon- struction quality is worsened, but the main task accuracy is (a) 4-party Backdoor MNIST (b) 4-party Backdoor CIFAR10 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Comparison of various kinds of defense methods against 4-party targeted backdoor attack on MNIST dataset and CIFAR10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' Defense Method CIFAR10 CIFAR100 PSNR Value Main ACC PSNR Value Main ACC No defense 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6015 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3296 MID, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5956 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4584 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3281 MID, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5944 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3249 MID, λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6929 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5920 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3235 MID, λ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5908 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3230 MID, λ = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5881 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3213 MID, λ = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5873 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9704 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3219 MID, λ = 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5844 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1711 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3209 DP-G, ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2754 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6617 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0525 DP-G, ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2742 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0497 DP-G, ϵ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2722 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0489 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' PSNR value for recovered data and main task accuracy of CAFE for CIFAR10 and CIFAR100 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' not affected as much as defense with DP-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='200 e-4 DP-L 18 GS 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='175 accuracy DG MID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 w/o defense 3e-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='125 Label recovery 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='050 Te-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 Y1e-7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='025 1e-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 Main task accuracy0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 DP-L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='25 GS e-4 accuracy DG MID 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='20 w/o defense 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 Label recovery 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 1e 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 e-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='48 e- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='00 1le-43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 Main task accuracy100 DP-G 95 DP-L Label recovery accuracy 90 GS 85 MARVELL MID 80 w/o defense 75 1e-4 70 65 10 60 X 55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 5e- Te: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 50 99 45 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='6 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='8 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 Main task accuracy100 DP-G 95 DP-L Label recovery accuracy 90 GS 85 MARVELL MID 80 w/o defense 1e-4 75 70 65 1e-47 60 90 55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 50 1e-3 95 99 98 5e-21e-3 45 80 82 84 86 88 90 92 Main task accuracy95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 100 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1e-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='11e-2 Backdoor task accuracy 80 w/o defense DP-G 60 DP-L GS 40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content='5 RVFR Te-!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itAzT4oBgHgl3EQfM_uk/content/2301.01142v1.pdf'} +page_content=' MID 3.' metadata={'source': 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index 0000000000000000000000000000000000000000..d0f361aaeb00a4d705d9793be9cd02e65ddbc5fb --- /dev/null +++ b/jNFJT4oBgHgl3EQfXSyN/content/tmp_files/2301.11521v1.pdf.txt @@ -0,0 +1,1409 @@ +arXiv:2301.11521v1 [cs.LO] 27 Jan 2023 +Stack-Aware Hyperproperties⋆ +Ali Bajwa2, Minjian Zhang1, Rohit Chadha2, and Mahesh Viswanathan1 +1 University of Illinois Urbana-Champaign, USA +2 University of Missouri in Columbia, USA +Abstract. A hyperproperty relates executions of a program and is used +to formalize security objectives such as confidentiality, non-interference, +privacy, and anonymity. Formally, a hyperproperty is a collection of al- +lowable sets of executions. A program violates a hyperproperty if the set +of its executions is not in the collection specified by the hyperproperty. +The logic HyperCTL* has been proposed in the literature to formally +specify and verify hyperproperties. The problem of checking whether +a finite-state program satisfies a HyperCTL* formula is known to be +decidable. However, the problem turns out to be undecidable for proce- +dural (recursive) programs. Surprisingly, we show that decidability can +be restored if we consider restricted classes of hyperproperties, namely +those that relate only those executions of a program which have the same +call-stack access pattern. We call such hyperproperties, stack-aware hy- +perproperties. Our decision procedure can be used as a proof method for +establishing security objectives such as noninference for recursive pro- +grams, and also for refuting security objectives such as observational +determinism. Further, if the call stack size is observable to the attacker, +the decision procedure provides exact verification. +Keywords: Hyperproperties · Temporal Logic · Recursive Programs · +Model Checking · Pushdown Systems · Visibly Pushdown Automata. +1 +Introduction +Temporal logics HyperLTL and HyperCTL* [5] were designed to express +and reason about security guarantees that are hyperproperties [6]. A hyper- +property [6] is a security guarantee that does not depend solely on individual +executions. Instead, a hyperproperty relates multiple executions. For example, +non-interference, a confidentiality property, states that any two executions of a +program that differ only in high-level security inputs must have the same low- +security observations. As pointed out in [6], several security guarantees are hy- +perproperties. The logic HyperCTL* subsumes HyperLTL, and the problem +of checking a finite-state system against a HyperCTL* formula is decidable [5]. +⋆ Ali Bajwa was partially supported by NSF CNS 1553548. Rohit Chadha was par- +tially supported by NSF CNS 1553548 and NSF SHF 1900924. Mahesh Viswanathan +and Minjian Zhang were partially supported by NSF SHF 1901069 and NSF SHF +2007428. + +2 +A. Bajwa et al. +In this paper, we consider the problem of model checking procedural (recur- +sive) programs against security hyperproperties. Recall recursive programs are +naturally modeled as a pushdown system. Unlike the case of finite-state tran- +sition systems, the problem of checking whether a pushdown system satisfies a +HyperCTL* formula is undecidable [16]. In contrast, CTL* model checking is +decidable for pushdown systems [3,18]. +Our contributions. We consider restricted classes of hyperproperties for re- +cursive programs, namely those that relate only those executions that have the +same call-stack access pattern. Intuitively, two executions have the same stack +access pattern if they access the call stack in the same manner at each step, i.e., +if in one execution there is a push (pop) at a point, then there is a push (pop) +at the same point in the other execution. Observe that if two executions have +the same stack access pattern, then their stack sizes are the same at all times. +We call such hyperproperties, stack-aware hyperproperties. +In order to specify stack-aware hyperproperties, we extend HyperCTL* to +sHCTL*. The logic sHCTL* has a two level syntax. At the first level, the +syntax is identical to HyperCTL* formulas, and is interpreted over executions +of the pushdown system with the same stack access pattern. At the top-level, +we quantify over different stack access patterns. The formula Eψ is true if for +some stack access pattern ρ of the system, the pushdown system restricted to +executions with stack access pattern ρ satisfies the HyperCTL* formula ψ. The +formula Aψ is true if for each stack access pattern ρ of the system, the pushdown +system restricted to executions with stack access pattern ρ satisfies the Hyper- +CTL* formula ψ. See Figure 1 on Page 8 for a side-by-side comparison of the +syntax for HyperCTL* and sHCTL*. HyperLTL is extended to sHLTL simi- +larly. Please note that sHCTL* subsumes sHLTL, and that sHCTL* (sHLTL) +coincides with HyperCTL* (HyperLTL) for finite state systems as all execu- +tions of finite state systems have the same stack access pattern. +We show that the model checking problem for sHCTL* is decidable. We +demonstrate three different ways this result can aid in verifying recursive pro- +grams. First, for security guarantees such as noninference [14], which are ex- +pressible in the ∀∃∗ fragment of HyperLTL, we can use the model checking +algorithm to establish that a recursive program satisfies the HyperLTL prop- +erty. Secondly, for the ∀∗ fragment of HyperLTL, the model checking algorithm +can be used to detect security flaws by establishing that a recursive program does +not satisfy security guarantees. Observational determinism [13,19] is an example +of such a property. Finally, when the attacker can observe stack access patterns +(or, equivalently, stack sizes), we can get exact verification for several proper- +ties. The assumption of the attacker observing stack access patterns holds, for +example, in the program counter security model [15] in which the attacker has +access to program counters at each step. As argued in [15], the program security +model is appropriate to capture control-flow side channels such as those arising +from timing behavior and/or disclosure of errors. +The decision procedure uses an automata-theoretic approach inspired by +the model checking algorithm for finite state systems and HyperCTL* given + +Stack-Aware Hyperproperties +3 +in [10]. Since stack-aware hyperproperties relate only executions with the same +stack access-pattern, a set of executions with the same stack access pattern +can be encoded as a word over a pushdown alphabet, 3 and the problem of +model checking a sHCTL* formula can be reduced to the problem of check- +ing emptiness of a non-deterministic visibly pushdown automaton (NVPA) over +infinite words [1]. The reduction of the model checking problem to the empti- +ness problem is based on a compositional construction of an automaton for each +sub-formula which accepts exactly the set of assignments to path variables that +satisfy the sub-formula. For this construction to be optimal, we carefully leverage +two equi-expressive classes of automata on infinite words, namely NVPAs and +1-way alternating jump automata (1-AJA) [4]. The model checking algorithm +for sHCTL* against procedural programs has a complexity that is very close to +the complexity of model checking finite state systems against HyperCTL*. If +g(k, n) denotes a tower of exponentials of height k, where the top most expo- +nent is poly(n), then for a formula with formula complexity r, 4 and a system +and formula whose size is bounded by n, our algorithm is in DTIME(g(⌈ r +2⌉, n)). +In comparison, model checking finite state systems against HyperCTL* is in +NSPACE(g(⌈ r +2⌉ − 1, n)). This slight difference in complexity is consistent with +checking other properties like invariants for finite state systems (NL) versus pro- +cedural programs (P). +We also prove that our model checking algorithm is optimal by proving a +matching lower bound. Our proof showing DTIME(g(⌈ r +2⌉, n)-hardness of the +model checking problem for formulas with (formula) complexity r, relies on re- +ducing the membership problem for g(⌈ r +2⌉ − 1, n) space bounded alternating +Turing machines (ATM) to the model checking problem. The reduction requires +identifying an encoding of computations of ATMs, which are trees, as strings +that can be guessed and generated by pushdown systems. The pushdown system +we construct for the model checking problem guesses potential computations +of the ATM, while the sHCTL* formula we construct checks if the guessed +computation is a valid accepting computation. +Related work. Clarkson and Schneider introduced hyperproperties [6] and +demonstrated their need to capture complex security properties. Temporal logics +HyperLTL and HyperCTL*, that describe hyperproperties, were introduced +by Clarkson et al. [5]. They also characterized the complexity of model checking +finite state transition systems against HyperCTL* specifications by a reduction +to the satisfiability problem of QPTL [17]. Subsequently, other model checking +algorithms for verifying finite state systems against HyperCTL* properties +have been proposed [10,7]. Tools that check satisfiability [8] and runtime verifi- +cation [9] for HyperLTL formulas have also been developed. Finkbeiner et al. +introduced the automata-theoretic approach to model checking HyperCTL* +for finite-state systems [10]. +3 A pushdown alphabet is an alphabet that is partitioned into three sets: a set of call +symbols, a set of internal symbols, and a set of return symbols. See Section 4.1. +4 Our definition of formula complexity is roughly double the usual notion of quantifier +alternation. For a precise definition, see Definition 4. + +4 +A. Bajwa et al. +The model checking problem for HyperLTL, and consequently Hyper- +CTL*, was shown to be undecidable for pushdown systems in [16]. For re- +stricted fragments of HyperLTL, Pommellet and Tayssir [16] introduced over- +approximations and under-approximations to establish/refute that a pushdown +system satisfies a HyperLTL formula in those fragments. Gutsfeld et al. intro- +duced stuttering Hµ, a linear time logic for checking asynchronous hyperprop- +erties for recursive programs in [12]. The authors present complexity results for +the model checking problem under an assumption of fairness, and a restriction of +well-alignment. While the restriction to paths with the same stack access pattern +is similar to the well-alignment restriction, we do not assume any fairness con- +dition to establish decidability. However, as sHCTL* is a branching time logic +and only considers synchronous hyperproperties, the two logics are not directly +comparable. It is also worth mentioning that the branching nature of sHCTL* +requires us to “copy” a potentially unbounded stack, from the most recently +quantified path variable, when assigning a path to the “current” quantified path +variable. In contrast, all path assignments in [12] start with an empty stack. +An extended abstract of this paper appears in the 29th International Con- +ference on Tools and Algorithms for the Construction and Analysis of Systems +(TACAS) [2]. +2 +Motivation +Clarkson and Schneider [6] argue that many important security guarantees are +expressible only as hyperproperties. We discuss two examples of security hyper- +properties, and the logics HyperLTL and HyperCTL* used to specify them. +Hyperproperties and temporal logics. We discuss two variants of non- +interference [11] that model confidentiality requirements. In non-interference, +the inputs of a system are partitioned into low-level input security variables and +high-level input security variables. The attacker is assumed to know the values of +low-level security inputs. During an execution, the attacker can observe parts of +the system configuration such as system outputs, or the memory usage. A system +satisfies non-interference if the attacker cannot deduce the values of high-level +inputs from the low-level observations. To formally specify the variants, we use +the logic HyperLTL [5], a fragment of the logic HyperCTL* [5]. The precise +syntax of HyperLTL and HyperCTL* is shown in Fig. 1. In the syntax, π is a +path variable and the formula aπ is true if the proposition a is true along the path +“π”. Intuitively, the formula ∃π. ψ is existential quantification over paths, and is +true if there is a path that can be assigned to π such that ψ is true. Universal +quantification (∀π. ψ), and other logical connectives such as conjunction (∧), +implication (→), equivalence (↔) and the temporal operators G and F can be +defined in the standard way. By having explicit path variables, HyperLTL and +HyperCTL* allow quantification over multiple paths simultaneously. +Example 1. The first variant, noninference [14], states that for each execution σ +of a program, there is another execution σ′ such that (a) σ′ is obtained from σ by + +Stack-Aware Hyperproperties +5 +replacing the high-level security inputs by a dummy input, and (b) σ and σ′ have +the same low-level observations. Noninference is a hyperliveness property [5,6]. +Let us assume that the low-level observations of a configuration are deter- +mined by the values of the propositions in L = {ℓ1, · · · ℓm}. As shown in [5], non- +inference is expressible by the HyperLTL formula: NI +def += ∀π. ∃π′.(G λπ′) ∧ π ≡L +π′. Here G λπ′ expresses that Globally (or in each configuration of the execution) +the high input of π′ is the dummy input λ, and π ≡L π′ def += G(∧ℓ∈L(ℓπ ↔ ℓπ′)) +expresses that π and π′ have the same low-level observations. +Example 2. The second variant, observational determinism [13,19], states that +any two executions that have the same low-level initial inputs, must have the +same low-level output observations. Observational determinism is a hypersafety +property [5,6], and is also expressible in HyperLTL using the formula [5]: OD def += +∀π. ∀π′.(π[0] ≡L,in π′[0]) → π ≡L,out π′. Here ≡L,in and ≡L,out express the fact +that π and π′ have the same low-security inputs and outputs respectively. +Procedural (recursive) programs and Stack-aware hyperproperties. +Pushdown systems model procedural programs that do not dynamically allo- +cate memory, and whose program variables take values in finite domains. Unlike +finite-state transition systems, the problem of checking whether a pushdown sys- +tem satisfies a HyperCTL* formula is undecidable [16]. However, we identify a +natural class of hyperproperties for which the model checking problem becomes +decidable. As we shall shortly see, this class of hyperproperties not only enjoys +decidability, but is also useful in reasoning about security hyperproperies such +as noninference and observational determinism. +We consider a restricted class of hyperproperties for recursive programs, +which relate only executions that access the call stack in the same manner, +i.e., push or pop at the same time. An execution of a pushdown system P is a +sequence of configurations (control state + stack) σ = c1c2 · · · , such that the +stacks of consecutive configurations ci and ci+1 differ only due to the possible +presence of an additional element at the top of the stack of either ci or ci+1. +For such a sequence, we can associate a sequence pr(σ) = o1o2 · · · such that +oi ∈ {call, int, ret} such that oi = call (ret respectively) if and only if the stack +in ci+1 has one more (less respectively) element than ci. The sequence pr(σ) is +said to be the stack access pattern of σ. Observe that the stack sizes of two +executions with the same stack access pattern evolve in a similar fashion. Thus, +equivalently, we can consider this class of hyperproperties to be the hyperprop- +erties that relate executions with identical memory usage. +To specify these hyperproperties, we propose the logic sHCTL* which ex- +tends HyperCTL*. sHCTL* has a two level syntax. At the innermost level, +the syntax is identical to that of HyperCTL* formulas, but is interpreted over +executions of the pushdown system with the same stack access pattern. At the +outer level, we quantify over different stack access patterns. Intuitively, the for- +mula Eψ is true if there is a stack access pattern ρ exhibited by the system such +that the set of executions with access pattern ρ satisfy the hyperproperty ψ. +The dual formula Aψ, defined as ¬E¬ψ, is true if for each stack access pattern + +6 +A. Bajwa et al. +ρ exhibited by the system, the set of all executions with stack access pattern ρ +satisfy ψ. The syntax of sHLTL is obtained from HyperLTL in a similar fash- +ion. Please see Fig. 1 on Page 8 for a side-by-side comparison of the syntax of +HyperCTL* (HyperLTL) and sHCTL* (sHLTL). Unlike HyperCTL*, we +show that the problem of checking sHCTL* is decidable for pushdown systems +(Theorem 3). Formal definitions of stack access patterns, syntax and semantics +of sHCTL* are in Section 3. +For the rest of the paper, hyperproperties expressible in sHCTL* will be +called stack-aware hyperproperties. Restricting to stack-aware hyperproperties is +useful in verifying security guarantees of recursive programs as discussed below. +Proving ∀∃∗ hyperproperties. The noninference property (Example 1) can +be expressed in HyperLTL as NI def += ∀π. ∃π.′(G λπ′) ∧ π ≡L π′. Consider the +sHLTL formula A(NI) obtained by putting an A in front NI. A pushdown sys- +tem satisfies A(NI) only if for each execution σ of the system, there is another +execution σ′ with the same stack access pattern as σ such that σ, σ′ together +satisfy (G λσ′) ∧ σ ≡L σ′. Thus, if the pushdown system satisfies the sHLTL +formula A(NI), then it also satisfies noninference. Thus, a decision procedure for +sHLTL can be used to prove that a recursive program satisfies noninference. +The above observation generalizes to HyperLTL formulas of the form ψ = +∀π.∃π1. . . . ∃πk.ψ′ — if a system satisfies the sHLTL formula Aψ then it must +also satisfy the HyperLTL formula ψ. Though the model checking problem +is undecidable for pushdown systems even when restricted to such HyperLTL +formulas, we gain decidability by restricting the search space for π, π1, . . . , πk. +Refuting ∀∗ hyperproperties. Observational determinism (Example 2) can +be written in HyperLTL as OD def += ∀π. ∀π′.(π[0] ≡L,in π′[0]) → π ≡L,out π′. +Consider the sHLTL formula A(OD). A pushdown system fails to satisfy the +sHLTL formula A(OD) only if there is a stack access pattern ρ and executions +σ1 and σ2 with stack access pattern ρ such that the pushdown system does not +satisfy (σ[0] ≡L,in σ′[0]) → σ ≡L,out σ′. +This observation generalizes to HyperLTL formulas of the form ψ = +∀π1. . . . ∀πk.ψ′ — if a pushdown system fails to satisfy the sHLTL formula +Aψ then it does not satisfy ψ. Even though model checking pushdown systems +against such restricted specifications is undecidable, our decision procedure can +be used to show that a recursive program does not meet such properties. +Exact verification when stack access pattern is observable. Often, it is +reasonable to assume that the attacker can observe the stack access pattern. For +example, in the program counter security model [15], the attacker has access to +the program counter transcript, i.e., the sequence of program counters during an +execution. Access to the program counter transcript implies that the attacker can +observe stack access pattern. The assumption that the program counter tran- +script is observable helps model control flow side channel attacks which include +timing attacks and error disclosure attacks [15]. sHCTL* can be used to verify +security guarantees in this security model. For example, the sHCTL* formula +A( NI) models noninference faithfully by introducing a unique proposition for + +Stack-Aware Hyperproperties +7 +each control state. Observational determinism can also be verified in this model +by suitably transforming the pushdown automaton. +Another scenario in which stack access patterns are observable is when the +attacker can observe the memory usage of a program in terms of stack size. +As observing stack size may lead to information leakage, stack size should be +considered a low-level observation. Since the stack size can be unbounded, it +cannot be modeled as a proposition. sHCTL*, however, can still be used to verify +security guarantees in this case. For example, A( NI) = A(∀π. ∃π.′(G λπ′) ∧ π ≡L +π′) faithfully models non-inference as semantics of sHCTL* forces π and π′ to +have the same call-stack size in addition to other low-level observations. Once +again, observational determinism can also be verified in this model by suitably +transforming the pushdown automaton. +3 +Stack-aware Hyper Computation Tree Logic (sHCTL*) +Stack-aware Hyper Computation Tree Logic (sHCTL*), and its sub-logic Stack- +aware Hyper Linear Temporal Logic (sHLTL) are formally presented. We begin +by establishing some conventions over strings. +Strings. A string/word w over a finite alphabet Σ is a sequence w = a0a1 · · · +of finite or infinitely many symbols from Σ, i.e., ai ∈ Σ for all i. The length +of a string w, denoted |w|, is the number of symbols appearing in it — if w = +a0a1 · · · an−1 is finite then |w| = n, and if w = a0a1 · · · is infinite then |w| = ω. +The unique string of length 0, the empty string, is denoted ε. For a string w = +a0a1 · · · ai · · · , w(i) = ai denotes the ith symbol, w[ : i] = a0a1 · · · ai−1 denotes +the prefix of length i, w[i : ] = aiai+1 · · · denotes the suffix of w starting at +position i, and w[i : j] = aiai+1 · · · aj−1 denotes the substring from position i +(included) to position j (not included). Thus w[0 : ] = w. By convention, when +i ≤ 0, we take w[ : i] = ε. Over Σ, the set of all finite strings is denoted Σ∗, and +the set of all infinite strings is denoted Σω. For a finite string u and a (finite or +infinite) string v, uv denotes the concatenation of u and v. +3.1 +Pushdown Systems +Pushdown systems naturally model for sequential recursive programs. Formally, +an AP-labeled pushdown system is a tuple P = (S, Γ, sin, ∆, L), where S is a +finite set of control states, Γ is a finite set of stack symbols, sin ∈ S is the initial +control state, L : S → 2AP is the labeling function, and ∆ is the transition +relation. The transition relation ∆ = ∆int ∪· ∆call ∪· ∆ret is the disjoint union of +internal transitions ∆int ⊆ S × S where the stack is unchanged, call transitions +∆call ⊆ S × (S × Γ) where a single symbol is pushed onto the stack, and return +transitions ∆ret ⊆ (S × Γ) × S where a single symbol is popped from the stack. +When AP is clear from the context, we simply refer to them as pushdown systems. +Transition System Semantics. We recall the standard semantics of a push- +down system as a transition system. Let us fix a pushdown system P = +(S, Γ, sin, ∆, L). A configuration c of P is a pair (s, α) where s ∈ S and α ∈ Γ ∗. + +8 +A. Bajwa et al. +a ∈ AP, π ∈ V +ψ ::= aπ | +¬ψ +| ψ ∨ ψ | Xψ +| ψ U ψ | ∃π. ψ +(a) HyperCTL* +θ ::= Eψ | ¬θ | θ ∨ θ +ψ ::= aπ | ¬ψ | ψ ∨ ψ | Xψ | ψ U ψ | ∃π. ψ +(b) sHCTL* +Fig. 1: BNF for HyperCTL* and sHCTL*. Let ∀ denote ¬∃¬ and A denote ¬E¬ψ. +HyperLTL is the set of HyperCTL* formulas Q1π1. · · · Qrπr.ψ where Qi ∈ {∃, ∀} +and ψ is quantifier-free. sHLTL is the set of sHCTL* formulas qϕ, where q ∈ {A, E} +and ϕ is in HyperLTL. +The set of all configurations of P will be denoted ConfP = S × Γ ∗. The labeled +transition system associated with P is �P� := (ConfP, cin, −→, AP, L) where +cin = (sin, ε) is the initial configuration, −→⊆ ConfP × ({call, ret, int} × S × +(Γ ∪{ε})×S)×ConfP is the transition relation, and L is the labeling function that +extends the labeling function of P to configurations as follows: L(s, α) = L(s). +The transition relation −→ is defined to capture the informal semantics of inter- +nal, call, and return transitions — for any α ∈ Γ ∗, (int) (s, α) +(int,s,ε,s′) +−−−−−−→ (s′, α) +iff (s, s′) ∈ ∆int; (call) (s, α) +(call,s,a,s′) +−−−−−−−→ (s′, aα) iff (s, (s′, a)) ∈ ∆call; and (ret) +(s, aα) +(ret,s,a,s′) +−−−−−−→ (s′, α) iff ((s, a), s′) ∈ ∆ret. +A path of �P� is an infinite sequence of configurations σ = c0, c1, . . . such that +for each i, ci +(o,s,a,s′) +−−−−−−→ ci+1 for some o ∈ {int, call, ret}, s, s′ ∈ S and a ∈ Γ ∪ {ε}. +The path σ is said to start in configuration c0 (the first configuration in the +sequence). We will use Paths(�P�, c) to denote the set of paths of �P� starting +in the configuration c and Paths(�P�) to denote all paths of �P�. +We conclude this section by introducing some notation on configurations. For +c = (s, α), its stack height is |α|, its control state is state(c) = s, and its top of +stack symbol is top(c) = a ∈ Γ if α = aα′ and is undefined if α = ε. +3.2 +Syntax of sHCTL* +Let us fix a set of atomic propositions AP, and a set of path variables, V. The BNF +grammar for sHCTL* formulas is given in Figure 1(b). In the BNF grammar, +a ∈ AP is an atomic proposition, π is a path variable, ψ is a cognate formula, and θ +is a sHCTL* formula. The syntax has two levels, with the inner level identical to +HyperCTL* formulas, while the outer level allows quantification over different +stack access patterns (see Section 3.3). Also, following [5,10], we assume that the +until operator U only occurs within the scope of a path quantifier. +Remark 1. We have chosen to not have A, the dual of E, and conjunction as +explicit logical operators to keep our exposition simple. This choice does makes +the automata constructions presented here less efficient for formulas involving + +Stack-Aware Hyperproperties +9 +conjunction. Adding them explicitly does not pose a technical challenge to our +setup and our automata constructions can be extended to handle them explicitly. +In addition, we will sometimes use other quantifiers and logical operators to write +formulas. Some standard examples include: θ1 ∧ θ2 = ¬(¬θ1 ∨ ¬θ2), where θi (i ∈ +{1, 2}) is either a sHCTL* or cognate formula; ∀π.ψ = ¬ ∃π. ¬ψ; F ψ = true U ψ, +where true = aπ ∨ ¬aπ; G ψ = ¬ F ¬ψ. +We call formulas of the form qψ (where q ∈ {A, E} and ψ is a cognate +formula) basic formulas. Observe that any sHCTL* formula is a Boolean com- +bination of basic formulas. A sHCTL* formula θ is a sentence if in each basic +sub-formula qψ, ψ is a sentence, i.e., every path variable appearing in ψ is +quantified. Without loss of generality, we assume that in any cognate formula ψ, +all bound variables in ψ are renamed to ensure that any path variable is quanti- +fied at most once. We will only consider sHCTL* sentences in this paper. The +logic sHLTL is the sub-logic of sHCTL* consisting of all formulas of the form +qQ1π1. · · · Qrπr.ψ where q ∈ {A, E}, Qi ∈ {∃, ∀} and ψ is quantifier free. +3.3 +Semantics of sHCTL* +The syntax of cognate formulas is identical to that HyperCTL* formulas. Their +semantics will be described in a similar manner, in a context where free path +variables in the formula are interpreted as executions of a system. However, we +will require that the interpretations of every path variable share a common stack +access pattern — hence the term cognate. Thus, before defining the semantics, +we will define what we mean by the stack access pattern of a path and a path +environment that assigns an interpretation to path variables. +For the rest of this section let us fix a pushdown system P = (S, Γ, sin, ∆, L). +A string w ∈ {call, int, ret}∗ is said to be well matched if either w = ε or w = +int or w = call u ret or w = uv, where u, v ∈ {call, int, ret}∗ are (recursively) +well matched. In a string ρ ∈ {call, int, ret}ω, ρ(i) is an unmatched return, if +ρ[ : i + 1] = w ret, where w is well matched. We are now ready to present the +definition of a stack access pattern. +Definition 1 (Stack access pattern). A string ρ ∈ {call, int, ret}ω is a stack +access pattern if the set {i ∈ N | ρ(i) is an unmatched return} is finite. +A path σ = c0c1c2 · · · ∈ Paths(�P�) is said to have a stack access pattern ρ = +o0o1 · · · (denoted pr(σ) = ρ) if for every i: (a) oi = call if and only if stack(ci+1) += top(ci+1) stack(ci), (b) oi = int if and only if stack(ci+1) = stack(ci), +and (c) oi = ret if and only if stack(ci) = top(ci) stack(ci+1). +We now present the definition of path environment that interprets the free +path variables in a cognate formula as paths of �P� such that they share a +common stack access pattern. This plays a key role in defining the semantics of +sHCTL*. For a set of path variables V, let V† be defined as the set V ∪· {†}. +Definition 2 (Path Environment). A path environment for pushdown sys- +tem P over variables V is function Π : V† → Paths(�P�) ∪{call, int, ret}ω such + +10 +A. Bajwa et al. +that Π(†) is a stack access pattern , and for every π ∈ V, Π(π) ∈ Paths(�P�) +with pr(Π(π)) = Π(†). When the pushdown system is clear from the context, we +will simply refer to it as a path environment over V. +When V = ∅, we additionally require that there is a path σ ∈ Paths(�P�, cin) +(where cin is the initial configuration of �P�) such that pr(σ) = Π(†). +We introduce some notation related to path environments. Let us fix a path +environment Π over variables V. Given a path σ ∈ Paths(�P�), Π[π �→ σ] denotes +the path environment over V ∪{π} such that Π[π �→ σ](π) = σ, and Π[π �→ +σ](π′) = Π(π′), for any π′ ∈ V† with π′ ̸= π. Finally, for i ∈ N, Π[i : ] denotes the +suffix path environment, where every variable is mapped to the suffix of the path +starting at position i. More formally, for every π′ ∈ V†, Π[i : ](π′) = Π(π′)[i : ]. +We now define when a pushdown system P satisfies a sHCTL* sentence θ, +denoted P |= θ. The definition of satisfaction of θ relies on a definition of satis- +faction for cognate formulas. To inductively to define the semantics of cognate +formulas, we will interpret free path variables using a path environment. Fi- +nally, as in HyperCTL*, it is important to track the most recently quantified +path variable because that influences the semantics of ∃π(·). Thus satisfaction of +cognate formulas takes the form P, Π, π′ |= ψ, where π′ is the most recently quan- +tified path variable, and Π is a path environment over the free variables of ψ. +Finally, by convention, we will take Paths(�P�, Π(†)(0)) to mean Paths(�P�, cin), +where cin is the initial configuration of �P� 5. Below, θ, θ1, and θ2 are sHCTL* +sentences, while ψ, ψ1, ψ2 are cognate formulas. +P |= ¬θ iff P ̸|= θ +P |= θ1 ∨ θ2 iff P |= θ1 or P |= θ2 +P |= Eψ iff for some path environment Π over ∅, P, Π, † |= ψ +P, Π, π′ |= aπ iff a ∈ L(Π(π)(0)) +P, Π, π′ |= ¬ψ iff P, Π, π′ ̸|= ψ +P, Π, π′ |= ψ1 ∨ ψ2 iff P, Π, π′ |= ψ1 or P, Π, π′ |= ψ2 +P, Π, π′ |= Xψ iff P, Π[1 : ], π′ |= ψ +P, Π, π′ |= ψ1 U ψ2 iff ∃i ≥ 0 : P, Π[i : ], π′ |= ψ2 and ∀j, 0 ≤ j < i, +P, Π[j : ], π′ |= ψ1 +P, Π, π′ |= ∃π. ψ iff ∃σ ∈ Paths(�P�, Π(π′)(0)) with pr(σ) = Π(†), +such that P, Π[π �→ σ], π |= ψ +4 +A Decision Procedure for sHCTL* +Given a pushdown system P and a sHCTL* sentence θ, we present an algorithm +that determines if P |= θ. Our approach is similar to the one in [10]. Given a finite +state transition system K and a HyperCTL* formula ϕ, Finkbeiner et. al. [10], +construct an alternating (finite state) Büchi automaton AK,ϕ, by induction on +ϕ, such that an input word σ is accepted by AK,ϕ if and only if σ is the encoding +5 The convention is needed because Π(†)(0) is not a configuration but an element of +the set {call, int, ret}. + +Stack-Aware Hyperproperties +11 +of a path environment Π such that K, Π |= ϕ. Determining if K |= ϕ then reduces +to checking if AK,ϕ accepts any string. +Extending these ideas to sHCTL* and pushdown systems, requires one to +answer two questions: (a) What is an encoding of path environments for cog- +nate formulas where path variables are mapped to sequences of configurations +(control state + stack)?; (b) Which automata models can capture the collection +of path environments satisfying a cognate formula with respect to a pushdown +system? We encode path environments for cognate formulas using strings over +a pushdown alphabet — pushdown tags on symbols adds structure that helps +encode sequences of configurations. And for automata, we consider automata +that process such strings and accept visibly pushdown languages. A natural gen- +eralization of the approach outlined in [10] would suggest the use of alternating +visibly pushdown automata (AVPA) on infinite strings [4]. However, using AV- +PAs results in an inefficient algorithm. To get a more efficient algorithm, we +instead rely on a careful use of nondeterministic visibly pushdown automata +(NVPA) [1] and 1-way alternating jump automata (1-AJA) [4]. The advantage +of using NVPA and 1-AJA can be seen in the case of existential quantification +(∃π.) which requires converting an alternating automaton to a nondeterministic +one [10]: Converting from 1-AJA to NVPA leads to exponential blowup while +converting AVPA to NVPA leads to a doubly exponential blowup [4]. +The rest of this section is organized as follows. We begin by introducing +the automata models on pushdown alphabets (Section 4.1). Next we present +our encoding of path environments, and finally our automata constructions that +establish the decidability result (Section 4.2). +4.1 +Automata on Pushdown Alphabets +A pushdown alphabet is a finite set Σ that is partitioned into three sets +Σcall ∪· Σint ∪· Σret, where Σcall is the set of call symbols, Σint is the set of inter- +nal symbols, and Σret is the set of return symbols. Automata models processing +strings over a pushdown alphabet are restricted to perform certain types of tran- +sitions based on whether the read symbol is a call, internal, or return symbol. +We introduce, informally, two such automata models next. Precise definition and +its semantics can be found in Appendix B and Appendix C. +Nondeterministic Visibly Pushdown Büchi Automata. A nondetermin- +istic visibly pushdown automaton (NVPA) [1] is like a pushdown system. It has +finitely many control states and uses an unbounded stack for storage. However, +unlike a pushdown system, it is an automaton that processes an infinite sequence +of input symbols from a pushdown alphabet Σ = Σcall ∪· Σint ∪· Σret. Transitions +are constrained to conform to pushdown alphabet — whenever a Σcall symbol +is read, a symbol onto the stack, whenever a Σret symbol is read, the top stack +symbol is popped, and whenever Σint symbol is read, the stack is unchanged. +1-way Alternating Jump Automata. Our second automaton model is 1- +way Alternating Parity Jump Automata (1-AJA) [4]. 1-AJA are computation- +ally equivalent to NVPAs (i.e., accept the same class of languages) but provide + +12 +A. Bajwa et al. +greater flexibility in describing algorithms. 1-AJAs are alternating automata, +which means that they can define acceptance based on multiple runs of the ma- +chine on an input word. Though they are finite state machines with no auxiliary +storage, their ability to spawn a computation thread that jumps to a future +portion of the input string on reading a symbol, allows them to have the same +computational power as a more conventional machine with storage (like NVPAs). +We present some useful properties of NVPA and 1-AJA. The two models are +equi-expressive with the size of automata constructed by the translation known. +Theorem 1 ([4]). For any NVPA N of size n, there is a 1-AJA AN of size +O(n2), such that L(AN) = L(N). Conversely, for any 1-AJA A of size n, there +is a NVPA NA of size 2O(n), such that L(NA) = L(A). Constructions can be +carried out in time proportional to the size of the resulting automaton. +Both 1-AJA and NVPAs are closed for language operations like complemen- +tation, union and prefixing. We also recall the following result. +Theorem 2. ([1]) For NVPAs, the emptiness problem is PTIME-complete. +4.2 +Algorithm for sHCTL* +Let us fix a pushdown system P = (S, Γ, sin, ∆, L) and a sHCTL* sentence θ. +Our goal is to decide if P |= θ. We will reduce this problem to checking the empti- +ness of multiple NVPAs (Theorem 2). Our approach is similar to [10] — for each +cognate sub-formula ψ (not necessarily sentence) of θ, we will compositionally +construct an automaton that accepts the path environments satisfying ψ. Path +environments will be encoded by strings over pushdown alphabets as follows. +For a path σ = c0c1c2 · · · of �P�, the trace of σ, denoted tr(σ), is the +(unique) sequence (o0, q0, a0, q1)(o1, q1, a1, q2) · · · such that for every i ∈ N, +ci +(oi,qi,ai,qi+1) +−−−−−−−−−→ ci+1 where oi ∈ {call, int, ret}, qi, qi+1 ∈ Q, and ai ∈ Γ ∪ {ε} 6. +While tr(σ) is uniquely determined by the path σ, the converse is not true +— different paths may have the same trace. To see this, consider the following +example. For configuration c and γ ∈ Γ ∗, let γ(c) denote the configuration +(state(c), stack(c)γ), i.e., the configuration with the same control state, but with +stack containing the symbols in γ at the bottom. Observe that, for any γ ∈ Γ ∗, +if σ = c0c1c2· is a path then so is γ(σ) = γ(c0)γ(c1)γ(c2) · · · . Additionally, +tr(σ) = tr(γ(σ)). Two paths σ1 and σ2 of �P� will be said to be equivalent if +tr(σ1) = tr(σ2) and will be denoted as σ1 ≃ σ2. Observe that equivalent paths +have the same stack access pattern , i.e. if σ1 ≃ σ2 then pr(σ1) = pr(σ2). The +semantics of sHCTL* doesn’t distinguish between equivalent paths. +6 Observe that even when σ is not a path in �P� (i.e., corresponds to an actual se- +quence of transitions of P), the trace of σ is uniquely defined as long as stacks of +successive configurations of σ can be obtained by leaving the stack unchanged, or +pushing/popping one symbol. + +Stack-Aware Hyperproperties +13 +Proposition 1. Let ϕ be a cognate formula with V as the set of free path vari- +ables. Let Π1 and Π2 be two path environments such that for every π ∈ V, +Π1(π) ≃ Π2(π). Then, P, Π1, π |= ϕ if and only if P, Π2, π |= ϕ. +The proof of Proposition 1 follows by induction on cognate formulas. Propo- +sition 1 establishes that the set of path environments satisfying a cognate for- +mula is a union of equivalence classes with respect to path equivalence. Thus, +instead of constructing automata that accept path environments, we will con- +struct automata that accept mappings from path variables to traces of paths. +For m ∈ N, let Σ[m] = Σ[m]call ∪· Σ[m]int ∪· Σ[m]ret be the pushdown alpha- +bet where Σ[m]call = {call} × Sm × Γ m, Σ[m]int = {int} × Sm × {ε}m, and +Σ[m]ret = {ret} × Sm × Γ m. Observe Σ[0] is (essentially) the set {int, call, ret}. +Definition 3 (Encoding Path Environments). Consider a set of m path +variables V = {π1, π2, . . . πm}. A string w ∈ Σ[m]ω where for any j ∈ N, w(j) = +(oj, (sj +1, sj +2, . . . sj +m), (aj +1, aj +2, . . . aj +m)) encodes all path environments Π such that +Π(†) = o0o1o2 · · · oj · · · +tr(Π(πi)) = (o0, s0 +i , a0 +i , s1 +i )(o1, s1 +i , a1 +i , s2 +i ) · · · +for any i ∈ {1, 2, . . .m}. The string encoding a path environment Π is denoted +as enc(Π) (= w, in this case). +Based on the definitions, the following observation about traces and encod- +ings can be concluded. +Proposition 2. For any path σ ∈ Paths(�P�) and i ∈ N, tr(σ[i : ]) = tr(σ)[i : ]. +For any path environment Π and i ∈ N, enc(Π[i : ]) = enc(Π)[i : ]. +The encoding of path environments as strings over Σ[m] (for an appropriate +value of m) is used in our decision procedure, which compositionally constructs +automata that accept path environments satisfying each cognate formula. The +size of our constructed automata, like in [10], will be tower of exponentials that +depends on the formula complexity of the cognate formula ϕ. +Definition 4 (Formula Complexity). The formula complexity of a sHCTL* +formula ϕ, denoted fc(ϕ), is inductively defined as follows. Let odd : N → N be the +function that maps a number n to the smallest odd number ≥ n, i.e., odd(n) = n +if n is odd and odd(n) = n + 1 if n is even. Similarly, even : N → N maps n +to the smallest even number ≥ n, i.e., even(n) = odd(n + 1) − 1. Below ψ1, ψ2 +denote cognate formulas, and θ1, θ2 denote sHCTL* sentences. +fc(aπ) = 0 +fc(¬ψ1) = even(fc(ψ1)) +fc(Xψ1) = fc(ψ1) +fc(ψ1 ∨ ψ2) = max(fc(ψ1), fc(ψ2)) +fc(ψ1 U ψ2) = even(max(fc(ψ1), fc(ψ2))) +fc(∃π. ψ1) = odd(fc(ψ1)) +fc(Eψ1) = odd(fc(ψ1)) +fc(¬θ1) = fc(θ1) +fc(θ1 ∨ θ2) = max(fc(θ1), fc(θ2)) +Observe the difference in the definition of fc(¬θ1) and fc(¬ψ1); for ¬θ1 there is +no change in formula complexity, while for ¬ψ1 we move to the next even level. + +14 +A. Bajwa et al. +Our main technical lemma is a compositional construction of an automaton +for cognate formulas ψ. Depending on the parity of fc(ψ), the automaton we +construct will either be a 1-AJA or a NVPA. Before presenting this lemma, we +define a function that is a tower of exponentials. For c, k, n ∈ N, the value gc(k, n) +is defined inductively on k as follows: gc(0, n) = cn log n, and gc(k + 1, n) = +2gc(k,n). We use gO(1)(k, n) to denote the family of functions {gc(k, n) | c ∈ N}. +Lemma 1. Consider pushdown system P = (S, Γ, sin, ∆, L) and sHCTL* sen- +tence θ. Let ψ be a cognate subformula of θ with free path variables in the set +V = {π1, . . . πm} for m ∈ N. We assume, without loss of generality, that the vari- +ables π1, . . . πm are in the order in which they are quantified in θ with πm being +the first free variable of ψ that will be quantified in the context θ. In addition, we +assume that the size of both ψ and P is bounded by n. There is an automaton +Aψ over pushdown alphabet Σ[m] such that for any path environment Π over V, +P, Π, πm |= ψ if and only if enc(Π) ∈ L(Aψ). 7 +The automaton Aψ is a NVPA if fc(ψ) is odd, and a 1-AJA if fc(ψ) is even. +The size of Aψ is at most gO(1)(⌈ fc(ψ) +2 ⌉, n)8. +Before presenting the proof of Lemma 1, we would like to highlight a subtlety +about its statement. The result guarantees that for valid path environments Π, +encoding enc(Π) is accepted by Aψ if and only if Π satisfies ψ. It says nothing +about path environments that are not valid. In particular, there may be functions +that map path variables to traces that do not correspond to actual paths of �P�, +but which are nonetheless accepted by Aψ. Notice, however, when ψ = ∃π. ψ1 is +a cognate sentence, a string over {call, int, ret} will, by conditions guaranteed in +Lemma 1, be accepted if and only if it corresponds to a stack access pattern of +a path from the initial state that satisfies ∃π. ψ1. +Proof (Sketch of Lemma 1). Our construction of Aψ will proceed inductively. +The type of automaton constructed will be consistent with the parity of fc(ψ), +i.e., an NVPA if fc(ϕ) is odd and a 1-AJA if fc(ψ) is even. We sketch the main +ideas here, with the full proof available in Appendix D. +For aπ, ¬ψ1, ψ1 ∨ ψ2, and Xψ1, the construction essentially proceeds by con- +verting Aψi (i ∈ {1, 2}) if needed, into the type (NVPA or 1-AJA) of the target +automaton using Theorem 1, and then using standard closure properties to com- +bine them to get the desired automaton. In case of ψ = ψ1 U ψ2, we first convert +(if needed) Aψi (i ∈ {1, 2}) into a 1-AJA. At each step, the automaton for ψ +will choose to either run Aψ2, or run Aψ1 and restart itself. Correctness relies +on the fact that our encoding for path environments satisfies Proposition 2. +The most interesting case is that of ψ = ∃π. ψ1. We will first convert (if +needed) the automaton for ψ1 into a NVPA A1. The automaton for ψ will +essentially guess the encoding of a path that is consistent with the transitions of +7 When m = 0, we take πm to be †. +8 When the size of the specification ψ is considered constant, the size of Aψ is at most +gO(1)(⌈ fc(ψ) +2 +⌉ − 1, n) + +Stack-Aware Hyperproperties +15 +P, and check if assigning the guessed path to variable π satisfies ψ1 by running +the automaton A1. The additional requirement we have is that the guessed path +start at the same configuration as the current configuration of the path assigned +to variable πm which introduces some subtle challenges. In order to be able to +guess a path, Aψ will keep track of P’s control state in its control state, and use +its stack to track P’s stack operations along the guessed path. Since the stacks +of all paths are synchronized, it makes it possible for Aψ to use its (single stack) +to track the stack of both P and the stack of A1. +⊓⊔ +Using Lemma 1, we can establish the main result of this section. +Theorem 3. Given a P = (S, Γ, sin, ∆, L) and a sHCTL* sentence θ, the prob- +lem of determining if P |= θ is in ∪cDTIME(gc(⌈ fc(θ) +2 ⌉, n)), where n is a bound +on the size of P and θ. +Proof. Recall that a sHCTL* sentence is a Boolean combination of formulas of +the form Eψ, where ψ is a cognate sentence. Results on whether P |= Eψ for +each such subformula can be combined to determine whether P |= θ. Given this, +the time to determine if P |= θ is at most the time to decide if P satisfies each +subformula of the form Eψ plus O(n) (to compute the Boolean combination of +these results). Next, recall that the construction in Lemma 1 ensures that for +a cognate sentence of the form ∃π. ψ, L(A∃π. ψ) consists exactly of strings in +{call, int, ret}ω that encode a path environment over ∅ that satisfy ∃π. ψ. +Consider a sHCTL* sentence Eψ. Let π be a path variable that does not +appear in the sentence ψ. Based on the semantics of sHCTL* the following +observation holds: P |= Eψ if and only if for some path environment Π over +∅, P, Π, † |= ∃π. ψ. Which is equivalent to saying that P |= Eψ if and only if +L(A∃π. ψ) ̸= ∅. Since fc(Eψ) = fc(∃π. ψ), and the emptiness problem of NVPA +can be decided in polynomial time (Theorem 2), our theorem follows. +⊓⊔ +5 +Lower Bound +In this section, we establish a lower bound for the problem of model checking +sHCTL* sentences against pushdown systems. Our proof establishes a hardness +result for the sHLTL sub-fragment of sHCTL*. Before presenting this lower +bound, we introduce the function hc(·, ·), which is another tower of exponentials, +inductively defined as follows: hc(0, n) = n, and hc(k + 1, n) = hc(k, n) · chc(k,n). +Theorem 4. Let P be a pushdown system and θ be a sHLTL sentence such that +the sizes of both P and θ is bounded by n and fc(θ) = 2k − 1 for some k ∈ N. +The problem of checking if P |= θ is DTIME(hc(k, n))-hard, for every c ∈ N. +Proof (Sketch). We sketch the main intuitions behind the proof. To highlight the +novelties of this proof, it is useful to recall how NSPACE(hc(k−1, n))-hardness for +HyperLTL model checking is proved [5]. The idea is to reduce the language of +a nondeterministic hc(k−1, n) space bounded machine M to the model checking + +16 +A. Bajwa et al. +problem by constructing a finite state transition system that guesses a run of +M, and a HyperLTL formula that checks if the path is a valid accepting run. +To get the stricter bound of DTIME(hc(k, n)), we use the fact that we are +checking pushdown systems. The stack of the pushdown system can be used +to guess a tree, as opposed to a simple trace. Therefore, we reduce a hc(k − +1, n) space bounded alternating Turing machine, instead of a nondeterministic +machine. Since ASPACE(f(n)) = DTIME(2O(f(n))) for f(n) ≥ log n, the theorem +will follow if the reduction succeeds. +Recall that a run of an alternating Turing machine M is a rooted, labeled tree, +where vertices are labeled by configurations of M in a manner that is consistent +with the transition function of M. To faithfully encode a tree as a sequence +of symbols, we record the DFS traversal of the tree, making explicit the stack +operations performed during such a traversal. Consider a labeled, rooted tree T +with root r whose label is ℓ(r) with T1 as a the left sub-tree and T2 as the right +sub-tree. The DFS traversal of T will push ℓ(r), traverse T1 recursively, pop ℓ(r), +push ℓ(r), traverse T2, and then pop ℓ(r). We will use such a DFS traversal to +guess and encode runs of M. Popping and pushing ℓ(r) between the traversals +of T1 and T2 may seem redundant. Why not simply do nothing between the +traversals of T1 and T2? For T to be a valid run of M, the configuration labeling +of the root of T2 must be the result of taking one step from ℓ(r). Such checks +will be encoded in our sHLTL sentence, and for that to be possible, we need +successive configurations of M to be consecutive in the string encoding. +To highlight some additional consistency checks, let us continue with our +example tree T from the previous paragraph. For a string to be a correct encoding +of T , it is necessary that the string pushed before the traversal of Ti (i ∈ {1, 2}) +be the same as the string popped after the traversal. This can be ensured by the +pushdown system by actually pushing and popping those symbols. In addition, +the string popped after T1’s traversal must be the same as the string pushed +before T2’s traversal. Neither the stack nor the finite control of the pushdown +system can be used to ensure this. Instead this must be checked by the sHLTL +sentence we construct. But the symbols while popping ℓ(r) will be in reverse +order of the symbols being pushed, and it is challenging to perform this check +in the formula. To overcome this, we push/pop the label and its reverse at the +same time. This ensures that if we want to check if a string pushed is the same +as a string that was just popped, then we can check for string equality, and this +check is easier to do using formulas in sHLTL. Additional checks to ensure that +the tree encodes a valid accepting run are performed by the sHLTL sentence +using ideas from [17]. Full details can be found in Appendix E. +⊓⊔ +6 +Conclusions +In this paper, we introduced a branching time temporal logic sHCTL* that can +be used to specify synchronous hyperproperties for recursive programs modeled +as pushdown systems. The primary difference from the standard branching time +logic HyperCTL* for synchronous hyperproperties is that sHCTL* considers + +Stack-Aware Hyperproperties +17 +a restricted class of hyperproperties, namely, those that relate only executions +that the same stack access pattern. We call such hyperproperties stack-aware +hyperproperties. We showed that the problem of model checking pushdown sys- +tems sHCTL* specifications is decidable, and characterized its complexity. We +also showed how this result can potentially be used to aid security verification. +References +1. 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IEEE Computer Society (2003) +A +Observational determinism when the call stack size is +visible to the attacker +Observational determinism [13,19] states that any two executions that have the +same low-level initial inputs must have the same low-level output observations. +Observational determinism is a hypersafety property [5,6]. As [5] shows, ob- +servational determinism is also expressible in HyperLTL using the formula: +OD def += ∀π. ∀π′.(π[0] ≡L,in π′[0]) → π ≡L,out π′. Here ≡L,in and ≡L,out express +the fact that π and π′ have the same low-security inputs and outputs respectively. +In order to see how we can verify observational determinism when call stack +sizes are observable, let us consider the simple case when the call stack size is the +only low-security observation. Let P be the pushdown automaton corresponding +to a program. We can assume by loss of generality that the states of P encode if +the stack is empty, and that pop transitions occur only in states where the stack +is empty.9 Let ∆P be the pushdown automaton obtained from P which behaves +like P, except that it has additional nondeterministic transitions as follows. +∆P has three new states qcall, qint and qret. We also assume there are four new +propositions: new, call, int and ret. The proposition new is true in qcall, qint and +qret only. The only transitions in the new states are self-loops that do not change +the stack size. Whenever P has a push/internal/pop transition from a state q +to q′ in P, we add an internal nondeterministic transition from q to qcall/qint/qret +that does not change the stack size. Furthermore, the proposition call/int/ret is +9 Essentially, initially the stack is taken to be empty. When a symbol is pushed onto +an empty stack, it is “annotated ” with information that it is the bottom of the stack, +and the state remembers that stack is non-empty. When an “annotated ” symbol is +popped, the stack now remembers that the stack is empty. + +Stack-Aware Hyperproperties +19 +true in q in that case. Consider the formula +A ∀π. ∀π.′(π[0] ≡L,in π′[0]) → +((newπ ∨ newπ′) R +((callπ ⇒ callπ′ ∧ ¬intπ′ ∧ ¬retπ′) ∧ +(intπ ⇒ intπ′ ∧ ¬callπ′ ∧ ¬retπ′) ∧ +(retπ ⇒ retπ′ ∧ ¬callπ′ ∧ ¬intπ′))) +where R is the dual of U. More precisely, ψ R ψ′ is ¬(¬ψ′) U(¬ψ). +Observe that the above formula is not satisfied by the pushdown system ∆P +if and only if there are two finite executions σ1 and σ2 of P leading to states +q1 and q2 such that (a) the low-level inputs in σ1 and σ2 are the same, (b) σ1 +and σ2 have the same stack access pattern, and (c) a push/internal/pop can be +executed from only one of the states q1 and q2, but not the other. +B +Nondeterministic Visibly Pushdown Automata +(NVPA) +A nondeterministic visibly pushdown automaton (NVPA) [1] is like a push- +down system in that it has finitely many control states and uses an unbounded +stack for storage. However, unlike a pushdown system, it is an automaton +that processes an infinite sequence of input symbols from a pushdown alpha- +bet Σ = Σcall ∪· Σint ∪· Σret. Furthermore, transitions are constrained to conform +to pushdown alphabet — whenever the automaton reads from Σcall, it pushes a +symbol onto its stack, whenever it reads from Σret, it pops its top stack symbol, +and whenever it reads from Σint, it leaves its stack unchanged. +Definition 5 (NVPA). A nondeterministic visibly pushdown Büchi automa- +ton (NVPA) is a tuple N = (Q, qin, Σ, Γ, ⊥, δ, QF) where, Q is a finite set of +control states, qin ∈ Q is the initial control state, Σ = Σcall ∪· Σint ∪· Σret is a +pushdown alphabet that is used to encode inputs, Γ is a finite set called the stack +alphabet, ⊥ ̸∈ Γ is the bottom of stack symbol, QF ⊆ Q is the set of accepting +states, and δ is the transition relation. δ will be assumed to be partitioned into +three as δcall ∪· δret ∪· δint where δcall ⊆ (Q × Σcall × Q × Γ) is the transition on +call symbols, δint ⊆ (Q × Σint × Q) is the transition on internal symbols, and +δret ⊆ (Q × Σret × (Γ ∪ ⊥) × Q) is the transition on return symbols. +Let us fix an NVPA N = (Q, qin, Σ, Γ, ⊥, δ, QF). A configuration of N, like in +the case of a pushdown system, is a pair of the form (q, α⊥) where q ∈ Q +and α ∈ Γ ∗. The functions state(·), stack(·), and top(·) are defined in the same +manner as for pushdown system configurations. A run of N on an input string +w ∈ Σω is an infinite sequence of configurations c0, c1, . . . such that c0 = (qin, ⊥) +is the initial configuration of N, and successive configurations are consistent +with the input symbol read and the transition relation of N. More precisely, for +every i ∈ N, (a) if w(i) ∈ Σcall then (state(ci), w(i), state(ci+1), top(ci+1)) ∈ +δcall +and +stack(ci+1) += +top(ci+1)stack(ci); +(b) +if +w(i) +∈ +Σint +then + +20 +A. Bajwa et al. +(state(ci), w(i), state(ci+1)) +∈ +δint and stack(ci+1) += +stack(ci); and (c) if +w(i) ∈ Σret then (state(ci), w(i), top(ci), state(ci+1)) ∈ δret and additionally, +stack(ci+1) = stack(ci) = ⊥ if top(ci) = ⊥ and stack(ci) = top(ci)stack(ci+1) if +top(ci) ̸= ⊥. It is worth observing that transitions on call symbols push a symbol +onto the stack, transitions on internal symbols leave the stack unchanged, and +transitions on return symbols typically pop the top of the stack unless the stack +is empty (i.e. = ⊥) in which case they leave the stack unchanged. +A run c0, c1, · · · on input w is said to be accepting if it satisfies the Büchi +acceptance condition, i.e., for some q ∈ QF there are infinitely many i such that +state(ci) = q. Finally, the language accepted by NVPA N, denoted L(N), is +L(N) = {w ∈ Σω | N has an accepting run on w}. +C +1-way Alternating Jump Automata (1-AJA) +Our second model of an automaton with strings over a pushdown alphabet, is +1-way Alternating Parity Jump Automata (1-AJA) [4]. 1-AJA are computation- +ally equivalent to NVPAs (i.e., accept the same class of languages) but provide +greater flexibility in describing algorithms. 1-AJAs are alternating automata, +which means that they can define acceptance based on multiple runs of the ma- +chine on an input word. Though they are finite state machines with no auxiliary +storage, their ability to spawn a computation thread that jumps to a future +portion of the input string on reading a symbol, allows them to have the same +computational power as a more conventional machine with storage (like NVPAs). +We present this model after some necessary definitions. +Transitions of an alternating automata identify subsets of next steps that +must all be accepting for the machine to accept an input. We describe these +subsets of next steps is using Boolean functions. For a set X, let B+(X) denote +the set of positive Boolean expressions built using elements of X as propositions, +i.e., they consist of propositional logic formulas built using true, false, X, ∧ and +∨. Given ϕ ∈ B+(X) and A ⊆ X, we say A |= ϕ if ϕ evaluates to true under the +valuation that assigns true to the elements of A and false to everything else. The +dual of a formula ϕ ∈ B+(X), denoted dual(ϕ), is the formula obtained from +ϕ by replacing true by false, false by true, ∨ by ∧, and ∧ by ∨. More precisely, +dual(·) can be defined inductively as follows, where x ∈ X. +dual(true) = false +dual(false) = true +dual(x) = x +dual(ϕ ∧ ψ) = ϕ ∨ ψ +dual(ϕ ∨ ψ) = ϕ ∧ ψ +A 1-AJA is a finite state automaton that reads an input over a pushdown +alphabet. In other words, on reading an input symbol, a thread of computation +changes its control state based on its current state and symbol read. However, +one of the novel features of a 1-AJA is its ability to jump when it reads a call +symbol. Thus, a transition of a 1-AJA not only specifies what the next state is, +but also what the next symbol to be read is. When the symbol read is either an +internal symbol or a return symbol, the symbol to be read next is always the one + +Stack-Aware Hyperproperties +21 +immediately after; this is denoted as →. However, when the symbol read is a call +symbol, the automaton can choose to either read the next symbol or to read the +matching return symbol next; ↷ is used to indicate that the automaton should +read the matching return symbol next. We now have all the notation necessary +to define a 1-AJA formally. +Definition 6 (1-AJA). A 1-way Alternating Parity Jump Automaton (1-AJA) +is a tuple A = (Q, qin, Σ, δ, parity) where, Q is a finite set of control states, qin ∈ Q +is the initial control state, Σ = Σcall ∪· Σint ∪· Σret is a pushdown alphabet that is +used to encode inputs, δ : (Q × Σ) → B+(Q × {→, ↷}) is the transition relation +with the restriction that for any q ∈ Q and a ∈ Σint∪Σret, δ(q, a) ∈ B+(Q×{→}), +and parity : Q → N is the parity function. +In order to define a run of 1-AJA on an input string, we need to introduce +some notation. Let us fix a pushdown alphabet Σ = Σcall ∪· Σint ∪· Σret. We can +extend the notion of well matched strings to Σ∗ as follows. A string z ∈ Σ∗ is +said to be well matched if either z = ε, or z ∈ Σint, or z = cur, or z = uv, where +c ∈ Σcall, r ∈ Σret, and u, v ∈ Σ∗ are (recursively) well matched. Let us fix an +input string w ∈ Σω. The abstract successor of position i in w, denoted ↷ (i), +is given as follows. +↷ (i) = + + + + + + + +j +if w[i : j + 1] = c u r +where u is well matched, +c ∈ Σcall, r ∈ Σret +undefined otherwise +Notice that ↷ (i) is defined only if w(i) ∈ Σcall, and if defined, its value is +the position of its matching return symbol. The abstract successor identifies the +position of the next symbol that will be read if the 1-AJA decides to jump on a +call. Next, the local successor of position i in w, denoted → (i), is always defined +to be i + 1. +Let us fix a 1-AJA A = (Q, qin, Σ, δ, parity) and input string w ∈ Σω. A +run of A on w is a rooted, labeled tree R = (V, E, r, L), where V is the set of +vertices, E the set of edges, r ∈ V is the root, and L : V → (Q × N). The label +of vertex indicates the state of A and the position in w that is being read. We +require that L(r) = (qin, 0), i.e., the automaton is in the initial state and the +0th symbol is being read. Let v ∈ V be an arbitrary vertex with L(v) = (q, i) +and let C ⊆ V be the set vertices that are children of v in R. We require that +the labels of v and those of vertices in C are consistent with δ — there is a +set A ⊆ (Q × {→, ↷}) such that (a) A |= δ(q, w(i)); (b) for every (q′, d) ∈ A +such that d(i) is defined, there is a vertex n ∈ C with L(n) = (q′, d(i)); and +(c) for every vertex n ∈ C, there is (q′, d) ∈ A such that d(i) is defined and +L(n) = (q′, d(i)). A run R = (V, E, r, L) is accepting if every infinite path in R +satisfies the parity acceptance condition as defined by parity. That is, for every +infinite path σ in R, mp(σ) is even, where +mp(σ) = min{parity(q) | for infinitely many i, L(σ(i)) = (q, k) for some k}. + +22 +A. Bajwa et al. +Finally, as always, the language recognized by A is given by +L(A) = {w ∈ Σω | A has an accepting run on w}. +D +Proof of Lemma 1 +Before beginning the proof of Lemma 1, let us recall why NVPAs and 1-AJAs +are closed on standard operations on languages. We present these observations +with a focus on the size of the resulting automata. For a language A ⊆ Σω and +subset Γ ⊆ Σ, we take ΓA to be the language {aw | a ∈ Γ and w ∈ A}. +Proposition 3. Let Ai be a 1-AJA (NVPA) recognizing Li ⊆ Σω of size ni, +for i ∈ {1, 2}. Let Γ ⊆ Σ. Then the following automata can be constructed in +polynomial time. +1. There is a 1-AJA (NVPA) recognizing L1 ∪ L2 of size O(n1 + n2). +2. There is a 1-AJA (NVPA) recognizing ΓL1 of size O(n1). +3. There is a 1-AJA recognizing (Σω \ L1) of size O(n1). +Proof (Sketch). The automaton construction that establishes (1) chooses to ei- +ther run one of A1 or A2 nondeterministically to recognize L1 ∪ L2. (2) can be +established by checking if the first symbol belongs to Γ (and performing the nec- +essary stack operation demanded by the first symbol in the case of NVPA) and +then running the automaton for L1. Finally, for (3), the 1-AJA recognizing the +complement of L1 has the same set of states, the parity of each state is increased +by 1, and for any state q and symbol a, δ(q, a) = dual(δ1(q, a)), where δ1 and +δ are the transition functions of A1 and the automaton recognizing (Σω \ L1), +respectively. +Our construction of Aψ will proceed inductively based on the structure of ψ. +The type of automaton constructed will be consistent with the parity of fc(ψ), +i.e., an NVPA if fc(ϕ) is odd and a 1-AJA if fc(ψ) is even. +Case ψ = aπ: Let Γ ⊆ Σ[m] be the set of symbols ζ such that a is in the label +set of the πth control state in symbol ζ. We will construct a 1-AJA that accepts +Γ(Σ[m])ω using Proposition 3 (2). +Case ψ = ¬ψ1: We will convert Aψ1 into a 1-AJA (if needed) using Theorem 1, +and then use Proposition 3 (3) to construct a 1-AJA that accepts the language +L(Aψ1). +Case ψ = ψ1 ∨ ψ2: Without loss of generality, assume that fc(ψ1) ≥ fc(ψ2). We +convert (if needed) Aψ2 to an automaton of the same type as Aψ1 using The- +orem 1 and then use Proposition 3 (1), to construct an automaton recognizing +L(Aψ1) ∪ L(Aψ2). +Case ψ = Xψ1: We use Proposition 3 (2) to construct an automaton for +Σ[m]L(Aψ1). +Case ψ = ψ1 U ψ2: Using Theorem 1, we will construct (if needed) a 1-AJA +equivalent to Aψi and let this be Ai = (Qi, qini, Σ[m], δi, parityi), for i ∈ {1, 2}. + +Stack-Aware Hyperproperties +23 +The 1-AJA for ψ is given by Aψ = (Q1 ∪· Q2 ∪· {qin}, qin, Σ[m], δ, parity) where +δ(qin, ζ) = δ2(qin2, ζ) ∨((→, qin) ∧ δ1(qin1, ζ)), parity(qin) = 1, and for q ∈ Qi (i ∈ +{1, 2}), we have δ(q, ζ) = δi(q, ζ) and parity(q) = parityi(q). +Case ψ = ∃π. ψ1: Recall that our pushdown system P has control states S, stack +alphabet Γ, and transition relation ∆ = ∆int ∪· ∆call ∪· ∆ret. Using Theorem 1, +we will construct (if needed) a NVPA equivalent to Aψ1 and let this be A1 = +(Q1, qin1, Σ[m+1], Γ1, ⊥, δ1, QF1). Notice that the input alphabet of A1 is Σ[m+ +1], where the m + 1st component is an encoding of the path assigned to variable +π. The automaton for ψ will essentially guess the encoding of a path that is +consistent with the transitions of P, and check if assigning the guessed path to +variable π satisfies ψ1 by running the automaton A1. The additional requirement +we have is that the guessed path start at the same configuration as the current +configuration of the path assigned to variable πm. In order to be able to guess a +path, Aψ will keep track of P’s control state in its control state, and use its stack +to track P’s stack operations along the guessed path. Before defining Aψ formally +we need to introduce some notation that will be convenient. An element ζ ∈ +Σ[m] is of the form (o, (s1, s2, . . . sm), (a1, a2, . . . am)). We use op(ζ) to denote +o, state(ζ)|i to denote si, and stack(ζ)|i to denote ai, for i ∈ {1, 2, . . .m}. By +convention we take state(ζ)|0 = sin, and stack(ζ)|0 to be undefined. Finally, for +s ∈ S and a ∈ Γ, ζ + (s, a) is the symbol in alphabet Σ[m + 1] given by +(o, (s1, . . . sm, s), (a1, . . . am, a)). +The NVPA Aψ = ((Q1×S) ∪· {qin}, qin, Σ[m], (Γ1×Γ), ⊥, δ, (QF1 ×S)); notice +that the bottom of stack symbol (⊥) is same as the one in A1. The transition +relation δ = δcall ∪· δret ∪· δint is defined as follows. +– When op(ζ) = call, +δcall = {(qin, ζ, (q, s), (b, a)) | (qin1, ζ + (state(ζ)|m, a), q, b) ∈ δ1 and +(state(ζ)|m, (s, a)) ∈ ∆call} ∪ +{((q, s), ζ, (q′, s′), (b, a)) | (q, ζ + (s, a), q′, b) ∈ δ1, (s, (s′, a)) ∈ ∆call}. +– When op(ζ) = int, +δint = {(qin, ζ, (q, s)) | (qin1, ζ + (state(ζ)|m, ε), q) ∈ δ1, (state(ζ)|m, s) ∈ ∆int} +∪ {((q, s), ζ, (q′, s′)) | (q, ζ + (s, ε), q′) ∈ δ1, and (s, s′) ∈ ∆int}. +– When op(ζ) = ret, +δret = {(qin, ζ, ⊥, (q, s)) | (qin1, ζ + (state(ζ)|m, stack(ζ)|m), ⊥, q) ∈ δ1 and +((state(ζ)|m, stack(ζ)|m), s) ∈ ∆ret} ∪ +{((q, s), ζ, ⊥, (q′, s′)) | (q, ζ + (s, stack(ζ)|m), ⊥, q′) ∈ δ1, and +((s, stack(ζ)|m), s′) ∈ ∆ret} ∪ +{((q, s), ζ, (b, a), (q′, s′)) | (q, ζ + (s, a), b, q′) ∈ δ1, ((s, a), s′) ∈ ∆ret} +The requirement that the guessed path for π start in the same configuration as +the current configuration of the path assigned to πm, introduces a few points +in the definition of δ that are worth highlighting. Transitions from qin, whether + +24 +A. Bajwa et al. +they be call, int, or ret, pick a step in P that starts from the same state as the +one in the path assigned to πm. Stack symbols pushed by P along the guessed +path are pushed by Aψ onto its stack (see δcall). If the stack of Aϕ is ⊥ at a +ret-transition, that means on the guessed path the symbol popped must be from +what was on the stack at the start of the path. Since that matches with the +configuration on the path mapped to πm, this symbol must be the same as what +is popped for πm. This is reflected in δret for the cases with ⊥. +An important observation that we will exploit is that if ψ = ∃π. ψ1 is a +sentence, then the following stronger correctness guarantee holds: for any ρ ∈ +{call, int, ret}ω, ρ ∈ L(Aϕ) if any only if ρ is a stack access pattern and P, [† �→ +ρ], † |= ψ. The language of Aψ in this case consists of exactly the set of path +environments satisfying ψ. This stronger statement follows from the construction +of Aψ. +To complete the proof, we will bound the size of Aψ by induction on fc(ψ). +Recall that n is a bound on the size of P and ψ. +Base Case: Consider ψ such that fc(ψ) = 0. Based on the definition of fc(·) +(Definition 4), this means that ψ is built from propositions, ¬, ∨, X, and U; +in particular there are no path quantifiers in ψ in this case. Observe that the +construction for aπ is a constant sized automaton. Also, the constructions for ¬, +∨, X, and U add at most a constant factor to the size of the automaton. Given +these observations, size of Aψ in this case is bounded by O(n), which is bounded +by gO(1)(0, n). Thus the base case holds. +Induction Step: We break the induction step into two cases based on the +parity of fc(ψ). When fc(ψ) = 2k (for some k ≥ 1) then ψ is built using sub- +formulas of (formula) complexity at most 2k using Boolean operators (¬, ∨), X, +and U. Let us assume that we first construct 1-AJAs for each of the subformulas +with (formula) complexity < 2k. This results in at most a quadratic blowup +(Theorem 1), and so the size of the automata for each such subformula is at +most (gc(k, n))2 for some c. The constructions for ¬, ∨, X and U produce an +automaton that is at most a constant factor of the sum of the sizes of the +component automata. Thus, the size of Aψ is at most dn(gc(k, n))2 ≤ gc′(k, n) +for some c′, establishing the claim in this case. Now let us consider the case when +fc(ψ) = 2k+1 for some k ≥ 0. In this case ψ is built using ∨, X and ∃ to combine +subformulas of (formula) complexity ≤ 2k + 1. Again, we can convert automata +for each of the sub-formulas of (formula) complexity < 2k + 1 into NVPAs for +an exponential cost (Theorem 1). Thus, we can assume that the size of all of +these automata is bounded by gc(k + 1, n) for some c. Based on Proposition 3, +we can see that disjunction and X produce automata that grow by at most a +constant factor. Existential quantification are the only operators to have a non- +trivial blow-up. Based on the construction outlined in this proof, if ϕ has an +NVPA of size ℓ then the automaton of ∃π. ϕ has size O(nℓ) as n bounds the +size of P. Since there are at most n quantifiers, we can bound the size of Aψ +by dnn(gc(k + 1, n)) ≤ 2d′n log n+gc(k,n) ≤ gc′(k + 1, n); the last step is because +gc(k, n) ≥ cn log n. This establishes the induction step. + +Stack-Aware Hyperproperties +25 +E +Proof of Theorem 4 +We will show that every language L ∈ ASPACE(hc(k − 1, n)) can be reduced +to the model checking problem of sHLTL sentence with formula complexity +2k − 1. Since ASPACE(f(n)) = DTIME(2O(f(n))) for f(n) ≥ log n, the theorem +will follow. +Consider an arbitrary hc(k−1, n)-space bounded alternating Turing machine +(ATM) M. Since hc(k − 1, n) ≥ n, we may assume that M is a 1-tape machine. +Let M = (Q∃, Q∀, Σ, Γ, ⊔, qin, δ, qa), where Q∃ is the set of existential control +states, Q∀ is the set of universal control states, Σ is the input alphabet; Γ ⊇ Σ +is the tape alphabet, ⊔ ∈ Γ \ Σ is the blank symbol, qin ∈ Q∃ ∪ Q∀ and qa ∈ Q∃ +are the initial and accepting states, respectively, and δ is the transition function +of the Turing machine. We use Q = Q∃ ∪ Q∀ to denote the set of all states +of M. We assume that qa is a halting state (no transitions enabled) and so +δ : (Q\{qa})×Γ → 2(Q×Γ ×{→,←}), where given a state and current symbol being +read, the transition function identifies choices for the next state, the symbol to +be written, and the direction in which to move the tape head (→ for right, and ← +for left). We assume, without loss of generality, that for each pair (q, b) ∈ Q × Γ +that |δ(q, b)| ∈ {0, 2}, i.e., there are either no or two choices at each step. Also, +we assume that these choices are ordered in some fashion so we will often speak +of “choice i” for i ∈ {1, 2}. +A configuration c of M is a string in Γ ∗(Q × Γ)Γ ∗, where c = u(q, b)v with +u, v ∈ Γ ∗, b ∈ Γ and q ∈ Q, denotes that the tape of M is the string ubv, +the control state is q, and the head is reading the cell containing b. Since M is +hc(k − 1, n)-space bounded, we can assume any configuration c of M is a string +of length exactly hc(k − 1, n). The initial configuration of M on input bw of +length n is (qin, b)w⊔hc(k−1,n)−n. A configuration c = u(q, b)v is an existential +configuration if q ∈ Q∃, a universal configuration if q ∈ Q∀, and an accepting +configuration if q = qa. For a pair of configurations c and c′, we say c ⊢i c′ +if M’s configuration is c′ if it takes one step according to choice i ∈ {1, 2} +from configuration c. A run of M on input w is a finite, rooted binary tree +T = (V, E, r, ℓ), where V is the set of vertices, E is the set of edges oriented +away from the root, r ∈ V is the root, and ℓ is a function that maps each +vertex to a configuration of M. In addition, ℓ is required to satisfy the following +conditions. The root r is labeled by the initial configuration. For any internal +vertex v ∈ V , if ℓ(v) is an existential configuration then v has one child c such +that ℓ(v) ⊢i ℓ(c) for some i ∈ {1, 2}, and if ℓ(v) is a universal configuration, then +v has two children c1, c2 such that ℓ(v) ⊢i ℓ(ci) for i ∈ {1, 2}. Finally, a run T is +accepting if every leaf of T is labeled by an accepting configuration. An input w +is accepted by M, if M has an accepting run on w. +We will construct a reduction from L(M) (which by definition is in +ASPACE(hc(k − 1, n)) to the model checking problem for sHLTL. That is, given +input w, we will construct a pushdown system Pw and sHLTL sentence θw such +that Pw |= θw if and only if w ∈ L(M). The idea behind the reduction is to +construct a pushdown system Pw such that labels of paths starting from the +initial configuration in �Pw� encode possible computations of M on w. And θw + +26 +A. Bajwa et al. +is constructed to check if a path encodes a valid accepting run of M on w. To +formalize this intuition, we need to first identify a way to encode runs of M, +which are binary trees, as a string of labels. +Encoding ATM Runs. Recall that a run of M is a binary tree and we need +to find a way to encode the tree as string. One way to accomplish this faithfully, +is to have the encoding record the stack operations during a depth first search +(DFS) traversal of the tree. For example, if we have a tree T with root r, with +tree T1 as the left child and T2 as the right child, then during DFS, the algorithm +will first push r on the stack, perform DFS traversal on T1 (recursively), pop r +from the stack, push r back onto the stack, DFS traverse T2 (recursively), and +finally pop r. When encoding runs, what we need to push/pop is not the node +r, but rather its label. Notice that for a sequence of stack operations to conform +to the DFS traversal of a tree, it is necessary for the symbols being pushed and +popped be the label of the same node — for example, in the example tree before, +after traversing T1, we need the same node r to be popped and pushed. Since +the symbols when popping are in reverse order of when they are pushed, for long +labels (as in the case of configurations) this check is challenging. To overcome +this, we push/pop the label and its reverse at the same time. This ensures that if +we want to check if a string pushed is the same as a string that was just popped, +then we can check for string equality as opposed to one being the reverse of +another, and this check is easier to do using formulas in sHLTL. +We formalize the above discussion to give a precise definition of the encoding +of a binary tree as a string. We will abuse notation and overload enc(·) to refer to +multiple functions — the context will disambiguate which enc(·) we are referring +to. Let Λ = Γ ∪ (Q × Γ), the alphabet used to encode configurations of M. For +i ∈ {1, 2}, let [Λ]i = {[a]i | a ∈ Λ} be a “copy” of the alphabet Λ. For a string +c ∈ Λ∗ of length m, we define +enc(push, c) = (call, [c(0)]1, [c(m − 1)]2)(call, [c(1)]1, [c(m − 2)]2) · · · +(call, [c(i)]1, [c(m − i − 1)]2) · · · (call, [c(m − 1)]1, [c(0)]2) +enc(pop, c) = (ret, [c(m − 1)]1, [c(0)]2)(ret, [c(m − 2)]1, [c(1)]2) · · · +(ret, [c(i)]1, [c(m − i − 1)]2) · · · (ret, [c(0)]1, [c(m − 1)]2) +Essentially, enc(·) of a string encodes both the string and its reverse, has a +tag that indicates whether the string is being pushed or popped, and if it +is popped, the order of symbols is reversed. Consider a rooted, labeled bi- +nary tree T = (V, E, r, ℓ). Its encoding is inductively defined as follows. If +V = {r} and E = ∅ (i.e., T is a tree with only one vertex) then enc(T ) = +enc(push, ℓ(r))enc(pop, ℓ(r)). If r has only one child which is the subtree T1, then +enc(T ) = enc(push, ℓ(r))enc(T1)enc(pop, ℓ(r)), where enc(T1) is given recursively. +Finally, if r has T1 and T2 as left and right subtrees, respectively, then enc(T ) = +enc(push, ℓ(r))enc(T1)enc(pop, ℓ(r))enc(push, ℓ(r))enc(T2)enc(pop, ℓ(r)). +Before +moving on, let us highlight a subtle aspect of our encoding. In the case of a +tree where r has two sub-trees T1 and T2, we “pop” ℓ(r) and “push” ℓ(r) between +the traversals of T1 and T2. This may seem unnecessary on first reading. Notice +that for T to be a valid run, the label of the root of T2 must be the result of + +Stack-Aware Hyperproperties +27 +taking one step from ℓ(r). Such checks will be encoded in our sentence, and for +that to be possible, we need successive ATM configurations to consecutive in the +string encoding. +The Pushdown System. Labels of paths of our constructed pushdown system +will encode possible runs of M on the input w. At each step the pushdown system +guesses the next symbol of the possible run by moving to a control state whose +label corresponds to this symbol. The stack is used to ensure that when the +“pop” symbols in the encoding are encountered they match the symbols that +were “pushed” earlier in the guess. We can define this precisely as follows. Recall +that for Λ = Γ ∪ (Q × Γ), [Λ]i (i ∈ {1, 2}) refers to the “ith copy” of Λ. Let +us fix the set of atomic propositions AP = {call, ret, end} ∪ [Λ]1 ∪ [Λ]2 and let +S = {call, ret} × [Λ]1 × [Λ]2. Let P = (S ∪ {sin, se}, ([Λ]1 × [Λ]2) ∪ {⊥}, sin, ∆, L), +where L(sin) = ∅, L(se) = {end}, and L((o, [a]1, [b]2)) = {o, [a]1, [b]2}. The +transition relation ∆ = ∆int ∪· ∆call ∪· ∆ret is given as follows: ∆int = {(se, se)} +and +∆call = {(sin, (s, ⊥)) | s ∈ S} ∪ +{((call, [a]1, [b]2), (s, ([a]1, [b]2))) | s ∈ S and a, b ∈ Λ} +∆ret = {(((ret, [a]1, [b]2), ([a]1, [b]2)), s) | s ∈ S and a, b ∈ Λ} ∪ +{((s, ⊥), se) | s ∈ S} +Paths of �P� may not correspond to actual computations of M on w since P +doesn’t check for many properties that need to hold for such a string. On the +other hand, correct runs of M on w do correspond to paths of P that end in +(end)ω. Our final pushdown system will be a slight modification of P, in two ways. +First we will add some additional book-keeping to the states and stack symbols +to ensure that whenever a universal configuration is guessed, the computation +has transitions corresponding to both choices. Second, we will need to modify +the system to account for the specification, as we shall see towards the end of +this proof. +As mentioned before, for the labels of a path of P to correspond to the en- +coding of an accepting computation of M on w, we need to ensure that the labels +satisfy a few properties. These will be encoded in our sHLTL sentence. However, +instead of encoding these conditions in sHLTL, we will find it convenient to first +write them in QPTL, a logic introduced in [17]. We begin by introducing this +logic, showing how the properties of accepting runs can be encoded, and then +describing a way to translate them back to sHLTL. +QPTL. Quantified propositional temporal logic (QPTL) [17] extends LTL with +quantification over propositions. Fixing a set of atomic propositions AP, formulas +in the logic are given by the following BNF grammar; in what follows, a is an +element of AP. +ϕ ::= a | ¬ϕ | ϕ ∨ ϕ | Xϕ | F ϕ | ∃a. ϕ +Models of QPTL are the same as those for LTL, namely elements of (2AP)ω, and +the semantics of most of the constructs is similar. For w ∈ (2AP)ω, a holds if +a ∈ w(0); ¬ϕ holds if ϕ does not hold; ϕ1 ∨ ϕ2 holds if either ϕ1 or ϕ2 hold; + +28 +A. Bajwa et al. +Xϕ holds if ϕ holds on the suffix w[1 : ]; and F ϕ holds if ϕ holds in some suffix +w[i : ]. The only new operator is ∃a. ϕ which holds in w, if ϕ holds in some word +w′ which agrees with w in the evaluation of all propositions except possibly a. +Recall that F ϕ is equivalent to true U ϕ, G ϕ is a short hand for ¬ F(¬ϕ), and ∀a.ϕ +is ¬∃a. (¬ϕ). We can extend fc(·) to QPTL formulas, with the same definition; +for this definition ¬ will behave like negation for cognate formulas, rather than +negation for sHCTL*-sentences. Finally, it has been shown that every QPTL +formula is equivalent to one in prenex normal form, where all the quantifiers +have been pulled to the front of the formula. +Our QPTL formula ϕw that describes when a word encodes an accepting +run of M on w, will rely on formulas constructed in [17]. The first is formula +ϕc,k,n(p1, p2) (Lemma 4.4. in [17]) of size O(k + n) such that w |= ϕc,k,n(p1, p2) +if and only if propositions p1 and p2 are true exactly once in w, p2 is true +after p1, and they are separated by exactly hc(k, n) positions. Further more +fc(ϕc,k,n) = 2k − 1, and can be constructed O(log n) space. We will introduce +the other formulas as we describe the conditions needed. +The QPTL formula ϕw. We now describe ϕw with the property that a path +of �P� satisfies ϕw if and only if M accepts w; here a path c0c1 · · · of �P� satis- +fies ϕw, if L(c0)L(c1) · · · |= ϕw. Observe that the construction of P ensures that +symbols “popped” are the same as the symbols “pushed” and that every universal +configuration has two successors correspond to transitions corresponding to each +choice.. Therefore, the remaining conditions that need to be checked for a path +to be the encoding of an accepting computation of M on w are as follows. +1. Every configuration has length exactly hc(k − 1, n). +2. All symbols encoding a single configuration have the same tag, i.e., either +all call or all ret. +3. The first configuration is the initial configuration of M on w. +4. Successive “pushed” configurations correspond to a single step of M consis- +tent with its transition function. +5. “Leaves” of the run are accepting configurations. In other words, if a push +configuration is immediately followed by a pop configuration (they will be +identical thanks to P), then the control state must be qa. +6. If a pop configuration is immediately followed by a push configuration (i.e., +the DFS traversal of the run is exploring the right child) then they must be +the same configuration. +7. All the configurations are popped at the end. +If ϕ denotes the conjunction of all the above conditions written in QPTL, then +our desired formula ϕw is Xϕ since the initial state sin of P is not part of guessing +the encoding of the computation. +We describe how to encode each of these conditions in QPTL, often relying +on formulas from [17]. +1. To state property (1), we will have a proposition ⊲ (which will be exis- +tentially quantified) that marks the beginning of each configuration. The +fact that ⊲ marks the beginning of each configuration will be ensured by + +Stack-Aware Hyperproperties +29 +the other properties we write down. We will require that ⊲ is true exactly +hc(k−1, n) positions apart using the formula ϕc,k−1,n(·, ·) introduced before. +This is given by Equation (9) in [17] where r replaced by ⊲. +2. Since ⊲ marks the beginning of each configuration, saying that all symbols +in a configuration have the same tag, is equivalent to saying that if the tag +changes then ⊲ must be true when the tag changes. In other words, property +(2) can be written as +G((call ∧ Xret) → X⊲) ∧ G((ret ∧ Xcall) → X⊲) +3. Equation (10) in [17] states the property (3). It needs to be slightly modified +to also include the condition that the propositions from [Λ]2 encode the +reverse of the initial configuration of M. +4. Equation (11) in [17] states the property (4). It needs to be slightly to mod- +ified to also ensure that the reverse encodings using [Λ]2 are consistent with +the transitions of M, and this requirement is only imposed for successive +configurations that have the call tag. +5. Recall that corresponding symbols in successive configurations are hc(k − +1, n) apart. Property (5) can be written to say that if there are two positions +p and q that are hc(k−1, n) apart that have opposite tags, and if in addition +p corresponds to a position that is scanned by the tape head, then the control +state at p must be the accepting state qa. +∀p1∀p2. (ϕc,k−1,n(p1, p2) ∧ F(p1 ∧ call ∧ +� +a∈Q×Γ +[a]1) ∧ F(p2 ∧ ret)) → +F(p1 ∧ +� +a∈{qa}×Γ +[a]1) +6. Using the observations in property (5), property (6) can be written as +∀p1∀p2. ((ϕc,k−1,n(p1, p2) ∧ F(p1 ∧ ret) ∧ F(p2 ∧ call)) → +� +a,b∈Λ +(F(p1 ∧[a]1 ∧[b]2) ∧ F(p2 ∧[b]1 ∧[a]2)) +7. This property can be ensured by requiring that the path end in state s2. We +write this as F end. +Since formulas in QPTL can be written in prenex normal form, we can pull all +the universal quantifiers in the various formulas listed above to get a formula ϕw +of the form X ∃ ⊲ ∀p1∀p2. ϕ where ϕ is a Boolean combination of ¬ϕc,k−1,n(·, ·) +and simple formulas with temporal operators of (formula) complexity 0. Thus, +fc(ϕw) = 2k − 1. +Converting to sHLTL. Our QPTL formula ϕw is of the form X∃⊲ϕ′. We will +describe how to construct an “equivalent” formula in sHLTL; this will require +modifying our pushdown system P as well to obtain our final pushdown system +Pw. Construct a cognate sentence ψ′ as follows. Let a be a new proposition not + +30 +A. Bajwa et al. +appearing in ϕw. For every quantified proposition p in ϕw, replace every bound +occurrence of p by aπp and replace the quantification ∃p with ∃πp. Let π∗ be a +fresh path variable. Replace every free proposition p in ϕw by pπ∗. Let ψ be the +resulting cognate formula. Finally let θw = E∃π∗. ψ. +To finish the reduction, we need to modify P to account for the new proposi- +tion a introduced in θw. Our final pushdown system Pw is constructed as follows. +Recall that S = ({call, ret} × [Λ]1 × [Λ]2) and the states of P is S ∪ {sin, se}. Let +[S]1 and [S]2 be two copies of S. The states of Pw will be [S]1 ∪ [S]2 ∪ {sin, se}. +The two copies of state s ∈ S will have the same transitions into and out of it +and have the same labels for all the old propositions. The only difference will be +that a ∈ L([s]1) while a ̸∈ L([s]2). +It is easy to make the following observations: fc(θw) = fc(ϕw), and a path of +�P� satisfies ϕw if and only if Pw |= θw. This completes the proof of the hardness +result. + diff --git a/jNFJT4oBgHgl3EQfXSyN/content/tmp_files/load_file.txt b/jNFJT4oBgHgl3EQfXSyN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b6d0d7cc6f51f0f6cd4ae5538c13e7cae553187 --- /dev/null +++ b/jNFJT4oBgHgl3EQfXSyN/content/tmp_files/load_file.txt @@ -0,0 +1,1111 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf,len=1110 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='11521v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='LO] 27 Jan 2023 Stack-Aware Hyperproperties⋆ Ali Bajwa2, Minjian Zhang1, Rohit Chadha2, and Mahesh Viswanathan1 1 University of Illinois Urbana-Champaign, USA 2 University of Missouri in Columbia, USA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A hyperproperty relates executions of a program and is used to formalize security objectives such as confidentiality, non-interference, privacy, and anonymity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Formally, a hyperproperty is a collection of al- lowable sets of executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A program violates a hyperproperty if the set of its executions is not in the collection specified by the hyperproperty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The logic HyperCTL* has been proposed in the literature to formally specify and verify hyperproperties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The problem of checking whether a finite-state program satisfies a HyperCTL* formula is known to be decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' However, the problem turns out to be undecidable for proce- dural (recursive) programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Surprisingly, we show that decidability can be restored if we consider restricted classes of hyperproperties, namely those that relate only those executions of a program which have the same call-stack access pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We call such hyperproperties, stack-aware hy- perproperties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our decision procedure can be used as a proof method for establishing security objectives such as noninference for recursive pro- grams, and also for refuting security objectives such as observational determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Further, if the call stack size is observable to the attacker, the decision procedure provides exact verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Keywords: Hyperproperties · Temporal Logic · Recursive Programs · Model Checking · Pushdown Systems · Visibly Pushdown Automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 1 Introduction Temporal logics HyperLTL and HyperCTL* [5] were designed to express and reason about security guarantees that are hyperproperties [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A hyper- property [6] is a security guarantee that does not depend solely on individual executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Instead, a hyperproperty relates multiple executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For example, non-interference, a confidentiality property, states that any two executions of a program that differ only in high-level security inputs must have the same low- security observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' As pointed out in [6], several security guarantees are hy- perproperties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The logic HyperCTL* subsumes HyperLTL, and the problem of checking a finite-state system against a HyperCTL* formula is decidable [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ⋆ Ali Bajwa was partially supported by NSF CNS 1553548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Rohit Chadha was par- tially supported by NSF CNS 1553548 and NSF SHF 1900924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Mahesh Viswanathan and Minjian Zhang were partially supported by NSF SHF 1901069 and NSF SHF 2007428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In this paper, we consider the problem of model checking procedural (recur- sive) programs against security hyperproperties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Recall recursive programs are naturally modeled as a pushdown system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Unlike the case of finite-state tran- sition systems, the problem of checking whether a pushdown system satisfies a HyperCTL* formula is undecidable [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In contrast, CTL* model checking is decidable for pushdown systems [3,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We consider restricted classes of hyperproperties for re- cursive programs, namely those that relate only those executions that have the same call-stack access pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Intuitively, two executions have the same stack access pattern if they access the call stack in the same manner at each step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', if in one execution there is a push (pop) at a point, then there is a push (pop) at the same point in the other execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observe that if two executions have the same stack access pattern, then their stack sizes are the same at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We call such hyperproperties, stack-aware hyperproperties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In order to specify stack-aware hyperproperties, we extend HyperCTL* to sHCTL*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The logic sHCTL* has a two level syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' At the first level, the syntax is identical to HyperCTL* formulas, and is interpreted over executions of the pushdown system with the same stack access pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' At the top-level, we quantify over different stack access patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The formula Eψ is true if for some stack access pattern ρ of the system, the pushdown system restricted to executions with stack access pattern ρ satisfies the HyperCTL* formula ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The formula Aψ is true if for each stack access pattern ρ of the system, the pushdown system restricted to executions with stack access pattern ρ satisfies the Hyper- CTL* formula ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' See Figure 1 on Page 8 for a side-by-side comparison of the syntax for HyperCTL* and sHCTL*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' HyperLTL is extended to sHLTL simi- larly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Please note that sHCTL* subsumes sHLTL, and that sHCTL* (sHLTL) coincides with HyperCTL* (HyperLTL) for finite state systems as all execu- tions of finite state systems have the same stack access pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We show that the model checking problem for sHCTL* is decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We demonstrate three different ways this result can aid in verifying recursive pro- grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' First, for security guarantees such as noninference [14], which are ex- pressible in the ∀∃∗ fragment of HyperLTL, we can use the model checking algorithm to establish that a recursive program satisfies the HyperLTL prop- erty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Secondly, for the ∀∗ fragment of HyperLTL, the model checking algorithm can be used to detect security flaws by establishing that a recursive program does not satisfy security guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observational determinism [13,19] is an example of such a property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Finally, when the attacker can observe stack access patterns (or, equivalently, stack sizes), we can get exact verification for several proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The assumption of the attacker observing stack access patterns holds, for example, in the program counter security model [15] in which the attacker has access to program counters at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' As argued in [15], the program security model is appropriate to capture control-flow side channels such as those arising from timing behavior and/or disclosure of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The decision procedure uses an automata-theoretic approach inspired by the model checking algorithm for finite state systems and HyperCTL* given Stack-Aware Hyperproperties 3 in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since stack-aware hyperproperties relate only executions with the same stack access-pattern, a set of executions with the same stack access pattern can be encoded as a word over a pushdown alphabet, 3 and the problem of model checking a sHCTL* formula can be reduced to the problem of check- ing emptiness of a non-deterministic visibly pushdown automaton (NVPA) over infinite words [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The reduction of the model checking problem to the empti- ness problem is based on a compositional construction of an automaton for each sub-formula which accepts exactly the set of assignments to path variables that satisfy the sub-formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For this construction to be optimal, we carefully leverage two equi-expressive classes of automata on infinite words, namely NVPAs and 1-way alternating jump automata (1-AJA) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The model checking algorithm for sHCTL* against procedural programs has a complexity that is very close to the complexity of model checking finite state systems against HyperCTL*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' If g(k, n) denotes a tower of exponentials of height k, where the top most expo- nent is poly(n), then for a formula with formula complexity r, 4 and a system and formula whose size is bounded by n, our algorithm is in DTIME(g(⌈ r 2⌉, n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In comparison, model checking finite state systems against HyperCTL* is in NSPACE(g(⌈ r 2⌉ − 1, n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This slight difference in complexity is consistent with checking other properties like invariants for finite state systems (NL) versus pro- cedural programs (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We also prove that our model checking algorithm is optimal by proving a matching lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our proof showing DTIME(g(⌈ r 2⌉, n)-hardness of the model checking problem for formulas with (formula) complexity r, relies on re- ducing the membership problem for g(⌈ r 2⌉ − 1, n) space bounded alternating Turing machines (ATM) to the model checking problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The reduction requires identifying an encoding of computations of ATMs, which are trees, as strings that can be guessed and generated by pushdown systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The pushdown system we construct for the model checking problem guesses potential computations of the ATM, while the sHCTL* formula we construct checks if the guessed computation is a valid accepting computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Clarkson and Schneider introduced hyperproperties [6] and demonstrated their need to capture complex security properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Temporal logics HyperLTL and HyperCTL*, that describe hyperproperties, were introduced by Clarkson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' They also characterized the complexity of model checking finite state transition systems against HyperCTL* specifications by a reduction to the satisfiability problem of QPTL [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Subsequently, other model checking algorithms for verifying finite state systems against HyperCTL* properties have been proposed [10,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Tools that check satisfiability [8] and runtime verifi- cation [9] for HyperLTL formulas have also been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Finkbeiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' introduced the automata-theoretic approach to model checking HyperCTL* for finite-state systems [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 3 A pushdown alphabet is an alphabet that is partitioned into three sets: a set of call symbols, a set of internal symbols, and a set of return symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 4 Our definition of formula complexity is roughly double the usual notion of quantifier alternation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For a precise definition, see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The model checking problem for HyperLTL, and consequently Hyper- CTL*, was shown to be undecidable for pushdown systems in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For re- stricted fragments of HyperLTL, Pommellet and Tayssir [16] introduced over- approximations and under-approximations to establish/refute that a pushdown system satisfies a HyperLTL formula in those fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Gutsfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' intro- duced stuttering Hµ, a linear time logic for checking asynchronous hyperprop- erties for recursive programs in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The authors present complexity results for the model checking problem under an assumption of fairness, and a restriction of well-alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' While the restriction to paths with the same stack access pattern is similar to the well-alignment restriction, we do not assume any fairness con- dition to establish decidability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' However, as sHCTL* is a branching time logic and only considers synchronous hyperproperties, the two logics are not directly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' It is also worth mentioning that the branching nature of sHCTL* requires us to “copy” a potentially unbounded stack, from the most recently quantified path variable, when assigning a path to the “current” quantified path variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In contrast, all path assignments in [12] start with an empty stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' An extended abstract of this paper appears in the 29th International Con- ference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 2 Motivation Clarkson and Schneider [6] argue that many important security guarantees are expressible only as hyperproperties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We discuss two examples of security hyper- properties, and the logics HyperLTL and HyperCTL* used to specify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Hyperproperties and temporal logics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We discuss two variants of non- interference [11] that model confidentiality requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In non-interference, the inputs of a system are partitioned into low-level input security variables and high-level input security variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The attacker is assumed to know the values of low-level security inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' During an execution, the attacker can observe parts of the system configuration such as system outputs, or the memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A system satisfies non-interference if the attacker cannot deduce the values of high-level inputs from the low-level observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To formally specify the variants, we use the logic HyperLTL [5], a fragment of the logic HyperCTL* [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The precise syntax of HyperLTL and HyperCTL* is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In the syntax, π is a path variable and the formula aπ is true if the proposition a is true along the path “π”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Intuitively, the formula ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ is existential quantification over paths, and is true if there is a path that can be assigned to π such that ψ is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Universal quantification (∀π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ), and other logical connectives such as conjunction (∧), implication (→), equivalence (↔) and the temporal operators G and F can be defined in the standard way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' By having explicit path variables, HyperLTL and HyperCTL* allow quantification over multiple paths simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The first variant, noninference [14], states that for each execution σ of a program, there is another execution σ′ such that (a) σ′ is obtained from σ by Stack-Aware Hyperproperties 5 replacing the high-level security inputs by a dummy input, and (b) σ and σ′ have the same low-level observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Noninference is a hyperliveness property [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let us assume that the low-level observations of a configuration are deter- mined by the values of the propositions in L = {ℓ1, · · · ℓm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' As shown in [5], non- inference is expressible by the HyperLTL formula: NI def = ∀π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ∃π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (G λπ′) ∧ π ≡L π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Here G λπ′ expresses that Globally (or in each configuration of the execution) the high input of π′ is the dummy input λ, and π ≡L π′ def = G(∧ℓ∈L(ℓπ ↔ ℓπ′)) expresses that π and π′ have the same low-level observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The second variant, observational determinism [13,19], states that any two executions that have the same low-level initial inputs, must have the same low-level output observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observational determinism is a hypersafety property [5,6], and is also expressible in HyperLTL using the formula [5]: OD def = ∀π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ∀π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (π[0] ≡L,in π′[0]) → π ≡L,out π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Here ≡L,in and ≡L,out express the fact that π and π′ have the same low-security inputs and outputs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Procedural (recursive) programs and Stack-aware hyperproperties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Pushdown systems model procedural programs that do not dynamically allo- cate memory, and whose program variables take values in finite domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Unlike finite-state transition systems, the problem of checking whether a pushdown sys- tem satisfies a HyperCTL* formula is undecidable [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' However, we identify a natural class of hyperproperties for which the model checking problem becomes decidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' As we shall shortly see, this class of hyperproperties not only enjoys decidability, but is also useful in reasoning about security hyperproperies such as noninference and observational determinism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We consider a restricted class of hyperproperties for recursive programs, which relate only executions that access the call stack in the same manner, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', push or pop at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' An execution of a pushdown system P is a sequence of configurations (control state + stack) σ = c1c2 · · · , such that the stacks of consecutive configurations ci and ci+1 differ only due to the possible presence of an additional element at the top of the stack of either ci or ci+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For such a sequence, we can associate a sequence pr(σ) = o1o2 · · · such that oi ∈ {call, int, ret} such that oi = call (ret respectively) if and only if the stack in ci+1 has one more (less respectively) element than ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The sequence pr(σ) is said to be the stack access pattern of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observe that the stack sizes of two executions with the same stack access pattern evolve in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Thus, equivalently, we can consider this class of hyperproperties to be the hyperprop- erties that relate executions with identical memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To specify these hyperproperties, we propose the logic sHCTL* which ex- tends HyperCTL*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' sHCTL* has a two level syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' At the innermost level, the syntax is identical to that of HyperCTL* formulas, but is interpreted over executions of the pushdown system with the same stack access pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' At the outer level, we quantify over different stack access patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Intuitively, the for- mula Eψ is true if there is a stack access pattern ρ exhibited by the system such that the set of executions with access pattern ρ satisfy the hyperproperty ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The dual formula Aψ, defined as ¬E¬ψ, is true if for each stack access pattern 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ρ exhibited by the system, the set of all executions with stack access pattern ρ satisfy ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The syntax of sHLTL is obtained from HyperLTL in a similar fash- ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Please see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 1 on Page 8 for a side-by-side comparison of the syntax of HyperCTL* (HyperLTL) and sHCTL* (sHLTL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Unlike HyperCTL*, we show that the problem of checking sHCTL* is decidable for pushdown systems (Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Formal definitions of stack access patterns, syntax and semantics of sHCTL* are in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For the rest of the paper, hyperproperties expressible in sHCTL* will be called stack-aware hyperproperties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Restricting to stack-aware hyperproperties is useful in verifying security guarantees of recursive programs as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Proving ∀∃∗ hyperproperties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The noninference property (Example 1) can be expressed in HyperLTL as NI def = ∀π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='′(G λπ′) ∧ π ≡L π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Consider the sHLTL formula A(NI) obtained by putting an A in front NI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A pushdown sys- tem satisfies A(NI) only if for each execution σ of the system, there is another execution σ′ with the same stack access pattern as σ such that σ, σ′ together satisfy (G λσ′) ∧ σ ≡L σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Thus, if the pushdown system satisfies the sHLTL formula A(NI), then it also satisfies noninference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Thus, a decision procedure for sHLTL can be used to prove that a recursive program satisfies noninference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The above observation generalizes to HyperLTL formulas of the form ψ = ∀π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='∃π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ∃πk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='ψ′ — if a system satisfies the sHLTL formula Aψ then it must also satisfy the HyperLTL formula ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Though the model checking problem is undecidable for pushdown systems even when restricted to such HyperLTL formulas, we gain decidability by restricting the search space for π, π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' , πk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Refuting ∀∗ hyperproperties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observational determinism (Example 2) can be written in HyperLTL as OD def = ∀π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ∀π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (π[0] ≡L,in π′[0]) → π ≡L,out π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Consider the sHLTL formula A(OD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A pushdown system fails to satisfy the sHLTL formula A(OD) only if there is a stack access pattern ρ and executions σ1 and σ2 with stack access pattern ρ such that the pushdown system does not satisfy (σ[0] ≡L,in σ′[0]) → σ ≡L,out σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This observation generalizes to HyperLTL formulas of the form ψ = ∀π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ∀πk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='ψ′ — if a pushdown system fails to satisfy the sHLTL formula Aψ then it does not satisfy ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Even though model checking pushdown systems against such restricted specifications is undecidable, our decision procedure can be used to show that a recursive program does not meet such properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Exact verification when stack access pattern is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Often, it is reasonable to assume that the attacker can observe the stack access pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For example, in the program counter security model [15], the attacker has access to the program counter transcript, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', the sequence of program counters during an execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Access to the program counter transcript implies that the attacker can observe stack access pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The assumption that the program counter tran- script is observable helps model control flow side channel attacks which include timing attacks and error disclosure attacks [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' sHCTL* can be used to verify security guarantees in this security model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For example, the sHCTL* formula A( NI) models noninference faithfully by introducing a unique proposition for Stack-Aware Hyperproperties 7 each control state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observational determinism can also be verified in this model by suitably transforming the pushdown automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Another scenario in which stack access patterns are observable is when the attacker can observe the memory usage of a program in terms of stack size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' As observing stack size may lead to information leakage, stack size should be considered a low-level observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since the stack size can be unbounded, it cannot be modeled as a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' sHCTL*, however, can still be used to verify security guarantees in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For example, A( NI) = A(∀π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='′(G λπ′) ∧ π ≡L π′) faithfully models non-inference as semantics of sHCTL* forces π and π′ to have the same call-stack size in addition to other low-level observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Once again, observational determinism can also be verified in this model by suitably transforming the pushdown automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 3 Stack-aware Hyper Computation Tree Logic (sHCTL*) Stack-aware Hyper Computation Tree Logic (sHCTL*), and its sub-logic Stack- aware Hyper Linear Temporal Logic (sHLTL) are formally presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We begin by establishing some conventions over strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A string/word w over a finite alphabet Σ is a sequence w = a0a1 · · · of finite or infinitely many symbols from Σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', ai ∈ Σ for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The length of a string w, denoted |w|, is the number of symbols appearing in it — if w = a0a1 · · · an−1 is finite then |w| = n, and if w = a0a1 · · · is infinite then |w| = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The unique string of length 0, the empty string, is denoted ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For a string w = a0a1 · · · ai · · · , w(i) = ai denotes the ith symbol, w[ : i] = a0a1 · · · ai−1 denotes the prefix of length i, w[i : ] = aiai+1 · · · denotes the suffix of w starting at position i, and w[i : j] = aiai+1 · · · aj−1 denotes the substring from position i (included) to position j (not included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Thus w[0 : ] = w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' By convention, when i ≤ 0, we take w[ : i] = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Over Σ, the set of all finite strings is denoted Σ∗, and the set of all infinite strings is denoted Σω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For a finite string u and a (finite or infinite) string v, uv denotes the concatenation of u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='1 Pushdown Systems Pushdown systems naturally model for sequential recursive programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Formally, an AP-labeled pushdown system is a tuple P = (S, Γ, sin, ∆, L), where S is a finite set of control states, Γ is a finite set of stack symbols, sin ∈ S is the initial control state, L : S → 2AP is the labeling function, and ∆ is the transition relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The transition relation ∆ = ∆int ∪· ∆call ∪· ∆ret is the disjoint union of internal transitions ∆int ⊆ S × S where the stack is unchanged, call transitions ∆call ⊆ S × (S × Γ) where a single symbol is pushed onto the stack, and return transitions ∆ret ⊆ (S × Γ) × S where a single symbol is popped from the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' When AP is clear from the context, we simply refer to them as pushdown systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Transition System Semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We recall the standard semantics of a push- down system as a transition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let us fix a pushdown system P = (S, Γ, sin, ∆, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A configuration c of P is a pair (s, α) where s ∈ S and α ∈ Γ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' a ∈ AP, π ∈ V ψ ::= aπ | ¬ψ | ψ ∨ ψ | Xψ | ψ U ψ | ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ (a) HyperCTL* θ ::= Eψ | ¬θ | θ ∨ θ ψ ::= aπ | ¬ψ | ψ ∨ ψ | Xψ | ψ U ψ | ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ (b) sHCTL* Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 1: BNF for HyperCTL* and sHCTL*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let ∀ denote ¬∃¬ and A denote ¬E¬ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' HyperLTL is the set of HyperCTL* formulas Q1π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' · · · Qrπr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='ψ where Qi ∈ {∃, ∀} and ψ is quantifier-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' sHLTL is the set of sHCTL* formulas qϕ, where q ∈ {A, E} and ϕ is in HyperLTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The set of all configurations of P will be denoted ConfP = S × Γ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The labeled transition system associated with P is �P� := (ConfP, cin, −→, AP, L) where cin = (sin, ε) is the initial configuration, −→⊆ ConfP × ({call, ret, int} × S × (Γ ∪{ε})×S)×ConfP is the transition relation, and L is the labeling function that extends the labeling function of P to configurations as follows: L(s, α) = L(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The transition relation −→ is defined to capture the informal semantics of inter- nal, call, and return transitions — for any α ∈ Γ ∗, (int) (s, α) (int,s,ε,s′) −−−−−−→ (s′, α) iff (s, s′) ∈ ∆int;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (call) (s, α) (call,s,a,s′) −−−−−−−→ (s′, aα) iff (s, (s′, a)) ∈ ∆call;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' and (ret) (s, aα) (ret,s,a,s′) −−−−−−→ (s′, α) iff ((s, a), s′) ∈ ∆ret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A path of �P� is an infinite sequence of configurations σ = c0, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' such that for each i, ci (o,s,a,s′) −−−−−−→ ci+1 for some o ∈ {int, call, ret}, s, s′ ∈ S and a ∈ Γ ∪ {ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The path σ is said to start in configuration c0 (the first configuration in the sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We will use Paths(�P�, c) to denote the set of paths of �P� starting in the configuration c and Paths(�P�) to denote all paths of �P�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We conclude this section by introducing some notation on configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For c = (s, α), its stack height is |α|, its control state is state(c) = s, and its top of stack symbol is top(c) = a ∈ Γ if α = aα′ and is undefined if α = ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='2 Syntax of sHCTL* Let us fix a set of atomic propositions AP, and a set of path variables, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The BNF grammar for sHCTL* formulas is given in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In the BNF grammar, a ∈ AP is an atomic proposition, π is a path variable, ψ is a cognate formula, and θ is a sHCTL* formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The syntax has two levels, with the inner level identical to HyperCTL* formulas, while the outer level allows quantification over different stack access patterns (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Also, following [5,10], we assume that the until operator U only occurs within the scope of a path quantifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We have chosen to not have A, the dual of E, and conjunction as explicit logical operators to keep our exposition simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This choice does makes the automata constructions presented here less efficient for formulas involving Stack-Aware Hyperproperties 9 conjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Adding them explicitly does not pose a technical challenge to our setup and our automata constructions can be extended to handle them explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In addition, we will sometimes use other quantifiers and logical operators to write formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Some standard examples include: θ1 ∧ θ2 = ¬(¬θ1 ∨ ¬θ2), where θi (i ∈ {1, 2}) is either a sHCTL* or cognate formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ∀π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='ψ = ¬ ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ¬ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' F ψ = true U ψ, where true = aπ ∨ ¬aπ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' G ψ = ¬ F ¬ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We call formulas of the form qψ (where q ∈ {A, E} and ψ is a cognate formula) basic formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observe that any sHCTL* formula is a Boolean com- bination of basic formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A sHCTL* formula θ is a sentence if in each basic sub-formula qψ, ψ is a sentence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', every path variable appearing in ψ is quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Without loss of generality, we assume that in any cognate formula ψ, all bound variables in ψ are renamed to ensure that any path variable is quanti- fied at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We will only consider sHCTL* sentences in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The logic sHLTL is the sub-logic of sHCTL* consisting of all formulas of the form qQ1π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' · · · Qrπr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='ψ where q ∈ {A, E}, Qi ∈ {∃, ∀} and ψ is quantifier free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='3 Semantics of sHCTL* The syntax of cognate formulas is identical to that HyperCTL* formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Their semantics will be described in a similar manner, in a context where free path variables in the formula are interpreted as executions of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' However, we will require that the interpretations of every path variable share a common stack access pattern — hence the term cognate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Thus, before defining the semantics, we will define what we mean by the stack access pattern of a path and a path environment that assigns an interpretation to path variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For the rest of this section let us fix a pushdown system P = (S, Γ, sin, ∆, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A string w ∈ {call, int, ret}∗ is said to be well matched if either w = ε or w = int or w = call u ret or w = uv, where u, v ∈ {call, int, ret}∗ are (recursively) well matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In a string ρ ∈ {call, int, ret}ω, ρ(i) is an unmatched return, if ρ[ : i + 1] = w ret, where w is well matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We are now ready to present the definition of a stack access pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Definition 1 (Stack access pattern).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A string ρ ∈ {call, int, ret}ω is a stack access pattern if the set {i ∈ N | ρ(i) is an unmatched return} is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A path σ = c0c1c2 · · · ∈ Paths(�P�) is said to have a stack access pattern ρ = o0o1 · · · (denoted pr(σ) = ρ) if for every i: (a) oi = call if and only if stack(ci+1) = top(ci+1) stack(ci), (b) oi = int if and only if stack(ci+1) = stack(ci), and (c) oi = ret if and only if stack(ci) = top(ci) stack(ci+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We now present the definition of path environment that interprets the free path variables in a cognate formula as paths of �P� such that they share a common stack access pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This plays a key role in defining the semantics of sHCTL*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For a set of path variables V, let V† be defined as the set V ∪· {†}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Definition 2 (Path Environment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A path environment for pushdown sys- tem P over variables V is function Π : V† → Paths(�P�) ∪{call, int, ret}ω such 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' that Π(†) is a stack access pattern , and for every π ∈ V, Π(π) ∈ Paths(�P�) with pr(Π(π)) = Π(†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' When the pushdown system is clear from the context, we will simply refer to it as a path environment over V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' When V = ∅, we additionally require that there is a path σ ∈ Paths(�P�, cin) (where cin is the initial configuration of �P�) such that pr(σ) = Π(†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We introduce some notation related to path environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let us fix a path environment Π over variables V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Given a path σ ∈ Paths(�P�), Π[π �→ σ] denotes the path environment over V ∪{π} such that Π[π �→ σ](π) = σ, and Π[π �→ σ](π′) = Π(π′), for any π′ ∈ V† with π′ ̸= π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Finally, for i ∈ N, Π[i : ] denotes the suffix path environment, where every variable is mapped to the suffix of the path starting at position i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' More formally, for every π′ ∈ V†, Π[i : ](π′) = Π(π′)[i : ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We now define when a pushdown system P satisfies a sHCTL* sentence θ, denoted P |= θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The definition of satisfaction of θ relies on a definition of satis- faction for cognate formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To inductively to define the semantics of cognate formulas, we will interpret free path variables using a path environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Fi- nally, as in HyperCTL*, it is important to track the most recently quantified path variable because that influences the semantics of ∃π(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Thus satisfaction of cognate formulas takes the form P, Π, π′ |= ψ, where π′ is the most recently quan- tified path variable, and Π is a path environment over the free variables of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Finally, by convention, we will take Paths(�P�, Π(†)(0)) to mean Paths(�P�, cin), where cin is the initial configuration of �P� 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Below, θ, θ1, and θ2 are sHCTL* sentences, while ψ, ψ1, ψ2 are cognate formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' P |= ¬θ iff P ̸|= θ P |= θ1 ∨ θ2 iff P |= θ1 or P |= θ2 P |= Eψ iff for some path environment Π over ∅, P, Π, † |= ψ P, Π, π′ |= aπ iff a ∈ L(Π(π)(0)) P, Π, π′ |= ¬ψ iff P, Π, π′ ̸|= ψ P, Π, π′ |= ψ1 ∨ ψ2 iff P, Π, π′ |= ψ1 or P, Π, π′ |= ψ2 P, Π, π′ |= Xψ iff P, Π[1 : ], π′ |= ψ P, Π, π′ |= ψ1 U ψ2 iff ∃i ≥ 0 : P, Π[i : ], π′ |= ψ2 and ∀j, 0 ≤ j < i, P, Π[j : ], π′ |= ψ1 P, Π, π′ |= ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ iff ∃σ ∈ Paths(�P�, Π(π′)(0)) with pr(σ) = Π(†), such that P, Π[π �→ σ], π |= ψ 4 A Decision Procedure for sHCTL* Given a pushdown system P and a sHCTL* sentence θ, we present an algorithm that determines if P |= θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our approach is similar to the one in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Given a finite state transition system K and a HyperCTL* formula ϕ, Finkbeiner et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [10], construct an alternating (finite state) Büchi automaton AK,ϕ, by induction on ϕ, such that an input word σ is accepted by AK,ϕ if and only if σ is the encoding 5 The convention is needed because Π(†)(0) is not a configuration but an element of the set {call, int, ret}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Stack-Aware Hyperproperties 11 of a path environment Π such that K, Π |= ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Determining if K |= ϕ then reduces to checking if AK,ϕ accepts any string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Extending these ideas to sHCTL* and pushdown systems, requires one to answer two questions: (a) What is an encoding of path environments for cog- nate formulas where path variables are mapped to sequences of configurations (control state + stack)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (b) Which automata models can capture the collection of path environments satisfying a cognate formula with respect to a pushdown system?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We encode path environments for cognate formulas using strings over a pushdown alphabet — pushdown tags on symbols adds structure that helps encode sequences of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' And for automata, we consider automata that process such strings and accept visibly pushdown languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A natural gen- eralization of the approach outlined in [10] would suggest the use of alternating visibly pushdown automata (AVPA) on infinite strings [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' However, using AV- PAs results in an inefficient algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To get a more efficient algorithm, we instead rely on a careful use of nondeterministic visibly pushdown automata (NVPA) [1] and 1-way alternating jump automata (1-AJA) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The advantage of using NVPA and 1-AJA can be seen in the case of existential quantification (∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=') which requires converting an alternating automaton to a nondeterministic one [10]: Converting from 1-AJA to NVPA leads to exponential blowup while converting AVPA to NVPA leads to a doubly exponential blowup [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The rest of this section is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We begin by introducing the automata models on pushdown alphabets (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Next we present our encoding of path environments, and finally our automata constructions that establish the decidability result (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='1 Automata on Pushdown Alphabets A pushdown alphabet is a finite set Σ that is partitioned into three sets Σcall ∪· Σint ∪· Σret, where Σcall is the set of call symbols, Σint is the set of inter- nal symbols, and Σret is the set of return symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Automata models processing strings over a pushdown alphabet are restricted to perform certain types of tran- sitions based on whether the read symbol is a call, internal, or return symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We introduce, informally, two such automata models next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Precise definition and its semantics can be found in Appendix B and Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Nondeterministic Visibly Pushdown Büchi Automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A nondetermin- istic visibly pushdown automaton (NVPA) [1] is like a pushdown system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' It has finitely many control states and uses an unbounded stack for storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' However, unlike a pushdown system, it is an automaton that processes an infinite sequence of input symbols from a pushdown alphabet Σ = Σcall ∪· Σint ∪· Σret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Transitions are constrained to conform to pushdown alphabet — whenever a Σcall symbol is read, a symbol onto the stack, whenever a Σret symbol is read, the top stack symbol is popped, and whenever Σint symbol is read, the stack is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 1-way Alternating Jump Automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our second automaton model is 1- way Alternating Parity Jump Automata (1-AJA) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 1-AJA are computation- ally equivalent to NVPAs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', accept the same class of languages) but provide 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' greater flexibility in describing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 1-AJAs are alternating automata, which means that they can define acceptance based on multiple runs of the ma- chine on an input word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Though they are finite state machines with no auxiliary storage, their ability to spawn a computation thread that jumps to a future portion of the input string on reading a symbol, allows them to have the same computational power as a more conventional machine with storage (like NVPAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We present some useful properties of NVPA and 1-AJA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The two models are equi-expressive with the size of automata constructed by the translation known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Theorem 1 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For any NVPA N of size n, there is a 1-AJA AN of size O(n2), such that L(AN) = L(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Conversely, for any 1-AJA A of size n, there is a NVPA NA of size 2O(n), such that L(NA) = L(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Constructions can be carried out in time proportional to the size of the resulting automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Both 1-AJA and NVPAs are closed for language operations like complemen- tation, union and prefixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We also recall the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ([1]) For NVPAs, the emptiness problem is PTIME-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='2 Algorithm for sHCTL* Let us fix a pushdown system P = (S, Γ, sin, ∆, L) and a sHCTL* sentence θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our goal is to decide if P |= θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We will reduce this problem to checking the empti- ness of multiple NVPAs (Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our approach is similar to [10] — for each cognate sub-formula ψ (not necessarily sentence) of θ, we will compositionally construct an automaton that accepts the path environments satisfying ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Path environments will be encoded by strings over pushdown alphabets as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For a path σ = c0c1c2 · · · of �P�, the trace of σ, denoted tr(σ), is the (unique) sequence (o0, q0, a0, q1)(o1, q1, a1, q2) · · · such that for every i ∈ N, ci (oi,qi,ai,qi+1) −−−−−−−−−→ ci+1 where oi ∈ {call, int, ret}, qi, qi+1 ∈ Q, and ai ∈ Γ ∪ {ε} 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' While tr(σ) is uniquely determined by the path σ, the converse is not true — different paths may have the same trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To see this, consider the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For configuration c and γ ∈ Γ ∗, let γ(c) denote the configuration (state(c), stack(c)γ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', the configuration with the same control state, but with stack containing the symbols in γ at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observe that, for any γ ∈ Γ ∗, if σ = c0c1c2· is a path then so is γ(σ) = γ(c0)γ(c1)γ(c2) · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Additionally, tr(σ) = tr(γ(σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Two paths σ1 and σ2 of �P� will be said to be equivalent if tr(σ1) = tr(σ2) and will be denoted as σ1 ≃ σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observe that equivalent paths have the same stack access pattern , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' if σ1 ≃ σ2 then pr(σ1) = pr(σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The semantics of sHCTL* doesn’t distinguish between equivalent paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 6 Observe that even when σ is not a path in �P� (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', corresponds to an actual se- quence of transitions of P), the trace of σ is uniquely defined as long as stacks of successive configurations of σ can be obtained by leaving the stack unchanged, or pushing/popping one symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Stack-Aware Hyperproperties 13 Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let ϕ be a cognate formula with V as the set of free path vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let Π1 and Π2 be two path environments such that for every π ∈ V, Π1(π) ≃ Π2(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Then, P, Π1, π |= ϕ if and only if P, Π2, π |= ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The proof of Proposition 1 follows by induction on cognate formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Propo- sition 1 establishes that the set of path environments satisfying a cognate for- mula is a union of equivalence classes with respect to path equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Thus, instead of constructing automata that accept path environments, we will con- struct automata that accept mappings from path variables to traces of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For m ∈ N, let Σ[m] = Σ[m]call ∪· Σ[m]int ∪· Σ[m]ret be the pushdown alpha- bet where Σ[m]call = {call} × Sm × Γ m, Σ[m]int = {int} × Sm × {ε}m, and Σ[m]ret = {ret} × Sm × Γ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observe Σ[0] is (essentially) the set {int, call, ret}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Definition 3 (Encoding Path Environments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Consider a set of m path variables V = {π1, π2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' πm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A string w ∈ Σ[m]ω where for any j ∈ N, w(j) = (oj, (sj 1, sj 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' sj m), (aj 1, aj 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' aj m)) encodes all path environments Π such that Π(†) = o0o1o2 · · · oj · · · tr(Π(πi)) = (o0, s0 i , a0 i , s1 i )(o1, s1 i , a1 i , s2 i ) · · · for any i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The string encoding a path environment Π is denoted as enc(Π) (= w, in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Based on the definitions, the following observation about traces and encod- ings can be concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For any path σ ∈ Paths(�P�) and i ∈ N, tr(σ[i : ]) = tr(σ)[i : ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For any path environment Π and i ∈ N, enc(Π[i : ]) = enc(Π)[i : ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The encoding of path environments as strings over Σ[m] (for an appropriate value of m) is used in our decision procedure, which compositionally constructs automata that accept path environments satisfying each cognate formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The size of our constructed automata, like in [10], will be tower of exponentials that depends on the formula complexity of the cognate formula ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Definition 4 (Formula Complexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The formula complexity of a sHCTL* formula ϕ, denoted fc(ϕ), is inductively defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let odd : N → N be the function that maps a number n to the smallest odd number ≥ n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', odd(n) = n if n is odd and odd(n) = n + 1 if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Similarly, even : N → N maps n to the smallest even number ≥ n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', even(n) = odd(n + 1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Below ψ1, ψ2 denote cognate formulas, and θ1, θ2 denote sHCTL* sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' fc(aπ) = 0 fc(¬ψ1) = even(fc(ψ1)) fc(Xψ1) = fc(ψ1) fc(ψ1 ∨ ψ2) = max(fc(ψ1), fc(ψ2)) fc(ψ1 U ψ2) = even(max(fc(ψ1), fc(ψ2))) fc(∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ1) = odd(fc(ψ1)) fc(Eψ1) = odd(fc(ψ1)) fc(¬θ1) = fc(θ1) fc(θ1 ∨ θ2) = max(fc(θ1), fc(θ2)) Observe the difference in the definition of fc(¬θ1) and fc(¬ψ1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' for ¬θ1 there is no change in formula complexity, while for ¬ψ1 we move to the next even level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our main technical lemma is a compositional construction of an automaton for cognate formulas ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Depending on the parity of fc(ψ), the automaton we construct will either be a 1-AJA or a NVPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Before presenting this lemma, we define a function that is a tower of exponentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For c, k, n ∈ N, the value gc(k, n) is defined inductively on k as follows: gc(0, n) = cn log n, and gc(k + 1, n) = 2gc(k,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We use gO(1)(k, n) to denote the family of functions {gc(k, n) | c ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Consider pushdown system P = (S, Γ, sin, ∆, L) and sHCTL* sen- tence θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let ψ be a cognate subformula of θ with free path variables in the set V = {π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' πm} for m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We assume, without loss of generality, that the vari- ables π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' πm are in the order in which they are quantified in θ with πm being the first free variable of ψ that will be quantified in the context θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In addition, we assume that the size of both ψ and P is bounded by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' There is an automaton Aψ over pushdown alphabet Σ[m] such that for any path environment Π over V, P, Π, πm |= ψ if and only if enc(Π) ∈ L(Aψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 7 The automaton Aψ is a NVPA if fc(ψ) is odd, and a 1-AJA if fc(ψ) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The size of Aψ is at most gO(1)(⌈ fc(ψ) 2 ⌉, n)8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Before presenting the proof of Lemma 1, we would like to highlight a subtlety about its statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The result guarantees that for valid path environments Π, encoding enc(Π) is accepted by Aψ if and only if Π satisfies ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' It says nothing about path environments that are not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In particular, there may be functions that map path variables to traces that do not correspond to actual paths of �P�, but which are nonetheless accepted by Aψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Notice, however, when ψ = ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ1 is a cognate sentence, a string over {call, int, ret} will, by conditions guaranteed in Lemma 1, be accepted if and only if it corresponds to a stack access pattern of a path from the initial state that satisfies ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Proof (Sketch of Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our construction of Aψ will proceed inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The type of automaton constructed will be consistent with the parity of fc(ψ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', an NVPA if fc(ϕ) is odd and a 1-AJA if fc(ψ) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We sketch the main ideas here, with the full proof available in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For aπ, ¬ψ1, ψ1 ∨ ψ2, and Xψ1, the construction essentially proceeds by con- verting Aψi (i ∈ {1, 2}) if needed, into the type (NVPA or 1-AJA) of the target automaton using Theorem 1, and then using standard closure properties to com- bine them to get the desired automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In case of ψ = ψ1 U ψ2, we first convert (if needed) Aψi (i ∈ {1, 2}) into a 1-AJA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' At each step, the automaton for ψ will choose to either run Aψ2, or run Aψ1 and restart itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Correctness relies on the fact that our encoding for path environments satisfies Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The most interesting case is that of ψ = ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We will first convert (if needed) the automaton for ψ1 into a NVPA A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The automaton for ψ will essentially guess the encoding of a path that is consistent with the transitions of 7 When m = 0, we take πm to be †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 8 When the size of the specification ψ is considered constant, the size of Aψ is at most gO(1)(⌈ fc(ψ) 2 ⌉ − 1, n) Stack-Aware Hyperproperties 15 P, and check if assigning the guessed path to variable π satisfies ψ1 by running the automaton A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The additional requirement we have is that the guessed path start at the same configuration as the current configuration of the path assigned to variable πm which introduces some subtle challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In order to be able to guess a path, Aψ will keep track of P’s control state in its control state, and use its stack to track P’s stack operations along the guessed path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since the stacks of all paths are synchronized, it makes it possible for Aψ to use its (single stack) to track the stack of both P and the stack of A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ⊓⊔ Using Lemma 1, we can establish the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Given a P = (S, Γ, sin, ∆, L) and a sHCTL* sentence θ, the prob- lem of determining if P |= θ is in ∪cDTIME(gc(⌈ fc(θ) 2 ⌉, n)), where n is a bound on the size of P and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Recall that a sHCTL* sentence is a Boolean combination of formulas of the form Eψ, where ψ is a cognate sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Results on whether P |= Eψ for each such subformula can be combined to determine whether P |= θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Given this, the time to determine if P |= θ is at most the time to decide if P satisfies each subformula of the form Eψ plus O(n) (to compute the Boolean combination of these results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Next, recall that the construction in Lemma 1 ensures that for a cognate sentence of the form ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ, L(A∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ) consists exactly of strings in {call, int, ret}ω that encode a path environment over ∅ that satisfy ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Consider a sHCTL* sentence Eψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let π be a path variable that does not appear in the sentence ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Based on the semantics of sHCTL* the following observation holds: P |= Eψ if and only if for some path environment Π over ∅, P, Π, † |= ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Which is equivalent to saying that P |= Eψ if and only if L(A∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since fc(Eψ) = fc(∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ), and the emptiness problem of NVPA can be decided in polynomial time (Theorem 2), our theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ⊓⊔ 5 Lower Bound In this section, we establish a lower bound for the problem of model checking sHCTL* sentences against pushdown systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our proof establishes a hardness result for the sHLTL sub-fragment of sHCTL*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Before presenting this lower bound, we introduce the function hc(·, ·), which is another tower of exponentials, inductively defined as follows: hc(0, n) = n, and hc(k + 1, n) = hc(k, n) · chc(k,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let P be a pushdown system and θ be a sHLTL sentence such that the sizes of both P and θ is bounded by n and fc(θ) = 2k − 1 for some k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The problem of checking if P |= θ is DTIME(hc(k, n))-hard, for every c ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Proof (Sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We sketch the main intuitions behind the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To highlight the novelties of this proof, it is useful to recall how NSPACE(hc(k−1, n))-hardness for HyperLTL model checking is proved [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The idea is to reduce the language of a nondeterministic hc(k−1, n) space bounded machine M to the model checking 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' problem by constructing a finite state transition system that guesses a run of M, and a HyperLTL formula that checks if the path is a valid accepting run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To get the stricter bound of DTIME(hc(k, n)), we use the fact that we are checking pushdown systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The stack of the pushdown system can be used to guess a tree, as opposed to a simple trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Therefore, we reduce a hc(k − 1, n) space bounded alternating Turing machine, instead of a nondeterministic machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since ASPACE(f(n)) = DTIME(2O(f(n))) for f(n) ≥ log n, the theorem will follow if the reduction succeeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Recall that a run of an alternating Turing machine M is a rooted, labeled tree, where vertices are labeled by configurations of M in a manner that is consistent with the transition function of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To faithfully encode a tree as a sequence of symbols, we record the DFS traversal of the tree, making explicit the stack operations performed during such a traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Consider a labeled, rooted tree T with root r whose label is ℓ(r) with T1 as a the left sub-tree and T2 as the right sub-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The DFS traversal of T will push ℓ(r), traverse T1 recursively, pop ℓ(r), push ℓ(r), traverse T2, and then pop ℓ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We will use such a DFS traversal to guess and encode runs of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Popping and pushing ℓ(r) between the traversals of T1 and T2 may seem redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Why not simply do nothing between the traversals of T1 and T2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For T to be a valid run of M, the configuration labeling of the root of T2 must be the result of taking one step from ℓ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Such checks will be encoded in our sHLTL sentence, and for that to be possible, we need successive configurations of M to be consecutive in the string encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To highlight some additional consistency checks, let us continue with our example tree T from the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For a string to be a correct encoding of T , it is necessary that the string pushed before the traversal of Ti (i ∈ {1, 2}) be the same as the string popped after the traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This can be ensured by the pushdown system by actually pushing and popping those symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In addition, the string popped after T1’s traversal must be the same as the string pushed before T2’s traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Neither the stack nor the finite control of the pushdown system can be used to ensure this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Instead this must be checked by the sHLTL sentence we construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' But the symbols while popping ℓ(r) will be in reverse order of the symbols being pushed, and it is challenging to perform this check in the formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To overcome this, we push/pop the label and its reverse at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This ensures that if we want to check if a string pushed is the same as a string that was just popped, then we can check for string equality, and this check is easier to do using formulas in sHLTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Additional checks to ensure that the tree encodes a valid accepting run are performed by the sHLTL sentence using ideas from [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Full details can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ⊓⊔ 6 Conclusions In this paper, we introduced a branching time temporal logic sHCTL* that can be used to specify synchronous hyperproperties for recursive programs modeled as pushdown systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The primary difference from the standard branching time logic HyperCTL* for synchronous hyperproperties is that sHCTL* considers Stack-Aware Hyperproperties 17 a restricted class of hyperproperties, namely, those that relate only executions that the same stack access pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We call such hyperproperties stack-aware hyperproperties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We showed that the problem of model checking pushdown sys- tems sHCTL* specifications is decidable, and characterized its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We also showed how this result can potentially be used to aid security verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Alur, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', Madhusudan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=': Visibly pushdown languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' pp.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=': Model-checking HyperLTL for pushdown systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In: Model Checking Software - 25th International Symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 133–152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Springer (2018) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Sistla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', Vardi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', Wolper, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=': The complementation problem for büchi automata with appplications to temporal logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Theoretical Computer Science 49, 217–237 (1987) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Walukiewicz, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=': Pushdown processes: Games and model checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In: Computer Aided Verification, 8th International Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 62–74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Springer (1996) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Zdancewic, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', Myers, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=': Observational determinism for concurrent program security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In: Proceedings of the 16th IEEE Computer Security Foundations Work- shop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' IEEE Computer Society (2003) A Observational determinism when the call stack size is visible to the attacker Observational determinism [13,19] states that any two executions that have the same low-level initial inputs must have the same low-level output observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observational determinism is a hypersafety property [5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' As [5] shows, ob- servational determinism is also expressible in HyperLTL using the formula: OD def = ∀π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ∀π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (π[0] ≡L,in π′[0]) → π ≡L,out π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Here ≡L,in and ≡L,out express the fact that π and π′ have the same low-security inputs and outputs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In order to see how we can verify observational determinism when call stack sizes are observable, let us consider the simple case when the call stack size is the only low-security observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let P be the pushdown automaton corresponding to a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We can assume by loss of generality that the states of P encode if the stack is empty, and that pop transitions occur only in states where the stack is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='9 Let ∆P be the pushdown automaton obtained from P which behaves like P, except that it has additional nondeterministic transitions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ∆P has three new states qcall, qint and qret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We also assume there are four new propositions: new, call, int and ret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The proposition new is true in qcall, qint and qret only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The only transitions in the new states are self-loops that do not change the stack size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Whenever P has a push/internal/pop transition from a state q to q′ in P, we add an internal nondeterministic transition from q to qcall/qint/qret that does not change the stack size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Furthermore, the proposition call/int/ret is 9 Essentially, initially the stack is taken to be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' When a symbol is pushed onto an empty stack, it is “annotated ” with information that it is the bottom of the stack, and the state remembers that stack is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' When an “annotated ” symbol is popped, the stack now remembers that the stack is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Stack-Aware Hyperproperties 19 true in q in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Consider the formula A ∀π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ∀π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='′(π[0] ≡L,in π′[0]) → ((newπ ∨ newπ′) R ((callπ ⇒ callπ′ ∧ ¬intπ′ ∧ ¬retπ′) ∧ (intπ ⇒ intπ′ ∧ ¬callπ′ ∧ ¬retπ′) ∧ (retπ ⇒ retπ′ ∧ ¬callπ′ ∧ ¬intπ′))) where R is the dual of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' More precisely, ψ R ψ′ is ¬(¬ψ′) U(¬ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observe that the above formula is not satisfied by the pushdown system ∆P if and only if there are two finite executions σ1 and σ2 of P leading to states q1 and q2 such that (a) the low-level inputs in σ1 and σ2 are the same, (b) σ1 and σ2 have the same stack access pattern, and (c) a push/internal/pop can be executed from only one of the states q1 and q2, but not the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' B Nondeterministic Visibly Pushdown Automata (NVPA) A nondeterministic visibly pushdown automaton (NVPA) [1] is like a push- down system in that it has finitely many control states and uses an unbounded stack for storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' However, unlike a pushdown system, it is an automaton that processes an infinite sequence of input symbols from a pushdown alpha- bet Σ = Σcall ∪· Σint ∪· Σret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Furthermore, transitions are constrained to conform to pushdown alphabet — whenever the automaton reads from Σcall, it pushes a symbol onto its stack, whenever it reads from Σret, it pops its top stack symbol, and whenever it reads from Σint, it leaves its stack unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Definition 5 (NVPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A nondeterministic visibly pushdown Büchi automa- ton (NVPA) is a tuple N = (Q, qin, Σ, Γ, ⊥, δ, QF) where, Q is a finite set of control states, qin ∈ Q is the initial control state, Σ = Σcall ∪· Σint ∪· Σret is a pushdown alphabet that is used to encode inputs, Γ is a finite set called the stack alphabet, ⊥ ̸∈ Γ is the bottom of stack symbol, QF ⊆ Q is the set of accepting states, and δ is the transition relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' δ will be assumed to be partitioned into three as δcall ∪· δret ∪· δint where δcall ⊆ (Q × Σcall × Q × Γ) is the transition on call symbols, δint ⊆ (Q × Σint × Q) is the transition on internal symbols, and δret ⊆ (Q × Σret × (Γ ∪ ⊥) × Q) is the transition on return symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let us fix an NVPA N = (Q, qin, Σ, Γ, ⊥, δ, QF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A configuration of N, like in the case of a pushdown system, is a pair of the form (q, α⊥) where q ∈ Q and α ∈ Γ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The functions state(·), stack(·), and top(·) are defined in the same manner as for pushdown system configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A run of N on an input string w ∈ Σω is an infinite sequence of configurations c0, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' such that c0 = (qin, ⊥) is the initial configuration of N, and successive configurations are consistent with the input symbol read and the transition relation of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' More precisely, for every i ∈ N, (a) if w(i) ∈ Σcall then (state(ci), w(i), state(ci+1), top(ci+1)) ∈ δcall and stack(ci+1) = top(ci+1)stack(ci);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (b) if w(i) ∈ Σint then 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (state(ci), w(i), state(ci+1)) ∈ δint and stack(ci+1) = stack(ci);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' and (c) if w(i) ∈ Σret then (state(ci), w(i), top(ci), state(ci+1)) ∈ δret and additionally, stack(ci+1) = stack(ci) = ⊥ if top(ci) = ⊥ and stack(ci) = top(ci)stack(ci+1) if top(ci) ̸= ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' It is worth observing that transitions on call symbols push a symbol onto the stack, transitions on internal symbols leave the stack unchanged, and transitions on return symbols typically pop the top of the stack unless the stack is empty (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' = ⊥) in which case they leave the stack unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A run c0, c1, · · · on input w is said to be accepting if it satisfies the Büchi acceptance condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', for some q ∈ QF there are infinitely many i such that state(ci) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Finally, the language accepted by NVPA N, denoted L(N), is L(N) = {w ∈ Σω | N has an accepting run on w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' C 1-way Alternating Jump Automata (1-AJA) Our second model of an automaton with strings over a pushdown alphabet, is 1-way Alternating Parity Jump Automata (1-AJA) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 1-AJA are computation- ally equivalent to NVPAs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', accept the same class of languages) but provide greater flexibility in describing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 1-AJAs are alternating automata, which means that they can define acceptance based on multiple runs of the ma- chine on an input word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Though they are finite state machines with no auxiliary storage, their ability to spawn a computation thread that jumps to a future portion of the input string on reading a symbol, allows them to have the same computational power as a more conventional machine with storage (like NVPAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We present this model after some necessary definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Transitions of an alternating automata identify subsets of next steps that must all be accepting for the machine to accept an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We describe these subsets of next steps is using Boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For a set X, let B+(X) denote the set of positive Boolean expressions built using elements of X as propositions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', they consist of propositional logic formulas built using true, false, X, ∧ and ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Given ϕ ∈ B+(X) and A ⊆ X, we say A |= ϕ if ϕ evaluates to true under the valuation that assigns true to the elements of A and false to everything else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The dual of a formula ϕ ∈ B+(X), denoted dual(ϕ), is the formula obtained from ϕ by replacing true by false, false by true, ∨ by ∧, and ∧ by ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' More precisely, dual(·) can be defined inductively as follows, where x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' dual(true) = false dual(false) = true dual(x) = x dual(ϕ ∧ ψ) = ϕ ∨ ψ dual(ϕ ∨ ψ) = ϕ ∧ ψ A 1-AJA is a finite state automaton that reads an input over a pushdown alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In other words, on reading an input symbol, a thread of computation changes its control state based on its current state and symbol read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' However, one of the novel features of a 1-AJA is its ability to jump when it reads a call symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Thus, a transition of a 1-AJA not only specifies what the next state is, but also what the next symbol to be read is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' When the symbol read is either an internal symbol or a return symbol, the symbol to be read next is always the one Stack-Aware Hyperproperties 21 immediately after;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' this is denoted as →.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' However, when the symbol read is a call symbol, the automaton can choose to either read the next symbol or to read the matching return symbol next;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ↷ is used to indicate that the automaton should read the matching return symbol next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We now have all the notation necessary to define a 1-AJA formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Definition 6 (1-AJA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A 1-way Alternating Parity Jump Automaton (1-AJA) is a tuple A = (Q, qin, Σ, δ, parity) where, Q is a finite set of control states, qin ∈ Q is the initial control state, Σ = Σcall ∪· Σint ∪· Σret is a pushdown alphabet that is used to encode inputs, δ : (Q × Σ) → B+(Q × {→, ↷}) is the transition relation with the restriction that for any q ∈ Q and a ∈ Σint∪Σret, δ(q, a) ∈ B+(Q×{→}), and parity : Q → N is the parity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In order to define a run of 1-AJA on an input string, we need to introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let us fix a pushdown alphabet Σ = Σcall ∪· Σint ∪· Σret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We can extend the notion of well matched strings to Σ∗ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A string z ∈ Σ∗ is said to be well matched if either z = ε, or z ∈ Σint, or z = cur, or z = uv, where c ∈ Σcall, r ∈ Σret, and u, v ∈ Σ∗ are (recursively) well matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let us fix an input string w ∈ Σω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The abstract successor of position i in w, denoted ↷ (i), is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ↷ (i) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 j if w[i : j + 1] = c u r where u is well matched, c ∈ Σcall, r ∈ Σret undefined otherwise Notice that ↷ (i) is defined only if w(i) ∈ Σcall, and if defined, its value is the position of its matching return symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The abstract successor identifies the position of the next symbol that will be read if the 1-AJA decides to jump on a call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Next, the local successor of position i in w, denoted → (i), is always defined to be i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let us fix a 1-AJA A = (Q, qin, Σ, δ, parity) and input string w ∈ Σω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A run of A on w is a rooted, labeled tree R = (V, E, r, L), where V is the set of vertices, E the set of edges, r ∈ V is the root, and L : V → (Q × N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The label of vertex indicates the state of A and the position in w that is being read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We require that L(r) = (qin, 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', the automaton is in the initial state and the 0th symbol is being read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let v ∈ V be an arbitrary vertex with L(v) = (q, i) and let C ⊆ V be the set vertices that are children of v in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We require that the labels of v and those of vertices in C are consistent with δ — there is a set A ⊆ (Q × {→, ↷}) such that (a) A |= δ(q, w(i));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (b) for every (q′, d) ∈ A such that d(i) is defined, there is a vertex n ∈ C with L(n) = (q′, d(i));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' and (c) for every vertex n ∈ C, there is (q′, d) ∈ A such that d(i) is defined and L(n) = (q′, d(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A run R = (V, E, r, L) is accepting if every infinite path in R satisfies the parity acceptance condition as defined by parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' That is, for every infinite path σ in R, mp(σ) is even, where mp(σ) = min{parity(q) | for infinitely many i, L(σ(i)) = (q, k) for some k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 22 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Finally, as always, the language recognized by A is given by L(A) = {w ∈ Σω | A has an accepting run on w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' D Proof of Lemma 1 Before beginning the proof of Lemma 1, let us recall why NVPAs and 1-AJAs are closed on standard operations on languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We present these observations with a focus on the size of the resulting automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For a language A ⊆ Σω and subset Γ ⊆ Σ, we take ΓA to be the language {aw | a ∈ Γ and w ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let Ai be a 1-AJA (NVPA) recognizing Li ⊆ Σω of size ni, for i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let Γ ⊆ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Then the following automata can be constructed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' There is a 1-AJA (NVPA) recognizing L1 ∪ L2 of size O(n1 + n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' There is a 1-AJA (NVPA) recognizing ΓL1 of size O(n1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' There is a 1-AJA recognizing (Σω \\ L1) of size O(n1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Proof (Sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The automaton construction that establishes (1) chooses to ei- ther run one of A1 or A2 nondeterministically to recognize L1 ∪ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (2) can be established by checking if the first symbol belongs to Γ (and performing the nec- essary stack operation demanded by the first symbol in the case of NVPA) and then running the automaton for L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Finally, for (3), the 1-AJA recognizing the complement of L1 has the same set of states, the parity of each state is increased by 1, and for any state q and symbol a, δ(q, a) = dual(δ1(q, a)), where δ1 and δ are the transition functions of A1 and the automaton recognizing (Σω \\ L1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our construction of Aψ will proceed inductively based on the structure of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The type of automaton constructed will be consistent with the parity of fc(ψ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', an NVPA if fc(ϕ) is odd and a 1-AJA if fc(ψ) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Case ψ = aπ: Let Γ ⊆ Σ[m] be the set of symbols ζ such that a is in the label set of the πth control state in symbol ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We will construct a 1-AJA that accepts Γ(Σ[m])ω using Proposition 3 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Case ψ = ¬ψ1: We will convert Aψ1 into a 1-AJA (if needed) using Theorem 1, and then use Proposition 3 (3) to construct a 1-AJA that accepts the language L(Aψ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Case ψ = ψ1 ∨ ψ2: Without loss of generality, assume that fc(ψ1) ≥ fc(ψ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We convert (if needed) Aψ2 to an automaton of the same type as Aψ1 using The- orem 1 and then use Proposition 3 (1), to construct an automaton recognizing L(Aψ1) ∪ L(Aψ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Case ψ = Xψ1: We use Proposition 3 (2) to construct an automaton for Σ[m]L(Aψ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Case ψ = ψ1 U ψ2: Using Theorem 1, we will construct (if needed) a 1-AJA equivalent to Aψi and let this be Ai = (Qi, qini, Σ[m], δi, parityi), for i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Stack-Aware Hyperproperties 23 The 1-AJA for ψ is given by Aψ = (Q1 ∪· Q2 ∪· {qin}, qin, Σ[m], δ, parity) where δ(qin, ζ) = δ2(qin2, ζ) ∨((→, qin) ∧ δ1(qin1, ζ)), parity(qin) = 1, and for q ∈ Qi (i ∈ {1, 2}), we have δ(q, ζ) = δi(q, ζ) and parity(q) = parityi(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Case ψ = ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ1: Recall that our pushdown system P has control states S, stack alphabet Γ, and transition relation ∆ = ∆int ∪· ∆call ∪· ∆ret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Using Theorem 1, we will construct (if needed) a NVPA equivalent to Aψ1 and let this be A1 = (Q1, qin1, Σ[m+1], Γ1, ⊥, δ1, QF1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Notice that the input alphabet of A1 is Σ[m+ 1], where the m + 1st component is an encoding of the path assigned to variable π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The automaton for ψ will essentially guess the encoding of a path that is consistent with the transitions of P, and check if assigning the guessed path to variable π satisfies ψ1 by running the automaton A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The additional requirement we have is that the guessed path start at the same configuration as the current configuration of the path assigned to variable πm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In order to be able to guess a path, Aψ will keep track of P’s control state in its control state, and use its stack to track P’s stack operations along the guessed path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Before defining Aψ formally we need to introduce some notation that will be convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' An element ζ ∈ Σ[m] is of the form (o, (s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' sm), (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' am)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We use op(ζ) to denote o, state(ζ)|i to denote si, and stack(ζ)|i to denote ai, for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' By convention we take state(ζ)|0 = sin, and stack(ζ)|0 to be undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Finally, for s ∈ S and a ∈ Γ, ζ + (s, a) is the symbol in alphabet Σ[m + 1] given by (o, (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' sm, s), (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' am, a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The NVPA Aψ = ((Q1×S) ∪· {qin}, qin, Σ[m], (Γ1×Γ), ⊥, δ, (QF1 ×S));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' notice that the bottom of stack symbol (⊥) is same as the one in A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The transition relation δ = δcall ∪· δret ∪· δint is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' – When op(ζ) = call, δcall = {(qin, ζ, (q, s), (b, a)) | (qin1, ζ + (state(ζ)|m, a), q, b) ∈ δ1 and (state(ζ)|m, (s, a)) ∈ ∆call} ∪ {((q, s), ζ, (q′, s′), (b, a)) | (q, ζ + (s, a), q′, b) ∈ δ1, (s, (s′, a)) ∈ ∆call}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' – When op(ζ) = int, δint = {(qin, ζ, (q, s)) | (qin1, ζ + (state(ζ)|m, ε), q) ∈ δ1, (state(ζ)|m, s) ∈ ∆int} ∪ {((q, s), ζ, (q′, s′)) | (q, ζ + (s, ε), q′) ∈ δ1, and (s, s′) ∈ ∆int}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' – When op(ζ) = ret,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' δret = {(qin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' s)) | (qin1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ζ + (state(ζ)|m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' stack(ζ)|m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' q) ∈ δ1 and ((state(ζ)|m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' stack(ζ)|m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' s) ∈ ∆ret} ∪ {((q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' s′)) | (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ζ + (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' stack(ζ)|m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' q′) ∈ δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' and ((s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' stack(ζ)|m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' s′) ∈ ∆ret} ∪ {((q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (q′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' s′)) | (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ζ + (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' q′) ∈ δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ((s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' s′) ∈ ∆ret} The requirement that the guessed path for π start in the same configuration as the current configuration of the path assigned to πm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' introduces a few points in the definition of δ that are worth highlighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Transitions from qin, whether 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' they be call, int, or ret, pick a step in P that starts from the same state as the one in the path assigned to πm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Stack symbols pushed by P along the guessed path are pushed by Aψ onto its stack (see δcall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' If the stack of Aϕ is ⊥ at a ret-transition, that means on the guessed path the symbol popped must be from what was on the stack at the start of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since that matches with the configuration on the path mapped to πm, this symbol must be the same as what is popped for πm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This is reflected in δret for the cases with ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' An important observation that we will exploit is that if ψ = ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ1 is a sentence, then the following stronger correctness guarantee holds: for any ρ ∈ {call, int, ret}ω, ρ ∈ L(Aϕ) if any only if ρ is a stack access pattern and P, [† �→ ρ], † |= ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The language of Aψ in this case consists of exactly the set of path environments satisfying ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This stronger statement follows from the construction of Aψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To complete the proof, we will bound the size of Aψ by induction on fc(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Recall that n is a bound on the size of P and ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Base Case: Consider ψ such that fc(ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Based on the definition of fc(·) (Definition 4), this means that ψ is built from propositions, ¬, ∨, X, and U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' in particular there are no path quantifiers in ψ in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observe that the construction for aπ is a constant sized automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Also, the constructions for ¬, ∨, X, and U add at most a constant factor to the size of the automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Given these observations, size of Aψ in this case is bounded by O(n), which is bounded by gO(1)(0, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Thus the base case holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Induction Step: We break the induction step into two cases based on the parity of fc(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' When fc(ψ) = 2k (for some k ≥ 1) then ψ is built using sub- formulas of (formula) complexity at most 2k using Boolean operators (¬, ∨), X, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let us assume that we first construct 1-AJAs for each of the subformulas with (formula) complexity < 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This results in at most a quadratic blowup (Theorem 1), and so the size of the automata for each such subformula is at most (gc(k, n))2 for some c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The constructions for ¬, ∨, X and U produce an automaton that is at most a constant factor of the sum of the sizes of the component automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Thus, the size of Aψ is at most dn(gc(k, n))2 ≤ gc′(k, n) for some c′, establishing the claim in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Now let us consider the case when fc(ψ) = 2k+1 for some k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In this case ψ is built using ∨, X and ∃ to combine subformulas of (formula) complexity ≤ 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Again, we can convert automata for each of the sub-formulas of (formula) complexity < 2k + 1 into NVPAs for an exponential cost (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Thus, we can assume that the size of all of these automata is bounded by gc(k + 1, n) for some c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Based on Proposition 3, we can see that disjunction and X produce automata that grow by at most a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Existential quantification are the only operators to have a non- trivial blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Based on the construction outlined in this proof, if ϕ has an NVPA of size ℓ then the automaton of ∃π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ϕ has size O(nℓ) as n bounds the size of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since there are at most n quantifiers, we can bound the size of Aψ by dnn(gc(k + 1, n)) ≤ 2d′n log n+gc(k,n) ≤ gc′(k + 1, n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' the last step is because gc(k, n) ≥ cn log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This establishes the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Stack-Aware Hyperproperties 25 E Proof of Theorem 4 We will show that every language L ∈ ASPACE(hc(k − 1, n)) can be reduced to the model checking problem of sHLTL sentence with formula complexity 2k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since ASPACE(f(n)) = DTIME(2O(f(n))) for f(n) ≥ log n, the theorem will follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Consider an arbitrary hc(k−1, n)-space bounded alternating Turing machine (ATM) M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since hc(k − 1, n) ≥ n, we may assume that M is a 1-tape machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let M = (Q∃, Q∀, Σ, Γ, ⊔, qin, δ, qa), where Q∃ is the set of existential control states, Q∀ is the set of universal control states, Σ is the input alphabet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Γ ⊇ Σ is the tape alphabet, ⊔ ∈ Γ \\ Σ is the blank symbol, qin ∈ Q∃ ∪ Q∀ and qa ∈ Q∃ are the initial and accepting states, respectively, and δ is the transition function of the Turing machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We use Q = Q∃ ∪ Q∀ to denote the set of all states of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We assume that qa is a halting state (no transitions enabled) and so δ : (Q\\{qa})×Γ → 2(Q×Γ ×{→,←}), where given a state and current symbol being read, the transition function identifies choices for the next state, the symbol to be written, and the direction in which to move the tape head (→ for right, and ← for left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We assume, without loss of generality, that for each pair (q, b) ∈ Q × Γ that |δ(q, b)| ∈ {0, 2}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', there are either no or two choices at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Also, we assume that these choices are ordered in some fashion so we will often speak of “choice i” for i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A configuration c of M is a string in Γ ∗(Q × Γ)Γ ∗, where c = u(q, b)v with u, v ∈ Γ ∗, b ∈ Γ and q ∈ Q, denotes that the tape of M is the string ubv, the control state is q, and the head is reading the cell containing b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since M is hc(k − 1, n)-space bounded, we can assume any configuration c of M is a string of length exactly hc(k − 1, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The initial configuration of M on input bw of length n is (qin, b)w⊔hc(k−1,n)−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A configuration c = u(q, b)v is an existential configuration if q ∈ Q∃, a universal configuration if q ∈ Q∀, and an accepting configuration if q = qa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For a pair of configurations c and c′, we say c ⊢i c′ if M’s configuration is c′ if it takes one step according to choice i ∈ {1, 2} from configuration c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' A run of M on input w is a finite, rooted binary tree T = (V, E, r, ℓ), where V is the set of vertices, E is the set of edges oriented away from the root, r ∈ V is the root, and ℓ is a function that maps each vertex to a configuration of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In addition, ℓ is required to satisfy the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The root r is labeled by the initial configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For any internal vertex v ∈ V , if ℓ(v) is an existential configuration then v has one child c such that ℓ(v) ⊢i ℓ(c) for some i ∈ {1, 2}, and if ℓ(v) is a universal configuration, then v has two children c1, c2 such that ℓ(v) ⊢i ℓ(ci) for i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Finally, a run T is accepting if every leaf of T is labeled by an accepting configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' An input w is accepted by M, if M has an accepting run on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We will construct a reduction from L(M) (which by definition is in ASPACE(hc(k − 1, n)) to the model checking problem for sHLTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' That is, given input w, we will construct a pushdown system Pw and sHLTL sentence θw such that Pw |= θw if and only if w ∈ L(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The idea behind the reduction is to construct a pushdown system Pw such that labels of paths starting from the initial configuration in �Pw� encode possible computations of M on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' And θw 26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' is constructed to check if a path encodes a valid accepting run of M on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To formalize this intuition, we need to first identify a way to encode runs of M, which are binary trees, as a string of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Encoding ATM Runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Recall that a run of M is a binary tree and we need to find a way to encode the tree as string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' One way to accomplish this faithfully, is to have the encoding record the stack operations during a depth first search (DFS) traversal of the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For example, if we have a tree T with root r, with tree T1 as the left child and T2 as the right child, then during DFS, the algorithm will first push r on the stack, perform DFS traversal on T1 (recursively), pop r from the stack, push r back onto the stack, DFS traverse T2 (recursively), and finally pop r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' When encoding runs, what we need to push/pop is not the node r, but rather its label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Notice that for a sequence of stack operations to conform to the DFS traversal of a tree, it is necessary for the symbols being pushed and popped be the label of the same node — for example, in the example tree before, after traversing T1, we need the same node r to be popped and pushed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since the symbols when popping are in reverse order of when they are pushed, for long labels (as in the case of configurations) this check is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To overcome this, we push/pop the label and its reverse at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This ensures that if we want to check if a string pushed is the same as a string that was just popped, then we can check for string equality as opposed to one being the reverse of another, and this check is easier to do using formulas in sHLTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We formalize the above discussion to give a precise definition of the encoding of a binary tree as a string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We will abuse notation and overload enc(·) to refer to multiple functions — the context will disambiguate which enc(·) we are referring to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let Λ = Γ ∪ (Q × Γ), the alphabet used to encode configurations of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For i ∈ {1, 2}, let [Λ]i = {[a]i | a ∈ Λ} be a “copy” of the alphabet Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For a string c ∈ Λ∗ of length m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' we define enc(push,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' c) = (call,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(0)]1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(m − 1)]2)(call,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(1)]1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(m − 2)]2) · · · (call,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(i)]1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(m − i − 1)]2) · · · (call,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(m − 1)]1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(0)]2) enc(pop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' c) = (ret,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(m − 1)]1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(0)]2)(ret,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(m − 2)]1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(1)]2) · · · (ret,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(i)]1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(m − i − 1)]2) · · · (ret,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(0)]1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' [c(m − 1)]2) Essentially,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' enc(·) of a string encodes both the string and its reverse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' has a tag that indicates whether the string is being pushed or popped,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' and if it is popped,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' the order of symbols is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Consider a rooted, labeled bi- nary tree T = (V, E, r, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Its encoding is inductively defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' If V = {r} and E = ∅ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', T is a tree with only one vertex) then enc(T ) = enc(push, ℓ(r))enc(pop, ℓ(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' If r has only one child which is the subtree T1, then enc(T ) = enc(push, ℓ(r))enc(T1)enc(pop, ℓ(r)), where enc(T1) is given recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Finally, if r has T1 and T2 as left and right subtrees, respectively, then enc(T ) = enc(push, ℓ(r))enc(T1)enc(pop, ℓ(r))enc(push, ℓ(r))enc(T2)enc(pop, ℓ(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Before moving on, let us highlight a subtle aspect of our encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In the case of a tree where r has two sub-trees T1 and T2, we “pop” ℓ(r) and “push” ℓ(r) between the traversals of T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This may seem unnecessary on first reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Notice that for T to be a valid run, the label of the root of T2 must be the result of Stack-Aware Hyperproperties 27 taking one step from ℓ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Such checks will be encoded in our sentence, and for that to be possible, we need successive ATM configurations to consecutive in the string encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The Pushdown System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Labels of paths of our constructed pushdown system will encode possible runs of M on the input w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' At each step the pushdown system guesses the next symbol of the possible run by moving to a control state whose label corresponds to this symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The stack is used to ensure that when the “pop” symbols in the encoding are encountered they match the symbols that were “pushed” earlier in the guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We can define this precisely as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Recall that for Λ = Γ ∪ (Q × Γ), [Λ]i (i ∈ {1, 2}) refers to the “ith copy” of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let us fix the set of atomic propositions AP = {call, ret, end} ∪ [Λ]1 ∪ [Λ]2 and let S = {call, ret} × [Λ]1 × [Λ]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let P = (S ∪ {sin, se}, ([Λ]1 × [Λ]2) ∪ {⊥}, sin, ∆, L), where L(sin) = ∅, L(se) = {end}, and L((o, [a]1, [b]2)) = {o, [a]1, [b]2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The transition relation ∆ = ∆int ∪· ∆call ∪· ∆ret is given as follows: ∆int = {(se, se)} and ∆call = {(sin, (s, ⊥)) | s ∈ S} ∪ {((call, [a]1, [b]2), (s, ([a]1, [b]2))) | s ∈ S and a, b ∈ Λ} ∆ret = {(((ret, [a]1, [b]2), ([a]1, [b]2)), s) | s ∈ S and a, b ∈ Λ} ∪ {((s, ⊥), se) | s ∈ S} Paths of �P� may not correspond to actual computations of M on w since P doesn’t check for many properties that need to hold for such a string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' On the other hand, correct runs of M on w do correspond to paths of P that end in (end)ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our final pushdown system will be a slight modification of P, in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' First we will add some additional book-keeping to the states and stack symbols to ensure that whenever a universal configuration is guessed, the computation has transitions corresponding to both choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Second, we will need to modify the system to account for the specification, as we shall see towards the end of this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' As mentioned before, for the labels of a path of P to correspond to the en- coding of an accepting computation of M on w, we need to ensure that the labels satisfy a few properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' These will be encoded in our sHLTL sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' However, instead of encoding these conditions in sHLTL, we will find it convenient to first write them in QPTL, a logic introduced in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We begin by introducing this logic, showing how the properties of accepting runs can be encoded, and then describing a way to translate them back to sHLTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' QPTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Quantified propositional temporal logic (QPTL) [17] extends LTL with quantification over propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Fixing a set of atomic propositions AP, formulas in the logic are given by the following BNF grammar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' in what follows, a is an element of AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ϕ ::= a | ¬ϕ | ϕ ∨ ϕ | Xϕ | F ϕ | ∃a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ϕ Models of QPTL are the same as those for LTL, namely elements of (2AP)ω, and the semantics of most of the constructs is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For w ∈ (2AP)ω, a holds if a ∈ w(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ¬ϕ holds if ϕ does not hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ϕ1 ∨ ϕ2 holds if either ϕ1 or ϕ2 hold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 28 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Xϕ holds if ϕ holds on the suffix w[1 : ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' and F ϕ holds if ϕ holds in some suffix w[i : ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The only new operator is ∃a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ϕ which holds in w, if ϕ holds in some word w′ which agrees with w in the evaluation of all propositions except possibly a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Recall that F ϕ is equivalent to true U ϕ, G ϕ is a short hand for ¬ F(¬ϕ), and ∀a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='ϕ is ¬∃a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (¬ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We can extend fc(·) to QPTL formulas, with the same definition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' for this definition ¬ will behave like negation for cognate formulas, rather than negation for sHCTL*-sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Finally, it has been shown that every QPTL formula is equivalent to one in prenex normal form, where all the quantifiers have been pulled to the front of the formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our QPTL formula ϕw that describes when a word encodes an accepting run of M on w, will rely on formulas constructed in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The first is formula ϕc,k,n(p1, p2) (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' in [17]) of size O(k + n) such that w |= ϕc,k,n(p1, p2) if and only if propositions p1 and p2 are true exactly once in w, p2 is true after p1, and they are separated by exactly hc(k, n) positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Further more fc(ϕc,k,n) = 2k − 1, and can be constructed O(log n) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We will introduce the other formulas as we describe the conditions needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The QPTL formula ϕw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We now describe ϕw with the property that a path of �P� satisfies ϕw if and only if M accepts w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' here a path c0c1 · · · of �P� satis- fies ϕw, if L(c0)L(c1) · · · |= ϕw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Observe that the construction of P ensures that symbols “popped” are the same as the symbols “pushed” and that every universal configuration has two successors correspond to transitions corresponding to each choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='. Therefore, the remaining conditions that need to be checked for a path to be the encoding of an accepting computation of M on w are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Every configuration has length exactly hc(k − 1, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' All symbols encoding a single configuration have the same tag, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', either all call or all ret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The first configuration is the initial configuration of M on w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Successive “pushed” configurations correspond to a single step of M consis- tent with its transition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' “Leaves” of the run are accepting configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In other words, if a push configuration is immediately followed by a pop configuration (they will be identical thanks to P), then the control state must be qa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' If a pop configuration is immediately followed by a push configuration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=', the DFS traversal of the run is exploring the right child) then they must be the same configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' All the configurations are popped at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' If ϕ denotes the conjunction of all the above conditions written in QPTL, then our desired formula ϕw is Xϕ since the initial state sin of P is not part of guessing the encoding of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We describe how to encode each of these conditions in QPTL, often relying on formulas from [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To state property (1), we will have a proposition ⊲ (which will be exis- tentially quantified) that marks the beginning of each configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The fact that ⊲ marks the beginning of each configuration will be ensured by Stack-Aware Hyperproperties 29 the other properties we write down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We will require that ⊲ is true exactly hc(k−1, n) positions apart using the formula ϕc,k−1,n(·, ·) introduced before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This is given by Equation (9) in [17] where r replaced by ⊲.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since ⊲ marks the beginning of each configuration, saying that all symbols in a configuration have the same tag, is equivalent to saying that if the tag changes then ⊲ must be true when the tag changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' In other words, property (2) can be written as G((call ∧ Xret) → X⊲) ∧ G((ret ∧ Xcall) → X⊲) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Equation (10) in [17] states the property (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' It needs to be slightly modified to also include the condition that the propositions from [Λ]2 encode the reverse of the initial configuration of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Equation (11) in [17] states the property (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' It needs to be slightly to mod- ified to also ensure that the reverse encodings using [Λ]2 are consistent with the transitions of M, and this requirement is only imposed for successive configurations that have the call tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Recall that corresponding symbols in successive configurations are hc(k − 1, n) apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Property (5) can be written to say that if there are two positions p and q that are hc(k−1, n) apart that have opposite tags, and if in addition p corresponds to a position that is scanned by the tape head, then the control state at p must be the accepting state qa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ∀p1∀p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' (ϕc,k−1,n(p1, p2) ∧ F(p1 ∧ call ∧ � a∈Q×Γ [a]1) ∧ F(p2 ∧ ret)) → F(p1 ∧ � a∈{qa}×Γ [a]1) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Using the observations in property (5), property (6) can be written as ∀p1∀p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ((ϕc,k−1,n(p1, p2) ∧ F(p1 ∧ ret) ∧ F(p2 ∧ call)) → � a,b∈Λ (F(p1 ∧[a]1 ∧[b]2) ∧ F(p2 ∧[b]1 ∧[a]2)) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This property can be ensured by requiring that the path end in state s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We write this as F end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Since formulas in QPTL can be written in prenex normal form, we can pull all the universal quantifiers in the various formulas listed above to get a formula ϕw of the form X ∃ ⊲ ∀p1∀p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ϕ where ϕ is a Boolean combination of ¬ϕc,k−1,n(·, ·) and simple formulas with temporal operators of (formula) complexity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Thus, fc(ϕw) = 2k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Converting to sHLTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our QPTL formula ϕw is of the form X∃⊲ϕ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' We will describe how to construct an “equivalent” formula in sHLTL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' this will require modifying our pushdown system P as well to obtain our final pushdown system Pw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Construct a cognate sentence ψ′ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let a be a new proposition not 30 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Bajwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' appearing in ϕw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' For every quantified proposition p in ϕw, replace every bound occurrence of p by aπp and replace the quantification ∃p with ∃πp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let π∗ be a fresh path variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Replace every free proposition p in ϕw by pπ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let ψ be the resulting cognate formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Finally let θw = E∃π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' To finish the reduction, we need to modify P to account for the new proposi- tion a introduced in θw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Our final pushdown system Pw is constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Recall that S = ({call, ret} × [Λ]1 × [Λ]2) and the states of P is S ∪ {sin, se}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' Let [S]1 and [S]2 be two copies of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The states of Pw will be [S]1 ∪ [S]2 ∪ {sin, se}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The two copies of state s ∈ S will have the same transitions into and out of it and have the same labels for all the old propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' The only difference will be that a ∈ L([s]1) while a ̸∈ L([s]2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' It is easy to make the following observations: fc(θw) = fc(ϕw), and a path of �P� satisfies ϕw if and only if Pw |= θw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} +page_content=' This completes the proof of the hardness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFJT4oBgHgl3EQfXSyN/content/2301.11521v1.pdf'} diff --git a/jNFST4oBgHgl3EQfGzio/content/tmp_files/2301.13723v1.pdf.txt b/jNFST4oBgHgl3EQfGzio/content/tmp_files/2301.13723v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d989757b2994040da9f4ac6e017201757d66591 --- /dev/null +++ b/jNFST4oBgHgl3EQfGzio/content/tmp_files/2301.13723v1.pdf.txt @@ -0,0 +1,1182 @@ +arXiv:2301.13723v1 [cs.DS] 31 Jan 2023 +p-median location interdiction on trees +L. Leiß∗a, T. Hellerb, L. Sch¨aferc, M. Streicherd, and S. Ruzikaa +aDepartment of Mathematics, RPTU Kaiserslautern-Landau, 67663 Kaiserslautern, +Germany +bDepartment of Optimization, Fraunhofer Institute for Industrial Mathematics ITWM, +67663 Kaiserslautern, Germany +cComma Soft AG, 53229 Bonn, Germany +dPTV Group, 76131 Karlsruhe, Germany +Abstract +In p-median location interdiction the aim is to find a subset of edges +in a graph, such that the objective value of the p-median problem in +the same graph without the selected edges is as large as possible. +We prove that this problem is NP-hard even on acyclic graphs. +Restricting the problem to trees with unit lengths on the edges, unit +interdiction costs, and a single edge interdiction, we provide an al- +gorithm which solves the problem in polynomial time. Furthermore, +we investigate path graphs with unit and arbitrary lengths. For the +former case, we present an algorithm, where multiple edges can get +interdicted. Furthermore, for the latter case, we present a method to +compute an optimal solution for one interdiction step which can also +be extended to multiple interdicted edges. +Keywords: Network Interdiction, Location Planning, Median Problems, +Edge Interdiction, Network Location Planning +1 +Introduction +Location planning is a field of mathematical research which crosses our daily +life more often, than we might think at first sight. +The root of modern +∗Corresponding author, Email address: leiss@mathematik.uni-kl.de (L. Leiß) +1 + +location planning goes back to Pierre de Fermat and aims at finding the +point, which minimizes the sum of the Euclidean distances of three given +points to the new location (cf. [8]). A popular, more applied version of this +problem is the identification of a new location for a supplier of materials +for further industrial processing which has been stated in [30] and is called +Weber problem (cf. [8]). In a more general approach, a new location is to +be found which minimizes the sum of all - possibly weighted - distances from +all given locations to the new one. This problem is called median location +problem (cf. [19]). The underlying structure, on which location problems +can be analyzed, may vary. Mainly, we distinguish between planar location +problems and network location problems. In this article, we consider the +network case only. A further alteration of the main problem comes with the +number of new locations to be computed. We refer to the problem of placing +p new facilities as the p-median location problem. The decision version of +the p-median location problem is known to be NP-complete for variable p on +general networks, which can be shown by a reduction from the dominating +set problem ([18]). For this case, there are several heuristics known. A good +overview on this topic can for example be found in [22]. For a general graph +with unit length values on the edges and variable p, the problem still remains +NP-complete ([12]). In contrast, for fixed p, the decision problem is solvable +in polynomial time on general graphs by enumeration (cf. [12]). Due to the +hardness of the general case, research focused on particular graph structures. +For G being a tree, the authors in [18] present an O(n2p2)-algorithm to solve +the problem. A dynamic programming approach based on this result was +later proposed in [28], which runs in O(pn2) and got improved by [5] to an +algorithm which runs in O(n logp+2 n). The complexity for path graphs is +shown to be O(pn) in [16]. A linear time algorithm can be applied for the +1-median location problem on trees ([14]). +A good overview of location planning can for example be found in [8], [15] or +[19]. An overview of the solution methods for the p-median location problem +in particular can be found in [25]. +Interdiction problems pursue the question of how a system can be interrupted +in the worst possible way in terms of the original objective function. The +interruption itself can be caused by different acts, such as modification of the +edge lengths or deletion of entire edges as well as the vertices. +Interdiction problems have gained increasingly more attention lately (cf. [27]). +There are several different applications, that motivate research in this field. +The interdiction of a network can either have a desirable outcome, such as in +2 + +narrowing the spread of a disease (cf. [2]), interdicting smuggling routes (cf. +[23]) or – as for example done in [32] – the aim of (armed) forces to reduce the +amount of drugs and chemicals transported illegally via road or waterways – +possibly with limited resources. Also – on the contrary – attacks on networks +can be interpreted via interdiction steps. In this case, the analysis of these +problems might allow the determination of valuable edges or locations of the +original network. +Two main optimization problems, which have been studied in the context +of interdiction, are the shortest path problem as well as the maximum flow +problem. Notable research on the first topic has been done in [3], [4], [7] or +[17], for instance. Results on the latter problem might for example be found +in [13], [24], [26], [31] or [32]. In [27], one can find a recent overview of the +literature on interdiction problems. +Research concerning the combination of location and interdiction problems +on the other hand is quite scarce. +In this context, we mention the r- +interdiction median problem, which is defined as follows. Given a supply- +system with p existing locations, the interdictor wishes to find the subset +of r locations, which, when removed, yields the highest weighted distance +with respect to the median location function (cf. [6]). There are a few ex- +tensions to this problem, such as the possibility to fortify a fixed number +of the existing facilities, which in consequence cannot be interdicted by the +attacker anymore (cf. [21]). In [1], the authors alter the concept of fortifying +a specified number of locations, but rather introduce a restricted budget for +fortification. A further topic, gaining more interest, is the p-hub interdic- +tion problem, which is for example considered in [29]. Still, most of these +approaches have in common, that the interdictor intervenes the existing lo- +cations and not the underlying network. An exception to this concept can +be found in [10]. Here, the authors combine the covering problem with an +edge interdiction problem. In [11], the authors present a polynomial time al- +gorithm for the interdiction problem on trees, where an upfront chosen set of +facilities is given and the interdictor wishes to worsen the reachability within +the tree. In [9], the authors combine the median location problem with edge +interdiction. To the best of our knowledge, this is the only work on the p- +median interdiction problem with edge interdiction. There, they consider the +problem for different orders of action of the locator and the interdictor. For +the case, that the interdictor acts before the locator, they prove this problem +to be Σp +2-complete in the general case. In the same work, a bilevel mixed- +integer formulation is presented for the problem. This result motivates the +3 + +analysis of the problem for restricted cases. +Our contribution +Given the complexity analysis in [9], it remains open, if +efficient solution procedures can be found if the general problem is restricted. +Coming from the original p-median problem, which is solvable in polynomial +time on trees, an obvious variant of the corresponding interdiction problem +is to restrict the problem to trees (or even simpler structures) as well. For +such cases, complexity results and solution methods are not yet available. +In this article, we aim at closing this research gap. We consider the median +location problem in combination with an interdictor who can delete edges in +a given network and wishes to maximize the objective function value of the +locator. We assume, that the interdiction step is executed before the locator +places their median facility. Due to sigma-2-p hardness for the general case, +we consider the problem on particular graph structures. +We analyze the +complexity of the general problem on trees and present an algorithm to solve +the problem exactly for some variant. Furthermore, we present an algorithm +for graphs with path structure for both cases of unit and arbitrary lengths +on the edges. +Outline +The remainder of this article is structured as follows. In Section 2, +we give a short overview of the concepts needed for the article and define the +investigated problem. Section 3 deals with the complexity of the median +location interdiction problem. Section 4 studies the strategy for solving the +interdiction median problem on paths with unit length values on the edges. +Furthermore, we present a strategy for paths with arbitrary lengths. The +next Section 5 focuses on a tree with unit length values. We state an algo- +rithm, which solves the problem exactly in polynomial time. Section 6 then +summarizes the paper and proposes further directions of research. +2 +Preliminaries and problem formulation +Let G = (V (G), E(G)) be an undirected graph with vertex set V (G) = +{v1, . . . , vn} and edge set E(G) = {e1, . . . , em}, where n ..= |V (G)| and m ..= +|E(G)|. If the underlying graph is known by the context, we refer to V (G) +as V and to E(G) as E. +Also, for better readability, instead of |V (G)|, +we may write |G|. For a given vertex v, we denote the number of incident +edges, i.e. its degree, by deg(v). Further, we assign a length value to each +4 + +edge e ∈ E, i.e., ℓ: E → Z+. Let Puv be the set of all paths P connecting +vertices u, v ∈ V . Then, the length of a shortest path between the vertices u +and v is denoted by d(u, v), i.e., d(u, v) ..= min +P ∈Puv ℓ(P). If the graph on which +the distance is measured is not clear from context we also write dG(u, v). +If G is a tree, let some vertex r ∈ V be the root of the breadth-first-graph +of G ([20]). We denote by Gv the subtree of G, which is rooted in vertex v +and contains all descendants of vertex v in the breadth-first-graph of G with +root r as well as their incident edges. +As stated, given an instance of the median problem, one aims at placing one +(or more) new location(s), which minimize(s) the sum of the shortest path +lengths of all existing locations to their nearest facility. In this article, we +consider the case, that the set of existing locations is the vertex set. A chosen +set X of locations therefore has the objective value: +f(X) ..= +� +v∈V +d(v, X), with d(v, X) ..= min +x∈X d(v, x). +Based on this objective, we state the p-median location problem as follows. +p-median location problem (p, �, G) +Instance: Undirected graph G = (V, E), edge lengths ℓ: E → Z+, and +number of locations p ∈ Z+. +Task: +Find a set X ⊆ V of p new locations such that the objective +function of the p-median location problem is minimal, i.e. +minimize +� +v∈V +d(vi, X). +The optimal solution is denoted by X∗, while the optimal objective function +value is denoted by OPT(p, �, G). +Network interdiction problems involve an additional opposing force, called +the interdictor. Said interdictor wishes to worsen the objective function value +of the locator. In this article, the interdictor is constrained by an interdiction +budget B ∈ Z+, while each edge e ∈ E is associated with an interdiction cost +b(e) ∈ Z+, i.e., b: E → Z+. Consequently, the set of all feasible interdiction +strategies, denoted by Γ, can be expressed as follows: +Γ ..= +� +γ = (γe)e∈E ∈ {0, 1}m | +� +e∈E +b(e) · γe ≤ B +� +, +5 + +where γe equals one, if edge e is interdicted or zero, if not. +The set of +optimal interdiction strategies is denoted by Γ∗ = {γ∗ ∈ Γ | γ∗ optimal}. In +what follows, each interdiction strategy γ ∈ Γ induces an undirected graph +G(γ) ..= (V ′, E′) with V ′ = V and E′ = E \ E(γ), where E(γ) ..= {e ∈ E | +γe = 1}. In this article, the locator places their facility on G(γ), i.e. after +the interdiction step. Based on this, we define the decision version of the +p-median location interdiction problem as follows. +Decision version of the p-median location interdiction problem +Instance: Undirected graph G = (V, E), edge lengths ℓ: E → Z+, in- +terdiction costs b: E → Z+, interdiction budget B ∈ Z+, +number of locations p ∈ Z+, and decision parameter K ∈ Z+. +Question: Does there exist an interdiction strategy γ ∈ Γ such that +min +X⊆V,|X|=p +� +v∈V +dG(γ)(v, X) ≥ K ? +In the optimization version of the stated problem, we aim to find the maxi- +mum K for which the decision version is a yes-instance. +3 +Complexity results +It is well known that the p-median location problem is NP-complete, cf. [18]. +Therefore it would be surprising for the corresponding interdiction problem +to be polynomial time solvable. In fact Fr¨ohlich [9] proved that the decision +version of the p-median location interdiction problem is Σp +2-complete. How- +ever, there are several restrictions of the p-median location problem which +are proven to lead to polynomial time solvability, as mentioned in the intro- +duction. Among the restricted versions, that are polynomial time solvable is +the p-median location problem on trees, cf. [28]. In this section we prove that +adding the interdiction layer to the problem makes the problem significantly +harder: The p-median location interdiction problem is NP-complete even on +trees. +For this, we consider the knapsack problem with bounded profit ratio of 2 +(K-BPR2). An instance is given by a set M of m items with associated +weights wi ≥ 0 and profits pi ≥ 0 for all i ∈ {1, . . . , m}. For the profits +6 + +it holds pi +pj ≤ 2 for all pi, pj with i ̸= j. We now show that this problem is +NP-complete. +Lemma 3.1. The knapsack problem with bounded profit ratio of 2 is NP- +complete. +Proof. We reduce the equal partition problem to K-BPR2. Let an instance of +the equal partition problem, which is known to be NP-complete (cf. [12]), be +given with a set I = { ˜w1, . . . , ˜wn} such that �n +i=1 ˜wi = B ∈ 2 Z, ˜w ≥ 0. For +the equal partition problem we ask for a partitioning of the elements of I into +two subsets I1, I2 such that � +i∈I1 ˜wi = � +i∈I2 ˜wi = B +2 and |I1| = |I2| = n +2. +Define pi = wi ..= ˜wi + B + 1 for all i. According to this definition, the +smallest pi is at least B + 1 and the biggest pi is not greater than 2B + 1 +such that the ratio pi +pj of all pairs pi, pj, i ̸= j is bounded by 2. +Suppose we are given a solution I1 to the equal partition problem, we ask for +a solution to the knapsack problem with profit +P ≥ B +2 + n +2(B + 1) +and total weight +W ≤ B +2 + n +2(B + 1). +We state, that the selection I1 is such a solution. For the profit, it is P = +� +i∈I1 pi = � +i∈I1( ˜wi + B + 1) = +B +2 + n +2(B + 1). Also, W = � +i∈W1 wi = +� +i∈I1( ˜wi + B + 1) = B +2 + n +2(B + 1). +On the other hand, given a solution J ∈ I to the knapsack problem for which +P ≥ B +2 + n +2(B +1) and W ≤ B +2 + n +2(B +1) we show that J together with I \J +is a solution to the equal partition problem. To prove this statement, we +need to show that |J| = n +2. Suppose that |J| < n +2. Then for the profit, it is +� +j∈J pj = � +j∈J( ˜wj+B+1) ≤ B+( n +2−1)(B+1) = n +2(B+1)−1 < P which is a +contradiction to the solution of the knapsack problem. Analogously, suppose +that |J| > n +2. Then for the weight it is � +j∈J wj = � +j∈J( ˜wj + B + 1) ≥ +B + ( n +2 + 1)(B + 1) = 2B + 1 + n +2(B + 1) > W. Therefore, we have |J| = n +2. +Finally, calculate +� +j∈J +pj = +� +j∈J +˜wj + n +2(B + 1) ≥ B +2 + n +2(B + 1) +7 + +vx +. . . +ℓi = 0 +bi = B + 1 +ℓi = 0 +bi = wi +ℓi = pi +bi = B + 1 +ℓi = 0 +bi = B + 1 +(a) Red color depicts the edges with lengths +ℓi = pi ≥ 0; green color depicts the edges with +individual interdiction costs bi = wi. +. . . +(b) One optimal locator’s solu- +tion where the placed centers +are depicted in blue. +Figure 1: The tree graphs used in the proof of Theorem 3.2. +and also +� +j∈J +wi = +� +j∈J +˜wj + n +2(B + 1) ≤ B +2 + n +2 (B + 1). +Therefore, we get that � +j∈J ˜wj = B +2 . +Clearly, given a solution to the K-BPR2, we can verify this solution in poly- +nomial time. Thus, the K-BPR2 is NP-complete. +Theorem 3.2. +The p-median location interdiction problem, where the un- +derlying graph is given by a tree, is NP-complete. +Proof. We show this by reducing K-BPR2 to the location interdiction prob- +lem. +Let an instance of K-BPR2 be given by a set M of m items with +associated weights wi and profits pi for all i ∈ {1, . . . , m}. For the profit it +holds for all pi, pj with i ̸= j that pi +pj ≤ 2. Furthermore, let W, P ∈ Z+ be two +integers. Consider a tree with lengths ℓi and interdiction costs bi for every +edge ei as depicted in Figure 1a. Note, that the construction provides m +paths of four vertices emerging from vertex vx and one path of two vertices. +Let B = W be the interdiction budget. For every selection of interdicted +edges, one optimal strategy to choose m + 1 centers is depicted in Figure 1b. +8 + +vx +u2 +v2 +u1 +v1 +ℓ1 +ℓ2 +0 +0 +0 +0 +0 +0 +Figure 2: Excerpt of the tree from Figure 1. +To prove this fact, we need to show that every single path emerging from +vertex vx must have one center, and furthermore, that the center in the paths +with 4 vertices must be below the edges where ℓi = pi. +In case that every outgoing edge of vx which is part of the m paths of 4 vertices +(depicted in green in Figure 1a) is interdicted, the stated solution is clearly +optimal. Now consider the case where w.l.o.g. the leftmost interdictable edge +is interdicted and let the second outgoing edge of vx be non interdicted. The +notation used can be found in Figure 2. +It is clear, that every path with interdicted starting edge (the edge outgoing +of vx) needs at least one center for feasibility. Also, this center has to be +placed below the edge with length greater than zero – in the considered case +u1 or a vertex below. Now, the only possibility for the locator to change the +objective function value is to spare the center of the non interdicted paths +– u2 in the example – and instead place it at v1. Then, the length ℓ1 does +not appear in the calculation of the objective function value. But instead, +all vertices in the second path then need to be covered via center vx. That +means, the edge with length ℓ2 needs to be crossed 3 times. Even if we as- +sume that the lengths have the maximal possible factor ℓ1 = 2 · ℓ2, it would +still be better for the locator to place the center at vertex u2. The same +explanation holds for the shifting of the center in vx to a vertex in an inter- +dicted path. In this case, the vertices above vx need to be covered by another +center below which is – by the same estimation as before – worse than the +9 + +provided solution of Figure 1b. Note that this solution is not unique. In fact +– as stated before – the center in the m paths of 4 vertices can be placed +at any vertex below the edges where ℓi = pi. Also, the center at vx can be +shifted up to two vertices up. +Assume we are given a solution of the knapsack problem such that the se- +lection I ∈ M of items fulfills � +i∈I wi ≤ W and � +i∈I pi ≥ P. Now consider +the optimal solution of the locator as stated above after interdiction of the +edges ei, i ∈ I. For the interdiction costs, it is � +i∈I bi = � +i∈I wi ≤ W. Fur- +thermore, the corresponding objective function value calculates as � +i∈I ℓi = +� +i∈I pi ≥ P. +On the other hand, if we are given a selection of interdicted edges J ∈ EM +such that � +j∈J bj ≤ B and � +j∈J ℓj ≥ P, we show that J is a solution to +K-BPR2. Firstly, � +j∈J wj = � +j∈J bj ≤ B = W. Also, the profit calculates +as � +j∈J pj = � +j∈J ℓj ≥ P. +Given an interdiction strategy γ, the corresponding p-median problem on +G(γ) can be solved in polynomial time as G(γ) still does not contain cycles, +cf. [18]. Thus, the p-median interdiction problem on trees is contained in NP +and therefore NP-complete. +The p-median location interdiction problem is NP-complete, even on trees in +general. In the remainder of the article, we further restrict the problem in +order to get a better impression on what makes the problem hard. +A direct consequence of the p-median location interdiction problem on trees +beeing contained in NP, is that regarding the interdiction budget as a con- +stant makes the problem polynomial time solvable. This holds true, as hav- +ing a constant interdiction budget makes the number of possible interdiction +strategies polynomial in the instance size. Thus, the procedure of solving a +p-median problem for all interdiction strategies and thereby finding the best +strategy runs in polynomial time. +Observation 3.3. The p-median interdiction problem on trees is polynomial +time solvable if the interdiction budget is considered a constant. +Further possible restriction are to simplify the graph structure even more, +making the edges have unit interdiction costs, making the edges have constant +or even unit lengths, or restricting the number p of locations to be placed. +Any combinations of these restrictions is also interesting. As analyzing all +10 + +possible and interesting combinations would exceed the scope of a single +article, we focus on the subcase of unit interdiction cost and regard different +further restrictions. +In the next section, we consider the p-median location interdiction problem +on paths with unit interdiction costs and afterwards move back to the same +problem on trees. +4 +Interdicting a path +In this section, let G = (V, E) be a graph with V = {v1, . . . , vn} and E = +{ei = {vi, vi+1}: i = {1, . . . , n − 2}. The resulting graph has the structure of +a path consisting of n vertices and n − 1 edges. For the remainder, we refer +to these types of graphs as paths. +In this section we tackle the p-median location interdiction problem on paths, +where we additionally assume unit interdiction costs and p = B+1, i.e. every +component emerging from the interdiction can be equipped with exactly one +location. Note that for unit interdiction costs, we may assume B ≤ n−1, as a +path only contains n−1 edges. We first elaborate on paths with unit lengths. +In this case the interdictor can use a simple method to worsen the situation +for the locator. This procedure is initially analyzed for one interdiction step +(B = 1) and is then generalized to arbitrary interdiction budgets B > 1. +After that, we show, how paths with arbitrary lengths can be handled for +B = 1. +We want to briefly present the idea of Goldman’s algorithm (cf [14]), since +it is needed in the remainder of the article. For a tree T, we start at an +arbitrary leaf and compare the weight of that leaf to the total summarized +weight of all vertices. As long as the weight of the leaf is less than half of +the total weight, we delete the leaf and update the weight of the adjacent +vertex by adding the weight of the deleted leaf. In that manner, we iterate +over the leaves until we find one with a weight greater or equal to the half +of the total weight. This vertex is the 1-median of the original graph. With +this method, we are also able to state the 1-median location(s) on a path, +which is dependent on the number of the vertices. The optimal solution(s) +on a path is to place the new location(s) at vertex vn/2−1 or vn/2 for n even or +at vertex v⌈n/2⌉ for n odd. +11 + +4.1 +Paths with unit edge weigths +Let the graph be a path P = (v1, e1, v2, . . . , en−1, vn) with ℓ ≡ 1 and b ≡ 1 as +described above. We first examine the case for B = 1. +Lemma 4.1. Let P be a path with ℓ ≡ 1 and b ≡ 1. The optimal interdiction +strategies for the p-median location interdiction problem are to interdict e1 +or en−1, yielding in an isolated vertex and a new path of length n − 2, i.e. +Γ∗ = {γ∗ +1 = (1, 0, . . . , 0), γ∗ +2 = (0, . . . , 0, 1)}, +where the order of γ∗ +i , i = 1, 2 is induced by the order of the edges. +Proof. As described above, the optimal solution(s) for the 1-median problem +are at vertex vn/2−1 or vn/2 for n even or at vertex v⌈n/2⌉ for n odd. +The +resulting objective function value OPT(1, �, P) can then be computed via: +OPT(1, +� +, P) = +� +n2 +4 +n even +n2−1 +4 +else +Every interdiction strategy with B = 1 results in two components of the +original path. Since p = 2, i.e. one new location in each component, we +compute the optimal objective function value for the 2-median problem un- +der the given setting by adding up both objective function values computed +separately for every component. Therefore, let et be the interdicted edge +for some t ∈ {1, . . . , n − 1} yielding a separation of the original path P +into Pt = (v1, e1, v2, . . . , et−1, vt) and Pn−t = (vt+1, et+1, . . . , en−1, vn). Then, +the objective function for the overall problem of placing one new location in +either part is +z∗ = OPT(1, +� +, Pt) + OPT(1, +� +, Pn−t) = 1 +4(2t2 + n2 − 2nt − a) +(1) +with a ∈ {0, 1, 2}. The goal of the interdictor is to choose t such that z∗ +is maximal. For a given case, i.e. where n and a are fixed, we can reduce +equation 1 to the following expression +t2 − nt = +� +t − n +2 +�2 +− n2 +4 +for the computation of the maximum. Since we aim to maximize the latter +expression, it can again be reduced to +� +t − n +2 +�2 +. +12 + +. . . +. . . +. . . +. . . +vq +vr +vr+1 +vs +Figure 3: Detail of interdicted path P; interdicted edges are depicted in red. +For t ∈ {1, . . . , n − 1}, the maximum is found at t1 = 1 or t2 = n − 1, which +proves the claim. +This result allows to expand the considerations to B ≤ n − 1. +Lemma 4.2. Let a path P = (v1, e1, v2, . . . , en−1, vn) be given with ℓ ≡ 1, +b ≡ 1. The optimal interdiction strategy under an interdiction budget B ≤ +n − 1 is to successively interdict the edges incident to leaves, thus resulting +in B single vertices and one path of length n − 1 − B. +Proof. Consider an optimal interdiction strategy γ′, where at least one of +the interdicted edges does not cut off a leaf as depicted in Figure 3. Let er +be the described edge. Given there exists an edge with the stated proper- +ties, there also exist sets of vertices Vq = {vq ∈ V : q < r, deg(vq) = 1} and +Vs = {vs ∈ V : r < s, deg(vs) = 1}. Let vq ∈ Vq be the vertex with the +biggest index and vs ∈ Vs the vertex with the smallest index. This require- +ment ensures that the component Pqs = (vq, eq, . . . , es−1, vs) of the original +path is only interdicted once at er. +Now consider Pqs, which is interdicted at er with strategy γ′, resulting in +two new paths. Using the result of Lemma 4.1, the optimal objective func- +tion value for the interdiction of path Pqs does not decrease by interdicting +eq instead of er. Therefore, successively using this method of shifting the +interdicted edges to the leftmost edge of the respective components will also +not decrease the overall objective value of γ′ yielding an optimal interdiction +strategy γ∗ as stated. +4.2 +Paths with arbitrary lengths +For the remainder of the section, we assume an arbitrary length function ℓ +to be given. +One obvious strategy is to interdict all edges successively. In each step at +a time, we compute the optimal objective function value for the locator +by solving two median location problems on the remaining paths after the +current interdiction step. +Evaluating over all obtained objective function +13 + +values yields the best edge to interdict. We present an approach which effi- +ciently iterates over all edges by using an interesting structure of a matrix, +which helps computing the locators objective function values. Let a path +P = (v1, e1, v2, . . . , en−1, vn) be given. The median location is found at ver- +tex v⌈n/2⌉ for n odd or at vertex vn/2 or vn/2+1 for n even. As stated in Section +2, the objective function value for the 1−median problem on the given path +is determined by the total number of times, each edge is crossed to reach +all vertices from the median location. Given the structure of a path, these +numbers are bounded by ⌊n/2⌋ if n is odd and by n/2 if n is even. More pre- +cisely, these bounds hold for the edges e⌊n/2⌋ and e⌈n/2⌉ incident to the median +location (n odd) or the edge en/2 (n even), respectively. Furthermore, this +number decreases by one the closer the edges are to the leaves of the path. +Example 4.3 shows the case for a path of length 7. +Example 4.3. Let P = (v1, e1, . . . , e6, v7) a path with length values ℓi, i = +{1, . . . , 6} as depicted. With n odd and the observations above, we get that the +median is located at v⌈n/2⌉ = v4. Furthermore, we can determine the number +of times si the edge ei is represented in the objective function value (depicted +in green). +v1 +v2 +v3 +v5 +v6 +v7 +v4 +1 +l1 +2 +l2 +3 +l3 +3 +l4 +2 +l5 +1 +l6 +The information of how often an edge length contributes to the objective +function value can be stored in a vector S ∈ Nn−1, where each entry repre- +sents one edge. Multiplying S with the length vector ℓ yields the optimal +objective function value for the path. +We use this scheme for the construction of the matrix calculating the ob- +jective function values for different interdicted edges. +Assume that some +edge et, t ∈ {1, . . . , n − 1} gets interdicted. This results in the two paths +Pt,1 = (v1, e1, v2, . . . , et−1, vt) and Pt,2 = (vt+1, et+1, . . . , en−1, vn), for which +the objective function values can be calculated separately as stated via the +vectors St,1 = (s1, . . . , st−1) ∈ for P1 and St,2 = (st+1, . . . , sn−1) for P2. The +union yields a new vector St = (St,1, 0, St,2). Assuming that we only interdict +once, we can proceed to build St for all edges et, t = 1, . . . , n sequentially. The +matrix S = (St)t ∈ N(n−1)×(n−1) can again be multiplied with ℓ. Evaluat- +ing for the biggest objective function value solves the 1−interdiction median +problem. Example 4.4 shows the matrix for the path of Example 4.3. +14 + +Example 4.4. Let P be the path of Example 4.3. Furthermore, let the length +vector ℓ be given. The matrix obtained via the presented method is as follows: + + + + +0 +1 +2 +3 +2 +1 +1 +0 +1 +2 +2 +1 +1 +1 +0 +1 +2 +1 +1 +2 +1 +0 +1 +1 +1 +2 +2 +1 +0 +1 +1 +2 +3 +2 +1 +0 + + + + +As stated, the matrix S is of size (n − 1) × (n − 1), where each row +i ∈ {1, . . . , n − 1} represents the objective function value if edge ei gets +interdicted. +Therefore, the procedure explained runs in time O(n2) since +the multiplication of matrix S with the given length vector ℓ is of stated +time. We want to mention, that an increase of interdiction steps adds factor +n to the running time for each single interdiction. This is due to the fact, +that we need to consider every possible combination for the edges, that get +interdicted. +5 +Interdicting a tree +In this section, we show, how to interdict a tree T = (V, E) with ℓ ≡ 1, b ≡ 1, +B = 1 and p = B + 1. +Theorem 5.1. For T = (V, E) with ℓ ≡ 1, b ≡ 1, B = 1 and p = B + 1, the +optimal interdiction strategy is as follows: among all leaves of T, find one, +that has the least distance to at least one optimal solution of the 1−median +location problem on T. Interdict the edge incident to this leaf. +Proof. Since the proof is constructive, consider the exemplary tree in Figure 4 +which illustrates the used structures. +Let r ∈ V (T) be an optimal solution to the 1−median location problem on +T. Also, let T be rooted in r. Let f = (v1, v2) ∈ E(T) be an edge with v2 +being a leaf that fulfills +d(r, v2) = min +r∗∈X∗ +l leaf +d(r∗, l). +(2) +For all neighbors w ∈ N(r), it holds that +|Tw| ≤ |T − Tw| . +(3) +15 + +r +u∗ +1 +u∗ +2 +u1′ +u2′ +w1 +w2 +w3 +e∗ +e′ +Figure 4: Exemplary tree for Theorem 5.1. +Assume, that |Tw| > |T − Tw|. Then, w would yield a better objective func- +tion value for the 1−median location problem than r, which is a contradiction +to the assumption, that r is optimal for said problem. Now, for every edge +e = (u1, u2) ∈ E(T) with d(r, u1) < d(r, u2), it is +OPT(2, +� +, T − e) ≤ OPT(1, +� +, T) − (d(r, u2) · |Tu2|). +(4) +This is, since placing a second location in the tree Tu2 saves at least |Tu2| +times the distance d(r, u2) in comparison to the solution of the 1−median +problem on the original tree T. +We aim at finding e∗ = (u∗ +1, u∗ +2) such that d(r, u∗ +2) · +��Tu∗ +2 +�� = mine∈E(T) d(r, u2) · +|Tu2|. This leads to the biggest right hand side of inequality (4), thus provid- +ing an upper bound for the optimal objective function value OPT(2, �, T − +e), which the interdictor maximizes. We claim, that u∗ +2 is a leaf. +Case 1: d(r, u∗ +2) > 1. Suppose, u∗ +2 is not a leaf and let instead e′ = (u′ +1, u′ +2) ∈ +Tu∗ +2 with u′ +2 being a leaf. Then, +��Tu∗ +2 +�� ≥ d(r, u′ +2) − d(r, u∗ +2) + 1. Using +16 + +this, we get +d(r, u∗ +2) · +��Tu∗ +2 +�� ≥ d(r, u∗ +2) · d(r, u′ +2) − d2(r, u∗ +2) + d(r, u∗ +2) += d(r, u′ +2) +� +d(r, u∗ +2) − d(r, u∗ +2)(d(r, u∗ +2) − 1) +d(r, u′ +2) +� +> d(r, u′ +2) (d(r, u∗ +2) − (d(r, u∗ +2) − 1)) +(5) += d(r, u′ +2) = d(r, u′ +2) · +��Tu′ +2 +�� +where (5) follows from the fact that d(r,u∗ +2) +d(r,u′ +2) < 1 under the assumptions +made. This is a contradiction to d(r, u∗ +2) · +��Tu∗ +2 +�� = mine∈E(T) d(r, u2) · +|Tu2|. +Case 2: d(r, u∗ +2) = 1. In this case, (5) does not hold. We need to distinguish +between two cases. +Case 2.1: u∗ +2 is a leaf. ✓ +Case 2.2: Assume, that u∗ +2 is not a leaf. Again, we distinguish between +two cases. +Case 2.2.1: Assume, that Tu∗ +2 is not a path. Let u′ +2 ∈ Tu∗ +2 be a +leaf. There is at least one vertex u′′ ∈ Tu∗ +2 for which d(u′′) ≥ 2. +Thus, it holds that +d(u∗ +2, u′ +2) ≤ +��Tu∗ +2 +�� − 2 +⇐⇒ d(r, u′ +2) − 1 ≤ +��Tu∗ +2 +�� − 2 +⇐⇒ d(r, u′ +2) < +��Tu∗ +2 +�� +Since d(r, u∗ +2) = |Tu′| = 1, we get d(r, u′) · |Tu′| < d(r, u∗ +2) · Tu∗ +2, +which is a contradiction to d(r, u∗ +2)· +��Tu∗ +2 +�� = mine∈E(T) d(r, u2)· +|Tu2|. +Case 2.2.2: Assume Tu∗ +2 is a path. Then, we find that +mine∈E(T) d(r, u2) · |Tu2| = d(r, u∗ +2) · +��Tu∗ +2 +�� = d(r, u′ +2) · +��Tu′ +2 +�� with +u′ +2 ∈ Tu∗ +2 being a leaf. +Coming back to inequality (4) we conclude, that the right hand side is biggest, +if for e∗ = (u∗ +1, u∗ +2), vertex u∗ +2 is a leaf. +We now want to use the term on the right side of inequality (4) as an upper +bound for the objective function value of the interdiction problem. Therefore, +17 + +let f = (v1, v2) be the edge from Assumption (2), which gets interdicted. +Then, we find the following. +If the optimal solution r to the 1−median problem on T is part of the optimal +solution of the 2−median problem on (T − f) – together with the leaf v2 – +then it holds that +OPT(2, +� +, T − f) = OPT(1, +� +, T) − d(r, v2). +Now suppose, that the optimal solution r to the 1−median problem on T +is not part of the optimal solution of the 2−median problem on (T − f) +anymore. That means there exists s ∈ N(r) in the neighborhood of r which +is part of the optimal solution together with leaf v2. Using the fact, that +d(r, s) = 1, we get that +OPT(2, +� +, T − f) = OPT(1, +� +, T) − d(r, v2) − 1. +(6) +Note, that OPT(2, �, T − f) does not depend on the specific choice of f, +meaning the objective function value is the same for all f fulfilling (2). In +any case it holds true that +OPT(2, +� +, T − f) ≥ OPT(1, +� +, T) − d(r, v2) − 1. +(7) +Now we distinguish between two cases. +Case 1. Let f ′ = (v′ +1, v′ +2) such that v′ +2 is a leaf, but not fulfilling assump- +tion (2). Therefore, d(r, v′ +2) > d(r, v2). Then we get that +OPT(2, +� +, T − f ′) ≤ OPT(1, +� +, T) − d(r, v′ +2) +(8) +≤ OPT(1, +� +, T) − d(r, v2) − 1 +(9) +≤ OPT(2, +� +, T − f) +(10) +where (8) follows from inequality (4), (9) follows from the estimation +of the distances in the current case, and (10) is due to inequality (7). +Case 2. Let f ′ = (v′ +1, v′ +2) such that v′ +2 is not a leaf. Then, with q2 ∈ Tv′ +2 +18 + +being a leaf, it is +OPT(2, +� +, T − f ′) ≤ OPT(1, +� +, T) − d(r, v′ +2) · +��Tv′ +2 +�� +(11) +≤ OPT(1, +� +, T) − d(r, q2) − 1 +(12) +≤ OPT(1, +� +, T) − d(r, v2) − 1 +(13) +≤ OPT(2, +� +, T − f). +(14) +The first line (11) again follows from inequality (4), (12) is due to the +definition of q2, (13) follows from condition (2) and the last estima- +tion (14) follows from inequality (7). +Combining our results of cases 1 and 2, we find that choosing an edge f for +the interdiction step, which fulfills condition (2) yields the optimal objective +function value for the interdictor, namely the biggest objective function value +OPT(2, �, T − f). +Observation 5.2. In case 2.2.2, where d(r, u∗ +2) = 1 and Tu∗ +2 is a path, we +have seen that mine∈E(T) d(r, u2) · |Tu2| = d(r, u∗ +2) · +��Tu∗ +2 +�� = d(r, u′ +2) · +��Tu′ +2 +�� for +the leaf u′ +2. We stated, that we choose the edge connecting the leaf for our +interdiction strategy. In fact, this is mandatory in every case, except for u′ +2 +being the successor of u∗ +2. Let +��Tu∗ +2 +�� > 2. Then, if the edge e∗ is interdicted, +the locator can improve the objective function value of the remainder of the +path, which is not connected to the main tree containing r anymore, with the +placement of the new location. +Remark 5.3. The procedure described above needs to compute an optimal +solution of the 1-median location problem and the distance from all leaves to +this solution. The former computation can be done in time O(n) (cf. [14]), +whereas the latter can be done in time O(n2) by a breadth-first-search on +every vertex. Thus, in total, the procedure runs in O(n2). +Remark 5.4. It is not possible to apply the procedure to generalized problems. +There are several alterations possible, where one can change the input in one +parameter while the rest of the problem setup stays the same. This includes +a change of the interdiction budget (B > 1) or an arbitrary length function +on the edges. For the first problem, we immediately see, that simply choosing +the B leaves closest to the median location and interdict them in one step, +can lead to the following problems. First of all, it is not clear, whether B +19 + +leaves exist in the original graph. But even if there are, the procedure does +not necessarily give the optimal interdiction strategy as example 5.5 shows. +Example 5.5. Let a tree T = (V, E) be given as in Figure 5. Let the inter- +Figure 5: Exemplary tree, optimal median location depicted in blue. +diction budget be given with B = 3. Then, if we choose the 3 leaves closest to +the optimal location, we yield the graph in Figure 6a. The optimal objective +function value of the locator is 15 in this case, but there are better interdiction +strategies. One of the optimal strategies is depicted in Figure 6b and leaves +the locator with an optimal objective function value of 16. +(a) Exemplary tree after interdic- +tion step following the idea of The- +orem 5.1, median locations depicted +in blue. +(b) Exemplary tree after optimal in- +terdiction step, median locations de- +picted in blue. +Figure 6: Exemplary tree of Figure 5 after different interdiction strategies +For the case of arbitrary lengths on the edges, we find, that in case 1 of the +proof, the inequality +��Tu∗ +2 +�� ≥ d(r, u′ +2) − d(r, u∗ +2) + 1 relies on the fact, that the +distances can be calculated via the amount of vertices, which is only possible +because of the unit length values of the edges. In fact, consider the following +minimal example 5.6 illustrating that the proposed method does not work for +arbitrary lengths. +Example 5.6. Let a tree T = (V, E) be given as in Figure 7 and B = 1. +Following the procedure proposed in Theorem 5.1, one of the four edges in- +cident to the leaves gets interdicted resulting in the tree in Figure 8a. The +optimal objective function value of the locator is 35, while the optimal in- +terdiction strategy depicted in Figure 8b leaves the locator with an optimal +objective function value of 41. +20 + +10 +1 +1 +10 +10 +10 +Figure 7: Exemplary tree, optimal median location depicted in blue. +10 +1 +1 +10 +10 +(a) Exemplary tree after interdiction +step following Theorem 5.1, median +locations depicted in blue. +10 +1 +10 +10 +10 +(b) Exemplary tree after optimal in- +terdiction step, median locations de- +picted in blue. +Figure 8: Exemplary tree of Figure 7 after different interdiction strategies +6 +Conclusion +In this article, we introduced the p-median location interdiction problem +(MLIP). We proved MLIP to be NP-hard on trees in Theorem 3.2. We then +considered the MLIP where the underlying graph is given by a path. For the +case of unit lengths, we proved that interdicting the edge incident to a leaf is +an optimal interdiction strategy. This strategy can be applied iteratively, if +more than one edge can get interdicted. For the case of arbitrary lengths, we +showed that an optimal interdiction strategy can be computed in polynomial +time. Furthermore, we proposed a polynomial time algorithm for the case of +unit lengths on a tree. +The cases of arbitrary lengths on a tree graph as well as multiple interdiction +of edges may be considered in the future. It would also be interesting to +investigate the MLIP on different graph classes, e.g. on series-parallel graphs. +Moreover, other types of location problems may be investigated in the context +of interdiction. +Acknowledgments +This work is partially supported by the European Social Fund+ (ESF+), +project SchuMaMoMINT+, and by project Ageing Smart funded by Carl- +Zeiss-Stiftung. +21 + +References +[1] Deniz Aksen, Nuray Piyade, and Necati Aras. The budget constrained r- +interdiction median problem with capacity expansion. Central European +Journal of Operations Research, 18(3):269–291, 2010. +[2] Nikitas Assimakopoulos. A network interdiction model for hospital infec- +tion control. Computers in Biology and Medicine, 17(6):413–422, 1987. +[3] Michael O. Ball, Bruce L. Golden, and Rakesh V. Vohra. Finding the +most vital arcs in a network. Operations Research Letters, 8(2):73–76, +1989. +[4] Amotz Bar-Noy, Samir Khuller, and Baruch Schieber. 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Mathematical and +Computer Modelling, 17(2):1–18, 1993. +24 + diff --git a/jNFST4oBgHgl3EQfGzio/content/tmp_files/load_file.txt b/jNFST4oBgHgl3EQfGzio/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..157262c09690e001c103564c61a17c1a6fc3df92 --- /dev/null +++ b/jNFST4oBgHgl3EQfGzio/content/tmp_files/load_file.txt @@ -0,0 +1,682 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf,len=681 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='13723v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='DS] 31 Jan 2023 p-median location interdiction on trees L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Leiß∗a, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Hellerb, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Sch¨aferc, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Streicherd, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Ruzikaa aDepartment of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' RPTU Kaiserslautern-Landau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 67663 Kaiserslautern,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Germany bDepartment of Optimization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Fraunhofer Institute for Industrial Mathematics ITWM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 67663 Kaiserslautern,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Germany cComma Soft AG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 53229 Bonn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Germany dPTV Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 76131 Karlsruhe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Germany Abstract In p-median location interdiction the aim is to find a subset of edges in a graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' such that the objective value of the p-median problem in the same graph without the selected edges is as large as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We prove that this problem is NP-hard even on acyclic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Restricting the problem to trees with unit lengths on the edges, unit interdiction costs, and a single edge interdiction, we provide an al- gorithm which solves the problem in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Furthermore, we investigate path graphs with unit and arbitrary lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For the former case, we present an algorithm, where multiple edges can get interdicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Furthermore, for the latter case, we present a method to compute an optimal solution for one interdiction step which can also be extended to multiple interdicted edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Keywords: Network Interdiction, Location Planning, Median Problems, Edge Interdiction, Network Location Planning 1 Introduction Location planning is a field of mathematical research which crosses our daily life more often, than we might think at first sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The root of modern ∗Corresponding author, Email address: leiss@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='uni-kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='de (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Leiß) 1 location planning goes back to Pierre de Fermat and aims at finding the point, which minimizes the sum of the Euclidean distances of three given points to the new location (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' A popular, more applied version of this problem is the identification of a new location for a supplier of materials for further industrial processing which has been stated in [30] and is called Weber problem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In a more general approach, a new location is to be found which minimizes the sum of all - possibly weighted - distances from all given locations to the new one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This problem is called median location problem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The underlying structure, on which location problems can be analyzed, may vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Mainly, we distinguish between planar location problems and network location problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In this article, we consider the network case only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' A further alteration of the main problem comes with the number of new locations to be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We refer to the problem of placing p new facilities as the p-median location problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The decision version of the p-median location problem is known to be NP-complete for variable p on general networks, which can be shown by a reduction from the dominating set problem ([18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For this case, there are several heuristics known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' A good overview on this topic can for example be found in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For a general graph with unit length values on the edges and variable p, the problem still remains NP-complete ([12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In contrast, for fixed p, the decision problem is solvable in polynomial time on general graphs by enumeration (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Due to the hardness of the general case, research focused on particular graph structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For G being a tree, the authors in [18] present an O(n2p2)-algorithm to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' A dynamic programming approach based on this result was later proposed in [28], which runs in O(pn2) and got improved by [5] to an algorithm which runs in O(n logp+2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The complexity for path graphs is shown to be O(pn) in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' A linear time algorithm can be applied for the 1-median location problem on trees ([14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' A good overview of location planning can for example be found in [8], [15] or [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' An overview of the solution methods for the p-median location problem in particular can be found in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Interdiction problems pursue the question of how a system can be interrupted in the worst possible way in terms of the original objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The interruption itself can be caused by different acts, such as modification of the edge lengths or deletion of entire edges as well as the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Interdiction problems have gained increasingly more attention lately (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' There are several different applications, that motivate research in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The interdiction of a network can either have a desirable outcome, such as in 2 narrowing the spread of a disease (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [2]), interdicting smuggling routes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [23]) or – as for example done in [32] – the aim of (armed) forces to reduce the amount of drugs and chemicals transported illegally via road or waterways – possibly with limited resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Also – on the contrary – attacks on networks can be interpreted via interdiction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In this case, the analysis of these problems might allow the determination of valuable edges or locations of the original network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Two main optimization problems, which have been studied in the context of interdiction, are the shortest path problem as well as the maximum flow problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Notable research on the first topic has been done in [3], [4], [7] or [17], for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Results on the latter problem might for example be found in [13], [24], [26], [31] or [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In [27], one can find a recent overview of the literature on interdiction problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Research concerning the combination of location and interdiction problems on the other hand is quite scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In this context, we mention the r- interdiction median problem, which is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Given a supply- system with p existing locations, the interdictor wishes to find the subset of r locations, which, when removed, yields the highest weighted distance with respect to the median location function (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' There are a few ex- tensions to this problem, such as the possibility to fortify a fixed number of the existing facilities, which in consequence cannot be interdicted by the attacker anymore (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In [1], the authors alter the concept of fortifying a specified number of locations, but rather introduce a restricted budget for fortification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' A further topic, gaining more interest, is the p-hub interdic- tion problem, which is for example considered in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Still, most of these approaches have in common, that the interdictor intervenes the existing lo- cations and not the underlying network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' An exception to this concept can be found in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Here, the authors combine the covering problem with an edge interdiction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In [11], the authors present a polynomial time al- gorithm for the interdiction problem on trees, where an upfront chosen set of facilities is given and the interdictor wishes to worsen the reachability within the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In [9], the authors combine the median location problem with edge interdiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' To the best of our knowledge, this is the only work on the p- median interdiction problem with edge interdiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' There, they consider the problem for different orders of action of the locator and the interdictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For the case, that the interdictor acts before the locator, they prove this problem to be Σp 2-complete in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In the same work, a bilevel mixed- integer formulation is presented for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This result motivates the 3 analysis of the problem for restricted cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Our contribution Given the complexity analysis in [9], it remains open, if efficient solution procedures can be found if the general problem is restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Coming from the original p-median problem, which is solvable in polynomial time on trees, an obvious variant of the corresponding interdiction problem is to restrict the problem to trees (or even simpler structures) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For such cases, complexity results and solution methods are not yet available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In this article, we aim at closing this research gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We consider the median location problem in combination with an interdictor who can delete edges in a given network and wishes to maximize the objective function value of the locator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We assume, that the interdiction step is executed before the locator places their median facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Due to sigma-2-p hardness for the general case, we consider the problem on particular graph structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We analyze the complexity of the general problem on trees and present an algorithm to solve the problem exactly for some variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Furthermore, we present an algorithm for graphs with path structure for both cases of unit and arbitrary lengths on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Outline The remainder of this article is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In Section 2, we give a short overview of the concepts needed for the article and define the investigated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Section 3 deals with the complexity of the median location interdiction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Section 4 studies the strategy for solving the interdiction median problem on paths with unit length values on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Furthermore, we present a strategy for paths with arbitrary lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The next Section 5 focuses on a tree with unit length values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We state an algo- rithm, which solves the problem exactly in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Section 6 then summarizes the paper and proposes further directions of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 2 Preliminaries and problem formulation Let G = (V (G), E(G)) be an undirected graph with vertex set V (G) = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , vn} and edge set E(G) = {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , em}, where n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='.= |V (G)| and m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='.= |E(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' If the underlying graph is known by the context, we refer to V (G) as V and to E(G) as E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Also, for better readability, instead of |V (G)|, we may write |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For a given vertex v, we denote the number of incident edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' its degree, by deg(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Further, we assign a length value to each 4 edge e ∈ E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=', ℓ: E → Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let Puv be the set of all paths P connecting vertices u, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then, the length of a shortest path between the vertices u and v is denoted by d(u, v), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=', d(u, v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='.= min P ∈Puv ℓ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' If the graph on which the distance is measured is not clear from context we also write dG(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' If G is a tree, let some vertex r ∈ V be the root of the breadth-first-graph of G ([20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We denote by Gv the subtree of G, which is rooted in vertex v and contains all descendants of vertex v in the breadth-first-graph of G with root r as well as their incident edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' As stated, given an instance of the median problem, one aims at placing one (or more) new location(s), which minimize(s) the sum of the shortest path lengths of all existing locations to their nearest facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In this article, we consider the case, that the set of existing locations is the vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' A chosen set X of locations therefore has the objective value: f(X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='.= � v∈V d(v, X), with d(v, X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='.= min x∈X d(v, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Based on this objective, we state the p-median location problem as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' p-median location problem (p, �, G) Instance: Undirected graph G = (V, E), edge lengths ℓ: E → Z+, and number of locations p ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Task: Find a set X ⊆ V of p new locations such that the objective function of the p-median location problem is minimal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' minimize � v∈V d(vi, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The optimal solution is denoted by X∗, while the optimal objective function value is denoted by OPT(p, �, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Network interdiction problems involve an additional opposing force, called the interdictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Said interdictor wishes to worsen the objective function value of the locator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In this article, the interdictor is constrained by an interdiction budget B ∈ Z+, while each edge e ∈ E is associated with an interdiction cost b(e) ∈ Z+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=', b: E → Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Consequently, the set of all feasible interdiction strategies, denoted by Γ, can be expressed as follows: Γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='.= � γ = (γe)e∈E ∈ {0, 1}m | � e∈E b(e) · γe ≤ B � , 5 where γe equals one, if edge e is interdicted or zero, if not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The set of optimal interdiction strategies is denoted by Γ∗ = {γ∗ ∈ Γ | γ∗ optimal}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In what follows, each interdiction strategy γ ∈ Γ induces an undirected graph G(γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='.= (V ′, E′) with V ′ = V and E′ = E \\ E(γ), where E(γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='.= {e ∈ E | γe = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In this article, the locator places their facility on G(γ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' after the interdiction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Based on this, we define the decision version of the p-median location interdiction problem as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Decision version of the p-median location interdiction problem Instance: Undirected graph G = (V, E), edge lengths ℓ: E → Z+, in- terdiction costs b: E → Z+, interdiction budget B ∈ Z+, number of locations p ∈ Z+, and decision parameter K ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Question: Does there exist an interdiction strategy γ ∈ Γ such that min X⊆V,|X|=p � v∈V dG(γ)(v, X) ≥ K ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In the optimization version of the stated problem, we aim to find the maxi- mum K for which the decision version is a yes-instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 3 Complexity results It is well known that the p-median location problem is NP-complete, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Therefore it would be surprising for the corresponding interdiction problem to be polynomial time solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In fact Fr¨ohlich [9] proved that the decision version of the p-median location interdiction problem is Σp 2-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' How- ever, there are several restrictions of the p-median location problem which are proven to lead to polynomial time solvability, as mentioned in the intro- duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Among the restricted versions, that are polynomial time solvable is the p-median location problem on trees, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In this section we prove that adding the interdiction layer to the problem makes the problem significantly harder: The p-median location interdiction problem is NP-complete even on trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For this, we consider the knapsack problem with bounded profit ratio of 2 (K-BPR2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' An instance is given by a set M of m items with associated weights wi ≥ 0 and profits pi ≥ 0 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For the profits 6 it holds pi pj ≤ 2 for all pi, pj with i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We now show that this problem is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The knapsack problem with bounded profit ratio of 2 is NP- complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We reduce the equal partition problem to K-BPR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let an instance of the equal partition problem, which is known to be NP-complete (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [12]), be given with a set I = { ˜w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , ˜wn} such that �n i=1 ˜wi = B ∈ 2 Z, ˜w ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For the equal partition problem we ask for a partitioning of the elements of I into two subsets I1, I2 such that � i∈I1 ˜wi = � i∈I2 ˜wi = B 2 and |I1| = |I2| = n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Define pi = wi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='.= ˜wi + B + 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' According to this definition, the smallest pi is at least B + 1 and the biggest pi is not greater than 2B + 1 such that the ratio pi pj of all pairs pi, pj, i ̸= j is bounded by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Suppose we are given a solution I1 to the equal partition problem, we ask for a solution to the knapsack problem with profit P ≥ B 2 + n 2(B + 1) and total weight W ≤ B 2 + n 2(B + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We state, that the selection I1 is such a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For the profit, it is P = � i∈I1 pi = � i∈I1( ˜wi + B + 1) = B 2 + n 2(B + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Also, W = � i∈W1 wi = � i∈I1( ˜wi + B + 1) = B 2 + n 2(B + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' On the other hand, given a solution J ∈ I to the knapsack problem for which P ≥ B 2 + n 2(B +1) and W ≤ B 2 + n 2(B +1) we show that J together with I \\J is a solution to the equal partition problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' To prove this statement, we need to show that |J| = n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Suppose that |J| < n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then for the profit, it is � j∈J pj = � j∈J( ˜wj+B+1) ≤ B+( n 2−1)(B+1) = n 2(B+1)−1 < P which is a contradiction to the solution of the knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Analogously, suppose that |J| > n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then for the weight it is � j∈J wj = � j∈J( ˜wj + B + 1) ≥ B + ( n 2 + 1)(B + 1) = 2B + 1 + n 2(B + 1) > W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Therefore, we have |J| = n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Finally, calculate � j∈J pj = � j∈J ˜wj + n 2(B + 1) ≥ B 2 + n 2(B + 1) 7 vx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' ℓi = 0 bi = B + 1 ℓi = 0 bi = wi ℓi = pi bi = B + 1 ℓi = 0 bi = B + 1 (a) Red color depicts the edges with lengths ℓi = pi ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' green color depicts the edges with individual interdiction costs bi = wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' (b) One optimal locator’s solu- tion where the placed centers are depicted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Figure 1: The tree graphs used in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' and also � j∈J wi = � j∈J ˜wj + n 2(B + 1) ≤ B 2 + n 2 (B + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Therefore, we get that � j∈J ˜wj = B 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Clearly, given a solution to the K-BPR2, we can verify this solution in poly- nomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Thus, the K-BPR2 is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The p-median location interdiction problem, where the un- derlying graph is given by a tree, is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We show this by reducing K-BPR2 to the location interdiction prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let an instance of K-BPR2 be given by a set M of m items with associated weights wi and profits pi for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For the profit it holds for all pi, pj with i ̸= j that pi pj ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Furthermore, let W, P ∈ Z+ be two integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Consider a tree with lengths ℓi and interdiction costs bi for every edge ei as depicted in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Note, that the construction provides m paths of four vertices emerging from vertex vx and one path of two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let B = W be the interdiction budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For every selection of interdicted edges, one optimal strategy to choose m + 1 centers is depicted in Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 8 vx u2 v2 u1 v1 ℓ1 ℓ2 0 0 0 0 0 0 Figure 2: Excerpt of the tree from Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' To prove this fact, we need to show that every single path emerging from vertex vx must have one center, and furthermore, that the center in the paths with 4 vertices must be below the edges where ℓi = pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In case that every outgoing edge of vx which is part of the m paths of 4 vertices (depicted in green in Figure 1a) is interdicted, the stated solution is clearly optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Now consider the case where w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' the leftmost interdictable edge is interdicted and let the second outgoing edge of vx be non interdicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The notation used can be found in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' It is clear, that every path with interdicted starting edge (the edge outgoing of vx) needs at least one center for feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Also, this center has to be placed below the edge with length greater than zero – in the considered case u1 or a vertex below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Now, the only possibility for the locator to change the objective function value is to spare the center of the non interdicted paths – u2 in the example – and instead place it at v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then, the length ℓ1 does not appear in the calculation of the objective function value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' But instead, all vertices in the second path then need to be covered via center vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' That means, the edge with length ℓ2 needs to be crossed 3 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Even if we as- sume that the lengths have the maximal possible factor ℓ1 = 2 · ℓ2, it would still be better for the locator to place the center at vertex u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The same explanation holds for the shifting of the center in vx to a vertex in an inter- dicted path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In this case, the vertices above vx need to be covered by another center below which is – by the same estimation as before – worse than the 9 provided solution of Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Note that this solution is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In fact – as stated before – the center in the m paths of 4 vertices can be placed at any vertex below the edges where ℓi = pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Also, the center at vx can be shifted up to two vertices up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Assume we are given a solution of the knapsack problem such that the se- lection I ∈ M of items fulfills � i∈I wi ≤ W and � i∈I pi ≥ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Now consider the optimal solution of the locator as stated above after interdiction of the edges ei, i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For the interdiction costs, it is � i∈I bi = � i∈I wi ≤ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Fur- thermore, the corresponding objective function value calculates as � i∈I ℓi = � i∈I pi ≥ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' On the other hand, if we are given a selection of interdicted edges J ∈ EM such that � j∈J bj ≤ B and � j∈J ℓj ≥ P, we show that J is a solution to K-BPR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Firstly, � j∈J wj = � j∈J bj ≤ B = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Also, the profit calculates as � j∈J pj = � j∈J ℓj ≥ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Given an interdiction strategy γ, the corresponding p-median problem on G(γ) can be solved in polynomial time as G(γ) still does not contain cycles, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Thus, the p-median interdiction problem on trees is contained in NP and therefore NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The p-median location interdiction problem is NP-complete, even on trees in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In the remainder of the article, we further restrict the problem in order to get a better impression on what makes the problem hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' A direct consequence of the p-median location interdiction problem on trees beeing contained in NP, is that regarding the interdiction budget as a con- stant makes the problem polynomial time solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This holds true, as hav- ing a constant interdiction budget makes the number of possible interdiction strategies polynomial in the instance size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Thus, the procedure of solving a p-median problem for all interdiction strategies and thereby finding the best strategy runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The p-median interdiction problem on trees is polynomial time solvable if the interdiction budget is considered a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Further possible restriction are to simplify the graph structure even more, making the edges have unit interdiction costs, making the edges have constant or even unit lengths, or restricting the number p of locations to be placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Any combinations of these restrictions is also interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' As analyzing all 10 possible and interesting combinations would exceed the scope of a single article, we focus on the subcase of unit interdiction cost and regard different further restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In the next section, we consider the p-median location interdiction problem on paths with unit interdiction costs and afterwards move back to the same problem on trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 4 Interdicting a path In this section, let G = (V, E) be a graph with V = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , vn} and E = {ei = {vi, vi+1}: i = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , n − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The resulting graph has the structure of a path consisting of n vertices and n − 1 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For the remainder, we refer to these types of graphs as paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In this section we tackle the p-median location interdiction problem on paths, where we additionally assume unit interdiction costs and p = B+1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' every component emerging from the interdiction can be equipped with exactly one location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Note that for unit interdiction costs, we may assume B ≤ n−1, as a path only contains n−1 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We first elaborate on paths with unit lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In this case the interdictor can use a simple method to worsen the situation for the locator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This procedure is initially analyzed for one interdiction step (B = 1) and is then generalized to arbitrary interdiction budgets B > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' After that, we show, how paths with arbitrary lengths can be handled for B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We want to briefly present the idea of Goldman’s algorithm (cf [14]), since it is needed in the remainder of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For a tree T, we start at an arbitrary leaf and compare the weight of that leaf to the total summarized weight of all vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' As long as the weight of the leaf is less than half of the total weight, we delete the leaf and update the weight of the adjacent vertex by adding the weight of the deleted leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In that manner, we iterate over the leaves until we find one with a weight greater or equal to the half of the total weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This vertex is the 1-median of the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' With this method, we are also able to state the 1-median location(s) on a path, which is dependent on the number of the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The optimal solution(s) on a path is to place the new location(s) at vertex vn/2−1 or vn/2 for n even or at vertex v⌈n/2⌉ for n odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='1 Paths with unit edge weigths Let the graph be a path P = (v1, e1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , en−1, vn) with ℓ ≡ 1 and b ≡ 1 as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We first examine the case for B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let P be a path with ℓ ≡ 1 and b ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The optimal interdiction strategies for the p-median location interdiction problem are to interdict e1 or en−1, yielding in an isolated vertex and a new path of length n − 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Γ∗ = {γ∗ 1 = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , 0), γ∗ 2 = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , 0, 1)}, where the order of γ∗ i , i = 1, 2 is induced by the order of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' As described above, the optimal solution(s) for the 1-median problem are at vertex vn/2−1 or vn/2 for n even or at vertex v⌈n/2⌉ for n odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The resulting objective function value OPT(1, �, P) can then be computed via: OPT(1, � , P) = � n2 4 n even n2−1 4 else Every interdiction strategy with B = 1 results in two components of the original path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Since p = 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' one new location in each component, we compute the optimal objective function value for the 2-median problem un- der the given setting by adding up both objective function values computed separately for every component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Therefore, let et be the interdicted edge for some t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , n − 1} yielding a separation of the original path P into Pt = (v1, e1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , et−1, vt) and Pn−t = (vt+1, et+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , en−1, vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then, the objective function for the overall problem of placing one new location in either part is z∗ = OPT(1, � , Pt) + OPT(1, � , Pn−t) = 1 4(2t2 + n2 − 2nt − a) (1) with a ∈ {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The goal of the interdictor is to choose t such that z∗ is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For a given case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' where n and a are fixed, we can reduce equation 1 to the following expression t2 − nt = � t − n 2 �2 − n2 4 for the computation of the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Since we aim to maximize the latter expression, it can again be reduced to � t − n 2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' vq vr vr+1 vs Figure 3: Detail of interdicted path P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' interdicted edges are depicted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , n − 1}, the maximum is found at t1 = 1 or t2 = n − 1, which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This result allows to expand the considerations to B ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let a path P = (v1, e1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , en−1, vn) be given with ℓ ≡ 1, b ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The optimal interdiction strategy under an interdiction budget B ≤ n − 1 is to successively interdict the edges incident to leaves, thus resulting in B single vertices and one path of length n − 1 − B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Consider an optimal interdiction strategy γ′, where at least one of the interdicted edges does not cut off a leaf as depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let er be the described edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Given there exists an edge with the stated proper- ties, there also exist sets of vertices Vq = {vq ∈ V : q < r, deg(vq) = 1} and Vs = {vs ∈ V : r < s, deg(vs) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let vq ∈ Vq be the vertex with the biggest index and vs ∈ Vs the vertex with the smallest index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This require- ment ensures that the component Pqs = (vq, eq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , es−1, vs) of the original path is only interdicted once at er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Now consider Pqs, which is interdicted at er with strategy γ′, resulting in two new paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Using the result of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='1, the optimal objective func- tion value for the interdiction of path Pqs does not decrease by interdicting eq instead of er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Therefore, successively using this method of shifting the interdicted edges to the leftmost edge of the respective components will also not decrease the overall objective value of γ′ yielding an optimal interdiction strategy γ∗ as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='2 Paths with arbitrary lengths For the remainder of the section, we assume an arbitrary length function ℓ to be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' One obvious strategy is to interdict all edges successively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In each step at a time, we compute the optimal objective function value for the locator by solving two median location problems on the remaining paths after the current interdiction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Evaluating over all obtained objective function 13 values yields the best edge to interdict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We present an approach which effi- ciently iterates over all edges by using an interesting structure of a matrix, which helps computing the locators objective function values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let a path P = (v1, e1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , en−1, vn) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The median location is found at ver- tex v⌈n/2⌉ for n odd or at vertex vn/2 or vn/2+1 for n even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' As stated in Section 2, the objective function value for the 1−median problem on the given path is determined by the total number of times, each edge is crossed to reach all vertices from the median location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Given the structure of a path, these numbers are bounded by ⌊n/2⌋ if n is odd and by n/2 if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' More pre- cisely, these bounds hold for the edges e⌊n/2⌋ and e⌈n/2⌉ incident to the median location (n odd) or the edge en/2 (n even), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Furthermore, this number decreases by one the closer the edges are to the leaves of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='3 shows the case for a path of length 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let P = (v1, e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , e6, v7) a path with length values ℓi, i = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , 6} as depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' With n odd and the observations above, we get that the median is located at v⌈n/2⌉ = v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Furthermore, we can determine the number of times si the edge ei is represented in the objective function value (depicted in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' v1 v2 v3 v5 v6 v7 v4 1 l1 2 l2 3 l3 3 l4 2 l5 1 l6 The information of how often an edge length contributes to the objective function value can be stored in a vector S ∈ Nn−1, where each entry repre- sents one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Multiplying S with the length vector ℓ yields the optimal objective function value for the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We use this scheme for the construction of the matrix calculating the ob- jective function values for different interdicted edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Assume that some edge et, t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , n − 1} gets interdicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This results in the two paths Pt,1 = (v1, e1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , et−1, vt) and Pt,2 = (vt+1, et+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , en−1, vn), for which the objective function values can be calculated separately as stated via the vectors St,1 = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , st−1) ∈ for P1 and St,2 = (st+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , sn−1) for P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The union yields a new vector St = (St,1, 0, St,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Assuming that we only interdict once, we can proceed to build St for all edges et, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , n sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The matrix S = (St)t ∈ N(n−1)×(n−1) can again be multiplied with ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Evaluat- ing for the biggest objective function value solves the 1−interdiction median problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='4 shows the matrix for the path of Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 14 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let P be the path of Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Furthermore, let the length vector ℓ be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The matrix obtained via the presented method is as follows: \uf8eb \uf8ec \uf8ec \uf8ed 0 1 2 3 2 1 1 0 1 2 2 1 1 1 0 1 2 1 1 2 1 0 1 1 1 2 2 1 0 1 1 2 3 2 1 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 As stated, the matrix S is of size (n − 1) × (n − 1), where each row i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' , n − 1} represents the objective function value if edge ei gets interdicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Therefore, the procedure explained runs in time O(n2) since the multiplication of matrix S with the given length vector ℓ is of stated time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We want to mention, that an increase of interdiction steps adds factor n to the running time for each single interdiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This is due to the fact, that we need to consider every possible combination for the edges, that get interdicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 5 Interdicting a tree In this section, we show, how to interdict a tree T = (V, E) with ℓ ≡ 1, b ≡ 1, B = 1 and p = B + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For T = (V, E) with ℓ ≡ 1, b ≡ 1, B = 1 and p = B + 1, the optimal interdiction strategy is as follows: among all leaves of T, find one, that has the least distance to at least one optimal solution of the 1−median location problem on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Interdict the edge incident to this leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Since the proof is constructive, consider the exemplary tree in Figure 4 which illustrates the used structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let r ∈ V (T) be an optimal solution to the 1−median location problem on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Also, let T be rooted in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let f = (v1, v2) ∈ E(T) be an edge with v2 being a leaf that fulfills d(r, v2) = min r∗∈X∗ l leaf d(r∗, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' (2) For all neighbors w ∈ N(r), it holds that |Tw| ≤ |T − Tw| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' (3) 15 r u∗ 1 u∗ 2 u1′ u2′ w1 w2 w3 e∗ e′ Figure 4: Exemplary tree for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Assume, that |Tw| > |T − Tw|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then, w would yield a better objective func- tion value for the 1−median location problem than r, which is a contradiction to the assumption, that r is optimal for said problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Now, for every edge e = (u1, u2) ∈ E(T) with d(r, u1) < d(r, u2), it is OPT(2, � , T − e) ≤ OPT(1, � , T) − (d(r, u2) · |Tu2|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' (4) This is, since placing a second location in the tree Tu2 saves at least |Tu2| times the distance d(r, u2) in comparison to the solution of the 1−median problem on the original tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We aim at finding e∗ = (u∗ 1, u∗ 2) such that d(r, u∗ 2) · ��Tu∗ 2 �� = mine∈E(T) d(r, u2) · |Tu2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This leads to the biggest right hand side of inequality (4), thus provid- ing an upper bound for the optimal objective function value OPT(2, �, T − e), which the interdictor maximizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We claim, that u∗ 2 is a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Case 1: d(r, u∗ 2) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Suppose, u∗ 2 is not a leaf and let instead e′ = (u′ 1, u′ 2) ∈ Tu∗ 2 with u′ 2 being a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then, ��Tu∗ 2 �� ≥ d(r, u′ 2) − d(r, u∗ 2) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Using 16 this, we get d(r, u∗ 2) · ��Tu∗ 2 �� ≥ d(r, u∗ 2) · d(r, u′ 2) − d2(r, u∗ 2) + d(r, u∗ 2) = d(r, u′ 2) � d(r, u∗ 2) − d(r, u∗ 2)(d(r, u∗ 2) − 1) d(r, u′ 2) � > d(r, u′ 2) (d(r, u∗ 2) − (d(r, u∗ 2) − 1)) (5) = d(r, u′ 2) = d(r, u′ 2) · ��Tu′ 2 �� where (5) follows from the fact that d(r,u∗ 2) d(r,u′ 2) < 1 under the assumptions made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This is a contradiction to d(r, u∗ 2) · ��Tu∗ 2 �� = mine∈E(T) d(r, u2) · |Tu2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Case 2: d(r, u∗ 2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In this case, (5) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We need to distinguish between two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='1: u∗ 2 is a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' ✓ Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='2: Assume, that u∗ 2 is not a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Again, we distinguish between two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='1: Assume, that Tu∗ 2 is not a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let u′ 2 ∈ Tu∗ 2 be a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' There is at least one vertex u′′ ∈ Tu∗ 2 for which d(u′′) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Thus, it holds that d(u∗ 2, u′ 2) ≤ ��Tu∗ 2 �� − 2 ⇐⇒ d(r, u′ 2) − 1 ≤ ��Tu∗ 2 �� − 2 ⇐⇒ d(r, u′ 2) < ��Tu∗ 2 �� Since d(r, u∗ 2) = |Tu′| = 1, we get d(r, u′) · |Tu′| < d(r, u∗ 2) · Tu∗ 2, which is a contradiction to d(r, u∗ 2)· ��Tu∗ 2 �� = mine∈E(T) d(r, u2)· |Tu2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='2: Assume Tu∗ 2 is a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then, we find that mine∈E(T) d(r, u2) · |Tu2| = d(r, u∗ 2) · ��Tu∗ 2 �� = d(r, u′ 2) · ��Tu′ 2 �� with u′ 2 ∈ Tu∗ 2 being a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Coming back to inequality (4) we conclude, that the right hand side is biggest, if for e∗ = (u∗ 1, u∗ 2), vertex u∗ 2 is a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We now want to use the term on the right side of inequality (4) as an upper bound for the objective function value of the interdiction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Therefore, 17 let f = (v1, v2) be the edge from Assumption (2), which gets interdicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then, we find the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' If the optimal solution r to the 1−median problem on T is part of the optimal solution of the 2−median problem on (T − f) – together with the leaf v2 – then it holds that OPT(2, � , T − f) = OPT(1, � , T) − d(r, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Now suppose, that the optimal solution r to the 1−median problem on T is not part of the optimal solution of the 2−median problem on (T − f) anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' That means there exists s ∈ N(r) in the neighborhood of r which is part of the optimal solution together with leaf v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Using the fact, that d(r, s) = 1, we get that OPT(2, � , T − f) = OPT(1, � , T) − d(r, v2) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' (6) Note, that OPT(2, �, T − f) does not depend on the specific choice of f, meaning the objective function value is the same for all f fulfilling (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In any case it holds true that OPT(2, � , T − f) ≥ OPT(1, � , T) − d(r, v2) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' (7) Now we distinguish between two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let f ′ = (v′ 1, v′ 2) such that v′ 2 is a leaf, but not fulfilling assump- tion (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Therefore, d(r, v′ 2) > d(r, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then we get that OPT(2, � , T − f ′) ≤ OPT(1, � , T) − d(r, v′ 2) (8) ≤ OPT(1, � , T) − d(r, v2) − 1 (9) ≤ OPT(2, � , T − f) (10) where (8) follows from inequality (4), (9) follows from the estimation of the distances in the current case, and (10) is due to inequality (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let f ′ = (v′ 1, v′ 2) such that v′ 2 is not a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then, with q2 ∈ Tv′ 2 18 being a leaf, it is OPT(2, � , T − f ′) ≤ OPT(1, � , T) − d(r, v′ 2) · ��Tv′ 2 �� (11) ≤ OPT(1, � , T) − d(r, q2) − 1 (12) ≤ OPT(1, � , T) − d(r, v2) − 1 (13) ≤ OPT(2, � , T − f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' (14) The first line (11) again follows from inequality (4), (12) is due to the definition of q2, (13) follows from condition (2) and the last estima- tion (14) follows from inequality (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Combining our results of cases 1 and 2, we find that choosing an edge f for the interdiction step, which fulfills condition (2) yields the optimal objective function value for the interdictor, namely the biggest objective function value OPT(2, �, T − f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Observation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='2, where d(r, u∗ 2) = 1 and Tu∗ 2 is a path, we have seen that mine∈E(T) d(r, u2) · |Tu2| = d(r, u∗ 2) · ��Tu∗ 2 �� = d(r, u′ 2) · ��Tu′ 2 �� for the leaf u′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We stated, that we choose the edge connecting the leaf for our interdiction strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In fact, this is mandatory in every case, except for u′ 2 being the successor of u∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let ��Tu∗ 2 �� > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then, if the edge e∗ is interdicted, the locator can improve the objective function value of the remainder of the path, which is not connected to the main tree containing r anymore, with the placement of the new location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The procedure described above needs to compute an optimal solution of the 1-median location problem and the distance from all leaves to this solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The former computation can be done in time O(n) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [14]), whereas the latter can be done in time O(n2) by a breadth-first-search on every vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Thus, in total, the procedure runs in O(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' It is not possible to apply the procedure to generalized problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' There are several alterations possible, where one can change the input in one parameter while the rest of the problem setup stays the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This includes a change of the interdiction budget (B > 1) or an arbitrary length function on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For the first problem, we immediately see, that simply choosing the B leaves closest to the median location and interdict them in one step, can lead to the following problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' First of all, it is not clear, whether B 19 leaves exist in the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' But even if there are, the procedure does not necessarily give the optimal interdiction strategy as example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='5 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let a tree T = (V, E) be given as in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let the inter- Figure 5: Exemplary tree, optimal median location depicted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' diction budget be given with B = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Then, if we choose the 3 leaves closest to the optimal location, we yield the graph in Figure 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The optimal objective function value of the locator is 15 in this case, but there are better interdiction strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' One of the optimal strategies is depicted in Figure 6b and leaves the locator with an optimal objective function value of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' (a) Exemplary tree after interdic- tion step following the idea of The- orem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='1, median locations depicted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' (b) Exemplary tree after optimal in- terdiction step, median locations de- picted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Figure 6: Exemplary tree of Figure 5 after different interdiction strategies For the case of arbitrary lengths on the edges, we find, that in case 1 of the proof, the inequality ��Tu∗ 2 �� ≥ d(r, u′ 2) − d(r, u∗ 2) + 1 relies on the fact, that the distances can be calculated via the amount of vertices, which is only possible because of the unit length values of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' In fact, consider the following minimal example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='6 illustrating that the proposed method does not work for arbitrary lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Let a tree T = (V, E) be given as in Figure 7 and B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Following the procedure proposed in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='1, one of the four edges in- cident to the leaves gets interdicted resulting in the tree in Figure 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The optimal objective function value of the locator is 35, while the optimal in- terdiction strategy depicted in Figure 8b leaves the locator with an optimal objective function value of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 20 10 1 1 10 10 10 Figure 7: Exemplary tree, optimal median location depicted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 10 1 1 10 10 (a) Exemplary tree after interdiction step following Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='1, median locations depicted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 10 1 10 10 10 (b) Exemplary tree after optimal in- terdiction step, median locations de- picted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Figure 8: Exemplary tree of Figure 7 after different interdiction strategies 6 Conclusion In this article, we introduced the p-median location interdiction problem (MLIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We proved MLIP to be NP-hard on trees in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' We then considered the MLIP where the underlying graph is given by a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For the case of unit lengths, we proved that interdicting the edge incident to a leaf is an optimal interdiction strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' This strategy can be applied iteratively, if more than one edge can get interdicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' For the case of arbitrary lengths, we showed that an optimal interdiction strategy can be computed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Furthermore, we proposed a polynomial time algorithm for the case of unit lengths on a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' The cases of arbitrary lengths on a tree graph as well as multiple interdiction of edges may be considered in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' It would also be interesting to investigate the MLIP on different graph classes, e.' metadata={'source': 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for the p-median problem: An annotated bibliography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Networks: An International Journal, 48(3):125–142, 10 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [26] Luca E Sch¨afer, Stefan Ruzika, Sven O Krumke, and Carlos M Fonseca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' On the bicriterion maximum flow network interdiction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content='02730, 2020.' metadata={'source': 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related problems on tree graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Operations Research Letters, 19(2):59–64, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [29] Thomas Ullmert, Stefan Ruzika, and Anita Sch¨obel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' On the p-hub inter- diction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Computers & Operations Research, 124:105056, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' [30] Alfred Weber and Georg Pick.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Deterministic network interdiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' Mathematical and Computer Modelling, 17(2):1–18, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} +page_content=' 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNFST4oBgHgl3EQfGzio/content/2301.13723v1.pdf'} diff --git a/lNFPT4oBgHgl3EQf2zXD/vector_store/index.faiss b/lNFPT4oBgHgl3EQf2zXD/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..38ccfa1c99fcfd3455bb37b547a5b1c5c1064a60 --- /dev/null +++ b/lNFPT4oBgHgl3EQf2zXD/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7b5a2b7e5712aa4500f546bb349a9c02acee5d14ce715b7d84189824b4703a29 +size 4980781 diff --git a/ltE2T4oBgHgl3EQfywiY/content/tmp_files/2301.04124v1.pdf.txt b/ltE2T4oBgHgl3EQfywiY/content/tmp_files/2301.04124v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f87c14893e9c123385fdeac1be45ad4d24b4fa02 --- /dev/null +++ b/ltE2T4oBgHgl3EQfywiY/content/tmp_files/2301.04124v1.pdf.txt @@ -0,0 +1,1686 @@ +Characterizing the Fundamental Bending Vibration of a Linear Polyatomic Molecule +for Symmetry Violation Searches +Arian Jadbabaie,1, ∗ Yuiki Takahashi,1, ∗ Nickolas H. Pilgram,2, † +Chandler J. Conn,1 Yi Zeng,1 Chi Zhang,1 and Nicholas R. Hutzler1 +1California Institute of Technology, Division of Physics, +Mathematics, and Astronomy. +Pasadena, CA 91125 +2California Institute of Technology, Division of Engineering and Applied Science. Pasadena, CA 91125 +(Dated: January 11, 2023) +Polyatomic molecules have been identified as sensitive probes of charge-parity violating and parity +violating physics beyond the Standard Model (BSM). For example, many linear triatomic molecules +are both laser-coolable and have parity doublets in the ground electronic ˜ +X2Σ+(010) state aris- +ing from the bending vibration, both features that can greatly aid BSM searches. Understanding +the ˜ +X2Σ+(010) state is a crucial prerequisite to precision measurements with linear polyatomic +molecules. Here, we characterize fundamental bending vibration of 174YbOH using high-resolution +optical spectroscopy on the nominally forbidden ˜ +X2Σ+(010) → ˜A2Π1/2(000) transition at 588 nm. +We assign 39 transitions originating from the lowest rotational levels of the ˜ +X2Σ+(010) state, and +accurately model the state’s structure with an effective Hamiltonian using best-fit parameters. Addi- +tionally, we perform Stark and Zeeman spectroscopy on the ˜ +X2Σ+(010) state and fit the molecule- +frame dipole moment to Dmol = 2.16(1) D and the effective electron g-factor to gS = 2.07(2). +Further, we use an empirical model to explain observed anomalous line intensities in terms of inter- +ference from spin-orbit and vibronic perturbations in the excited ˜A2Π1/2(000) state. Our work is +an essential step toward searches for BSM physics in YbOH and other linear polyatomic molecules. +I. +INTRODUCTION +Polyatomic molecules are at the frontier of advanced control over quantum complexity. Their additional rovibra- +tional degrees of freedom provide a large degree of control and tunability of both molecular structure and interactions +with a wide range of applications. Rapid progress [1–3] has been made in laser cooling molecules, including polyatomic +CaOH [4, 5], CaOCH3 [6], SrOH [7, 8], and YbOH [9]. Recently, CaOH was optically trapped and laser-cooled to +ultracold temperatures [10, 11]. Quantum control of polyatomic molecules will benefit next-generation searches for +new physics beyond the Standard Model [12–15], and will enable advances in quantum computation, simulation, and +chemistry [16–19]. +Currently, measurements of diatomic ThO and HfF+ bound charge-parity (CP) violating new physics at TeV energy +scales [20, 21]. These experiments benefit significantly from parity doubling, the occurrence of nearly-degenerate levels +of opposite parity. Molecules with parity doublets can be easily aligned in the lab frame with the application of modest +electric fields. Furthermore, when polarized, these molecules have both aligned and anti-aligned states. Known as +internal co-magnetometers, these states allow for reversal of CP-violating interactions without modifying the external +lab field [22]. This degree of control over molecular alignment is highly advantageous for robust systematic error +rejection in searches for CP violation. In diatomic molecules, parity doublets require orbital angular momentum, +which conflicts with electronic requirements for efficient laser cooling, especially for heavy molecules with enhanced +sensitivity to new physics [12, 15]. +Polyatomic molecules offer both generic parity doublets and laser cooling, and therefore provide a route to sig- +nificantly improve constraints on new CP-violating physics by multiple orders of magnitude [12]. A number of CP +∗ These authors contributed equally. +† Current affiliation: NIST Physical Measurement Laboratory. Gaithersburg, MD 20899 +arXiv:2301.04124v1 [physics.atom-ph] 10 Jan 2023 + +2 +violation searches are underway with laser-coolable diatomic molecules, such as BaF [23], YbF [24, 25], TlF [26], and +RaF [27, 28]. Without parity doublets in their ground states, these molecules require large electric fields (>10 kV/cm) +for significant polarization. By contrast, molecules with parity doublets offer similar polarization in much smaller +fields, and the variety of molecular orientations offer richer possibilities for state tuning. In polyatomic molecules, +parity doublets arise from rotation around the inter-nuclear axis and exist independently of the electronic structure +used for laser cooling [1, 2, 12]. Examples of polyatomic parity doublets include K doublets in rotations of symmetric +molecules, asymmetry doublets in the rotations of asymmetric molecules, and ℓ doublets in bending modes of linear +polyatomic molecules. +YbOH molecules in their doubly-degenerate bending mode have been identified as sensitive probes of CP-violating +physics [12]. The Yb-centered, core-penetrating valence electron provides both new physics sensitivity and optical +cycling, which was demonstrated with Sisyphus cooling of a YbOH beam to a transverse temperature of <600 µK [9]. +Meanwhile, the vibrational bending motion provides ℓ-type parity doublets that allow polarization control and internal +co-magnetometry in modest external fields. Furthermore, the multiple stable isotopes of Yb provide opportunities +for CP violation searches in both the hadronic and leptonic sectors of the Standard Model [12, 29–35]. +Finally, +other experiments leveraging the bending motion of linear triatomic molecules, including CP violation searches with +SrOH [36] and RaOH [12, 37], and parity-violation searches with linear triatomics [38], warrant further investigation +of these states, for which there is no previous, complete study of all molecular properties. +Here, we present a high-resolution, optical spectroscopy study of the fundamental bending vibration in the electronic +ground state of 174YbOH. The spectra are obtained by laser excitation on a rovibrationally forbidden electronic +transition in a cryogenic buffer gas beam (CBGB). By analyzing the field-free, Stark, and Zeeman spectra, we model +the rotational structure of the bending molecule, characterize the electric and magnetic tuning of the levels, and +extract the molecule-frame dipole moment. Our results demonstrate the high level of control available in polyatomic +molecules, which will be useful for future symmetry violation searches. +The structure of the paper is as follows. First, we provide a brief overview of the overall molecular structure in +section I A. The methods are described in section II, with section II A describing the experimental apparatus, and +section II B describing the effective Hamiltonians used to model the molecular states. In section III we describe our +experimental results and analysis. Section III A discusses the field-free spectrum and optimal state parameters, section +III C describes our model for the anomalous line intensities of the forbidden transition, and section III B presents the +Stark and Zeeman spectra and their analysis. We conclude in section IV. +A. +Molecular Structure +In this section, we briefly review the structure of linear polyatomic molecules, including states with bending vibra- +tion. We label the ground and excited state electronic states as ˜X and ˜A, respectively. Electronic states of linear +polyatomic molecules are labeled with the term symbol 2S+1ΛΩ(v1 vl +2 v3), where Λ = ⃗L · ˆn is the projection of elec- +tronic orbital angular momentum L on the internuclear axis ˆn, Σ = ⃗S · ˆn is the projection of the electron spin S, +Ω = Λ + Σ = ⃗J · ˆn is the total projection of the spin and rotational angular momentum J, and vi denotes the number +of quanta in the three vibrational modes of the molecule. For Λ = 0 states, an additional +/− subscript is used to +denote the parity of the electronic configuration, and the Ω subscript is sometimes dropped. In YbOH [12], the v1 +mode is the Yb-O stretch, the v3 mode the O-H stretch, and, due to the Yb mass, the doubly-degenerate vℓ +2 mode +can be viewed as the bending of the H atom relative to the Yb-O axis [39]. The additional ℓ label denotes the number +of quanta of vibrational angular momentum G projected on the internuclear axis, ℓ = ⃗G · ˆn. The degeneracy of ±ℓ +states are lifted by higher order perturbations, giving rise to parity doublets [40, 41]. +The above electronic labeling scheme treats the vibrational degrees of freedom separately. However, for states with +non-zero ℓ and Λ, interactions of the electrons with the bending vibration, known as Renner-Teller couplings [42, 43], +will cause rovibrational splittings for different states of K = Λ+ℓ = ⃗N·ˆn. Here is ⃗N = ⃗J−⃗S is the rovibrational angular +momentum of the electrons and nuclei, excluding spin. Note that N can receive contributions from multiple sources: +the end-over-end molecular rotation R, electronic orbital angular momentum L, and vibrational angular momentum + +3 +G. When both Renner-Teller and spin-orbit couplings are present, neither K nor Ω are completely conserved, and +instead the eigenstates have well defined projection quantum number P = ⃗J · ˆn = Λ + ℓ + Σ. We note that the total +angular momentum cannot be less than the projection angular momentum. For example, in a state with well-defined +N and K, we always have N ≥ |K|; a consequence relevant for this work is that the lowest rotational level of an ℓ = 1 +bending mode has N = 1. +We will restrict our discussion to states with v1 = v3 = 0 and v2 ∈ {1, 0}, allowing us to write vibronic term symbols +as 2S+1KP . Note that in the term symbols, both Λ and K are denoted as Σ, Π, ∆, . . . to indicate 0, 1, 2, . . ., similar +to the S, P, D, . . . notation in atoms. This can lead to confusion; for example the (010) vibrational state in the ground +electronic state is a 2Σ+ electronic state, but a 2Π vibronic state. Whenever we do not include the (v1 v2 v3) label, +we are referring to a vibronic term, unless otherwise noted. +In this work, we study the ˜X2Σ+ +1/2(0110) → ˜A2Π1/2(000) band of 174YbOH. This transition is nominally forbidden +in the dipole approximation, which requires ∆ℓ = 0, and it occurs via intensity borrowing in the excited state, +as we discuss later. We will neglect the other spin-orbit manifold, ˜A2Π3/2(000), which is located ∼40 THz above +˜A2Π1/2(000). The large spin-orbit coupling in YbOH means Ω is an approximately good quantum number, even in +bending states. For simplicity, we will abbreviate the ground state label as ˜X(010) and the excited state label as +˜A(000). +In 174YbOH, the 174Yb nucleus has no nuclear spin, and the hyperfine structure from the distant hydrogen nuclear +spin I is optically unresolved [44] and only contributes to broadening in the ground state. Therefore in this study we +neglect I, and label states with well-defined total angular momentum J. +Ground state quantum numbers are denoted with a double prime, e.g. N ′′, and excited states with a single prime, +e.g. J′. We denote rotational lines with notation similar to Ref. [45]. Given the parity doubling in both ˜X(010) and +˜A(000), we add an additional label to denote the parity of the ground state. We label transitions as ∆N∆JP′′ +F ′ +i ,F ′′ +i (N ′′). +Here, F ′ +i = 1 for the excited state, F ′′ +i = 1, 2 denotes ground states with J′′ = N ′′ ± S, and P′′ = ± denotes the +ground state parity. +II. +METHODS +A. +Experiment: Apparatus and Signals +The cryogenic buffer gas beam (CBGB) apparatus (Fig. 1a) is similar to that from our previous work [46, 47]. In +summary, the buffer gas cell is formed from a copper block with an interior cylindrical bore 7.5 cm long and 12.7 mm +in diameter, with windows on the sides for optical access. The cell is surrounded by radiation shields and cooled by +a pulse tube refrigerator down to ∼4 K. Helium buffer gas is introduced in the back of the cell via a 3.2 mm gas +inlet tube, and passes through a diffuser 3.2 mm downstream in the cell. Typical flow rates are 3 − 6 standard cubic +centimeters per minute (SCCM). The buffer gas exits the cell via a 5 mm diameter aperture at the front of the cell. +Activated charcoal fins on the interior surface of the 4 K radiation shields provide efficient cryo-pumping of the He +buffer gas. +YbOH molecules are produced by laser ablation of pressed powder targets made from a 1:1 stoichiometric mixture +of Yb(OH)3 powder and Yb powder (see supplementary materials). Laser ablation is performed by a Nd:YAG laser at +532 nm with ∼10 ns pulse length, 25−40 mJ pulse energy, and ∼9 Hz repetition rate. The ablation laser is focused with +a 300 or 400 mm lens placed approximately one focal length away from the target. Hot molecules produced via ablation +are subsequently thermalized by collisions with ∼4 K He buffer gas atoms [48]. We further increase YbOH yield by +around an order of magnitude by exciting atomic Yb to the excited 3P1 state [46]. Specifically, we send ∼300 mW of +556 nm light into the cell to resonantly drive the 1S0 → 3P1 transition of 174Yb. This technique significantly increases +the quantity of YbOH in excited vibrational states, including the ˜X(010) state, whose population is increased by a +factor of ∼10. +A few milliseconds after ablation, the He gas flow extracts the molecules out of the cell through the aperture. +Molecule density is monitored both in the cell and outside the cell aperture with 577 nm absorption probes resonant + +4 +Time After Ablation (ms) +Ablation +Skimmer +Collimator +Fluorescence probe +YbOH +4 K +PMT +Photodiodes +Target +Fill +line +Buffer +gas +Chemical +enhancement +Absorption +probes +ITO-coated +electrodes +Light +pipe +Optical +filters +Magnetic +field coils +(ii) +(iii) +(a) +in Cell +Front of Cell +in Beam +(b) +(i) +FIG. 1. +(a) Experimental schematic. YbOH molecules are produced in the 4 K cryogenic buffer gas cell (brown box) by +laser ablation (dark green triangle) of a solid pressed target. The molecules are thermalized by collisions with He buffer gas +continuously flowed into the cell. Chemical production of YbOH is enhanced by exciting Yb atoms using a laser (light green +line) resonant with the 1S0 → 3P1 atomic Yb transition. Some of the molecules are produced in the ˜ +X(010) bending mode. +The molecules are entrained in the He gas flow and extracted out of the cell. We detect the molecule number density in the +˜ +X(000) state via absorption spectroscopy (yellow lines) both in the cell (i) and in front of the cell (ii). The molecular beam is +collimated by a skimmer and collimators before entering the probe region with electric and magnetic fields. We apply magnetic +fields using coils outside the vacuum chamber, and apply electric fields using ITO coated glass electrodes inside the vacuum +chamber. In the center of the fields, molecules in the ˜ +X(010) state are excited by a laser (orange line) and their fluorescence +is collected through a light pipe to a PMT (iii). (b) Sample signals from the CBGB. (i) In-cell absorption on the RR11(0) +line of YbOH ˜ +X(000) → ˜A(000). The peak optical depth corresponds to a molecule density of ∼5×109 cm−3 in the ˜ +X(000), +N = 0 state. (ii) Front of cell absorption on the same RR11(0) line. The peak optical depth corresponds to a molecule density +of ∼2×109 cm−3. (iii) Fluorescence after excitation of the bending mode on a strong ˜ +X(010) → ˜A(000) line. The integrated +signal corresponds to ∼8300 photons detected on the PMT. +with the RR11(0) line of the ˜X(000) → ˜A(000) transition at 17325.0365 cm−1 [45]. The extracted beam is rotationally +and translationally cold, but can have significant excited vibrational population, a result of inefficient vibrational +thermalization from buffer gas collisions [49]. This provides a significant advantage, as we obtain ∼109 molecules +exiting the cell in the excited bending mode as a result. The molecular beam is collimated by a 6.4 mm diameter +skimmer 4.8 cm downstream from the cell aperture, a 9.5 mm diameter hole 11.4 cm downstream from the cell +aperture, and a 5 mm diameter hole 23.7 cm downstream from the cell aperture. The beam travels at 150 − 200 m/s +toward the laser-induced fluorescence (LIF) measurement region located ∼60 cm downstream from the cell. The +region is pumped by multiple turbomolecular pumps, and typical pressures when flowing He gas are 1−5×10−7 Torr. +In YbOH, the ˜A(000) → ˜X(010) transition has a vibrational branching ratio of r010 = 0.054(4)% [50], and the +lifetime of the ˜A2Π1/2 state is τ = 20(2) ns [51]. The excited state population primarily decays to the vibrational +ground state, ˜X(000), with r000 = 89.44% branching. +Therefore, in our experiment, the fluorescence signal will +saturate after roughly one photon scatter as the molecules are optically pumped out of the bending mode and +mostly into the ground state. With a ∼1 mm Gaussian laser beam intersecting a ∼200 m/s molecular beam, we + +5 +can estimate the saturation parameter required for a single photon scatter as s ≈ 1 × 10−2. Using the definition +of saturation intensity for a transition with branching ratio r as Is = πhc/(λ3τr) [52], we compute an intensity of +I ≈ 280 mW/cm2 required to optically pump the forbidden transition ˜X(010) → ˜A(000). For a 1 mm diameter +Gaussian laser beam, this requires ≳ 2 mW of optical power. While we have neglected rotational branching and other +experimental imperfections in this analysis, we observe the power requirements needed to produce fluorescence on +such a forbidden line are feasible. +Downstream in the LIF region, molecules in the ˜X(010) bending mode are excited by a 588 nm laser resonant with +the nominally forbidden ˜X(010) → ˜A(000) transition. The laser beam, with a ∼1 mm diameter and ∼40 mW of +power, is sent perpendicular to the molecular beam (see Fig 1a) through windows at Brewster’s angle. The resulting +577 nm fluorescence from decays to the ˜X(000) state is collected with a 19.4 mm diameter fused-quartz light pipe. +A 25.4 mm diameter, 19 mm focal length retroreflecting concave mirror opposite the light pipe improves collection +efficiency. We filter out the 588 nm scattered background light using a combination of interference and colored glass +filters on the exit of the light pipe, obtaining a signal-to-noise ratio of >10. The fluorescence signal is incident on a +photomultiplier tube (PMT) module (Hamamatsu H13543-300), and the resulting photocurrent is amplified with a +10−8 A/V trans-impedance amplifier with a 1.5 kHz low pass filter. +To obtain the field-free spectrum, we scan the 588 nm probe laser and record its frequency using a wavelength meter +(HighFinesse WS7-30) with an absolute accuracy of 30 MHz and a measurement resolution of 1 MHz. To improve the +absolute accuracy, we use the probe light to co-record sub-Doppler I2 spectra, obtained with amplitude modulated +saturated absorption spectroscopy [53]. Calibration of the laser frequency using the I2 spectra results in one standard +deviation error of 2.35 MHz in absolute frequency accuracy. +Figure 1b shows typical absorption and LIF signals obtained in a single shot. The LIF signal size typically varies +from shot to shot due to ablation yield fluctuations. +To construct the field-free spectrum, we scan the laser at +approximately 1-2 MHz per shot, average the LIF signal for 4 shots, integrate over the molecule pulse duration, and +plot the data against the calibrated probe frequency. The observed peaks are fit well by a Lorentzian function, with +fitting errors < 3 MHz. For the Stark and Zeeman spectra, we step the laser in 3 MHz increments, and average the +LIF signal for 10 shots at each step. +For Stark spectroscopy, we use two indium tin oxide (ITO) coated glass plates separated by a 4.99(3) mm gap +to apply fields up to 265 V/cm in the LIF region. Before entering the field region, the molecular beam is further +collimated with a 3 mm hole in a grounded aluminum plate. The molecules traveling through the ITO plates are then +excited by the 588 nm laser (see Fig. 1a). The resulting fluorescence is collected through the glass plates with the +setup described earlier. For Zeeman spectroscopy, we generate magnetic fields of 0 − 70 Gauss using two pairs of wire +coils outside the vacuum chamber (see Fig. 1a). The two coil pairs have a diameter of 21.4 cm with 500 windings each, +and are each symmetrically spaced from the LIF region with distances of 7.5(1) cm and 11.3(1) cm to the molecules. +B. +Theory: Effective Hamiltonian +The ground and excited states are modeled with an effective Hamiltonian approach [54]. The ˜A(000) state is well +described by a Hund’s case (a) Hamiltonian, using parameters from a previous optical study on a supersonic YbOH +beam [45]. Complete details of the effective Hamiltonian are provided in the supplementary materials. In the excited +state, strong spin-orbit interactions mean N is not a well-defined quantum number. Conversely, the molecule-frame +projection quantum numbers Λ, Σ, and Ω are well-defined in Hund’s case (a). Cross terms of spin-orbit and rotational +perturbations give rise to the Λ-doubling interaction, which mixes the projection quantum numbers. The resulting +Hund’s case (a) ˜A eigenstates are symmetric and anti-symmetric superpositions of projections with well defined parity +P: +|Λ; S, Σ; J, Ω, M, P = ±⟩ = +1 +√ +2(|Λ; S, Σ; J, Ω, M⟩ ± (−1)pa| − Λ; S, −Σ; J, −Ω, M⟩). +(1) + +6 +The phase factor pa = J−S−ℓ is connected to the convention for the action of the parity operator, P|Λ; S, Σ; J, Ω, M⟩ = +(−1)pa| − Λ; S, −Σ; J, −Ω, M⟩. +This phase convention is followed by Ref. [43, 55] (Details in the supplementary +materials). +We model the ground ˜X(010) state using a Hund’s case (b) effective Hamiltonian describing a 2Π vibronic state. +This approach has provided an accurate description of the vibrational bending modes in other metal hydroxide +molecules, such as CaOH and SrOH in optical [39] and millimeter wave [56] studies. The lack of first-order spin-orbit +interaction means the electron spin S is largely independent of the internuclear axis, and therefore both Σ and P +are undefined. Hund’s case (b) is the natural basis, with N and its projection ℓ as good quantum numbers. The +spin-rotation interaction then couples N with S to form well-defined J. Higher-order perturbations give rise to the +ℓ-doubling interaction, and the ˜X eigenstates of good parity are written as: +|ℓ; N, S, J, M, P = ±⟩ = +1 +√ +2(|ℓ; N, S, J, M⟩ ± (−1)pb| − ℓ; N, S, J, M⟩). +(2) +The phase factor in Hund’s case (b) is defined as pb = (−1)N−ℓ. The additional factor of ℓ = 1 means the action of +the parity operator on a singly excited bending mode is similar to that of a Σ− electronic state. While this phase +convention has physical basis (see supplementary materials) and has been used in literature [43, 55, 57–59], the choice +is not universal. The parity phase and the sign of the ℓ-doubling Hamiltonian together determine if the lowest energy +state is positive or negative parity. +We use an effective Hamiltonian for the ˜X(010) state given by +H ˜ +X(010) = B( ⃗N 2 − ℓ2) + γ( ⃗N · ⃗S − NzSz) + γGNzSz + pG +2 +� +N+S+e−i2φ + N−S−ei2φ� +− qG +2 +� +N 2 ++e−i2φ + N 2 +−ei2φ� +. (3) +This form was first derived in Ref. [60] and is presented in detail in Refs. [57, 58, 61]. Here, all subscripts on +angular momenta (z, ±) denote molecule-frame quantities. The azimuthal angle of the bending nuclear framework +is given by φ. +The first term gives the rotational energy of a symmetric top. +The next two terms describe the +spin-rotation interaction coupling N and S to form J. The last two terms describe ℓ-type parity doubling caused by +terms off-diagonal in the vibrational angular momentum G, and cause splittings of opposite parity states. +For the spin-rotation interaction we have modified the usual expression, γN · S, by subtracting γNzSz to account +for the bending motion. This modification is crucial for accurate description of low-N spectra (see supplementary +materials). Other perturbations can reintroduce this axial spin-rotation term into the Hamiltonian, labeled in the +literature with the coefficient γ′ [60] or γG [57, 61]. The first order contribution to γG arises from magnetic dipole +interactions [62] and is negligible for the Yb-centered electron in YbOH. At higher order, a combination of vibronic +coupling and spin-orbit interactions can contribute to γG by mixing states with Π electronic character, as observed +in NCO [63], CCH [64], and FeCO [65]. +In Eq. 3, the qG parity-doubling term is standard for a bending molecule in a 2Σ electronic state. This term arises +from Coriolis effects at second order, similar to the q term in Λ-doubling. The pG term, also in analogy with Λ- +doubling, is equivalent to a parity-dependent spin-rotation interaction. Owing to the weak coupling of the spin to the +internuclear axis in Σ electronic states, this term is small and has only been observed in submillimeter spectroscopy +of metal hydroxides [56, 66], ZnCN [67], and CrCN [68]. As with γG, this term receives higher-order contributions +from vibronic mixing with electronic Π states. In spherical harmonic notation [54], the ℓ-type doubling terms may be +written in the molecule frame as � +q=±1 e−2iqφ � +pGT 2 +2q(N, S) − qGT 2 +2q(N, N) +� +. +We are using a sign convention for the ℓ-type doubling Hamiltonian outlined by Brown [59, 61], where the ℓ-type +doubling Hamiltonian mirrors that used for Λ-doubling. However matrix elements of ℓ involve different phases than +Λ. As a result of the (−1)ℓ factor in our parity phase, we have the matrix elements ⟨ℓ = ±1|e±2iφ|ℓ′ = ∓1⟩ = 1, +differing from the azimuthal matrix elements for Λ-doubling. Matrix elements and complete details of the effective +Hamiltonian and conventions used are provided in the supplementary materials. +We construct the predicted spectrum by first separately diagonalizing the effective Hamiltonians for the ground +and excited states. The Hamiltonian basis is truncated at N ′′ = 6 for the ˜X(010) state and J′ = 15/2 for the ˜A(000) + +7 +* +* +* +* +FIG. 2. +Field-free spectrum over a ∼9 cm−1 range. Orange upper part is experimental observation and blue lower part is theory +prediction. Prediction is using effective model detailed in section III C with coefficients (cµ = 0.28, cκ = −0.49, cB = 0.83) and +a temperature of T = 2 K. Lines marked with * are unassigned and could arise from other isotopologues or bands. +state. Following Ref. [45], we include the P = 3/2 manifold when diagonalizing ˜A(000). After obtaining eigenvectors +and eigenvalues, we convert all eigenvectors to Hund’s case (a) and compute matrix elements of the transition dipole +moment (TDM) operator. Details of the TDM operator are given in section III C and in the supplementary materials. +For transitions with non-zero TDM, we compute the line position by taking the difference of excited and ground +eigenvalues. +III. +RESULTS +A. +Field-Free Spectrum +The observed spectrum (Fig 2) exhibits large splittings that match the excited state Λ-doubling and rotational +separation. +We perform combination-difference tests [54] with these splittings to obtain initial quantum number +assignments of transitions. With these assignments, we compute initial guesses for the B, γ, and qG Hamiltonian +parameters for the ˜X(010) state. Using these values and fixing the excited state parameters, we construct a predicted +spectrum and perform further line assignments (line notation is described in I A). With this analysis, we determined +the need for additional parameters pG and γG to accurately describe the full spectrum. +Without the pG term, various R and P branch features deviate from the prediction by a magnitude >20 MHz, +much larger than our frequency error. Specifically, in the region scanned in Fig. 2, without pG, lines with significant +residuals are: RR+ +11(2), RR− +11(3), OP + +12(4), P Q+ +12(5), and P P + +11(5). The magnitude and parity behavior of these residuals +cannot be explained by centrifugal distortion, but can be explained by a parity-dependent spin-rotation interaction, +namely pG. By introducing pG into the prediction, all of these residuals are reduced to values commensurate with +the experimental error. Furthermore, using the fit value of pG, we predicted and found the RR+ +11(4) and RR− +11(5) +lines (not visible in Fig. 2). These additional lines are added to the final fit and confirm the need for a pG term to +accurately model the full spectrum. +Unlike pG, the γG term does not scale with N ′′. However, we find this term necessary to describe the N ′′ = 1 +structure, which was crucial for accurate Stark and Zeeman analysis in section III B. In particular, we recorded multiple +field-free calibration scans of the QQ+ +11(1) and QR+ +12(1) lines. Since these lines share the same excited state, their +separation is insensitive to error in the ˜A(000) state parameters. We use the separation of these lines to determine +the N ′′ = 1+ spin-rotation splitting to be 61.8(20) MHz, and we add this value as an additional data point for our +analysis. By including the γG term in the spectral prediction, were we obtain an accurate prediction of the N ′′ = 1+ +splitting commensurate with our measurement error. +In total, we assigned 38 of the observed lines to 39 transitions originating from the N ′′ = 1 through N ′′ = 5 levels +of the ˜X(010) state. Note the QR− +12(1) and P Q− +12(5) lines are overlapped. To obtain optimal effective Hamiltonian + +8 +TABLE I. Spectroscopic parameters for the low-lying vibrational states of the ˜ +X2Σ+ manifold. +The ˜ +X(010) parameters are obtained from the current work. +Parameter +˜ +X(000) [44] +˜ +X(010) +˜ +X(100) [45] +T0/cm−1 +0 +319.90901(6) +529.3269(3) +B/MHz +7348.4005(3) +7328.64(15) +7305.37(24) +γ/MHz +−81.15(6) +−88.7(9) +−110.6(21) +γG/MHz +– +16(2) +– +qG/MHz +– +−12.0(2) +– +pG/MHz +– +−11(1) +– +parameters, we vary the ˜X(010) state parameters and hold fixed the ˜A(000) state parameters to the values given in +Ref. [45]. We construct predicted spectra and perform nonlinear least-squares minimization of the residuals between +the observed and predicted positions of all 39 assigned lines and the N ′′ = 1+ spin-rotation splitting. A full list of +line assignments is provided in the supplementary materials. +The best fit parameters are presented in Table I. The fit residuals have a standard deviation of 6.1 MHz, consistent +to order unity with the error reported in the previous optical study of the ˜A(000) state [45]. The rotational and spin +rotational ˜X(010) parameters are in good agreement with those for ˜X(000) and ˜X(100), also collected in Table I. +The location of the origin T0 is in excellent agreement with previous dispersed fluorescence studies [50, 51]. The +rotational constant B decreases in ˜X(010) as a result of vibrational corrections. +The increasingly negative spin- +rotation parameter γ between the three vibrational states is a result of second order spin-orbit perturbations from +low-lying electronic states with 4f 136s2 electronic configuration for the Yb centered electron, known as “4f hole” +states [44, 69]. +Vibronic mixing with electronic 2Π states can also explain the observed γG and pG parameters, which are not typical +for the bending mode of an isolated electronic 2Σ state. Vibronic mixing exchanges ℓ and Λ while preserving K. As a +result, the ˜X(010) state can acquire some Λ > 0 electronic character, inheriting spin-orbit and Λ-doubling interactions +from neighboring 2Π states. Specifically, in the effective Hamiltonian, these interactions can arise at third-order via a +combination of linear vibronic coupling and spin-orbit effects. This term was first described by Brown in the context +of spin-orbit corrections to electronic 2Π states as a result of mixing with other 2Σ or 2∆ states [70]. Neighboring +states that can contribute to γG and pG include both the ˜A manifold and the 4f hole states. The exact nature of +the 4f hole states and their vibronic mixing in YbOH is currently unknown and merits further study. However, their +proximity to the ground state and their large spin-orbit interactions could explain the significant magnitude of pG +and γG in YbOH compared to other metal hydroxides [56]. +The ℓ-type doubling parameter qG is a similar magnitude to that of other metal-hydroxide ˜X(010) states [39, 56], +and is in agreement with a recent theoretical calculation [71]. The parameter qG can be interpreted in terms of the +Coriolis coupling constants of a triatomic molecule [39, 41]: +qG = −(v2 + 1)B2 +ω2 +� +1 + +� +n=1,3 +ζ2 +2n +4ω2 +2 +ω2n − ω2 +2 +� +. +(4) +Here, v2 is the number of quanta in the bending vibration ω2, and ζ2n is the Coriolis coupling constant between +the bending mode and the vn stretch modes. To estimate ζ21, we can estimate the value of ω3 (O-H stretch) using +the CaOH value of 3778 cm−1 [72], and we set v2 = 1, ω2 ≈ T0, and ω1 ≈ 529.3 cm−1 [45]. Furthermore, we can +use the relationship ζ2 +21 + ζ2 +23 = 1 [41] to eliminate ζ2 +23. Using our values of B and qG, we then obtain a value of +ζ21 ≈ 0.137, slightly smaller than in CaOH (0.1969) [39] and SrOH (0.179) [73]. This is likely due to the break down +of the harmonic approximation ω2 ≈ T0 and the approximation of Be ≈ B. Further work is needed for a complete +vibrational characterization. +Using the parameters obtained from our analysis, we construct a field-free level diagram for the N = 1 manifold +of the ˜X(010) state, shown in Figure 3. As stated previously, N = 1 is the lowest rotational manifold in the ˜X(010) + +9 +−pG − 2qG +≈35 MHz +−3/4 (γ ＋ γG) ＋ pG/4 ＋ 2qG +≈28 MHz +pG/2 − 2qG +≈19 MHz +FIG. 3. +Field-free level structure of the N = 1 manifold in the ˜ +X(010) state. States are arranged vertically by energy and +horizontally by their MF angular momentum projection. States are labeled in the parity basis. The hyperfine structure was +not resolved in our work, and is instead approximated using parameters from a study of the ˜ +X(000) state [44]. +state, as we always have | ⃗N · ˆn| = 1. Due to their small parity splittings, N = 1 states are easily polarized, making +them useful for precision measurements [12]. The effect of the parity-dependent spin-rotation term, pG, is apparent +in the asymmetric parity-doubling of the J = 1/2 and J = 3/2 manifolds. Though we are not sensitive to hyperfine +splittings, for completeness we have included the H hyperfine structure using the parameters obtained for the ˜X(000) +state in a previous study [44]. The hyperfine structure is not expected to change significantly in the bending mode. +The recorded spectrum has lines present that could not be assigned with combination-differences using the ˜A(000) +structure, and are not observed in the prediction using the best-fit parameters. The lines are marked with * in Fig. 2. +We conclude that some of these lines are indeed from 174YbOH by comparing their chemical enhancement [46] when +using 1S0 → 3P1 transitions for different Yb isotopes. These lines could be unthermalized rotational states, or possibly +another overlapping ∆ℓ = ±1 band, such as the ˜X2Σ+(020,20) → ˜A2Π1/2(010) bands. +B. +Stark and Zeeman Spectra +After fitting the molecular structure with the field-free spectrum, we study the Stark and Zeeman spectra of +the molecule in the presence of static (DC) electric and magnetic fields, using the experimental setup described +in II. We obtain the spectra by scanning the 588 nm probe laser across two lines corresponding to the field-free +N ′′ = 1+ → J′ = +3 +2 +− transition, QQ+ +11(1) and QR+ +12(1). +The applied DC fields point along z, while the laser +polarization is along x. Spectra are taken with the E-field varied from 0 − 264 V/cm and with the applied B-field +varied from 0 − 70 G. Calibration spectra are taken with EZ = 0 V/cm and BZ < 0.5 G, and the observed line +positions are compared to the I2-corrected field-free positions to calibrate for frequency offsets. +The lines of interest are relatively well-isolated from other features, and the small N ′′ = 1 parity doubling allows us +to enter the linear stark regime with modest laboratory fields ≳100 V/cm. Since the parity splittings of the excited +˜A2Π1/2 state are >13 GHz, and its molecule frame dipole moment is D˜A = 0.43(10) D [45], at the fields we consider +the excited state Stark shifts are essentially negligible. Furthermore, given our frequency resolution and the natural +linewidth, we are only sensitive to the isotropic interaction of BZ with the electron spin magnetic moment. Curl-type +relationships [58] estimate anisotropic spin interactions at 6 × 10−3µB, and the nuclear magnetic moment is also +suppressed at a similar level, with both effects giving shifts below our resolution. + +10 +1/2+ +3/2+ +3/2− +3/2− +1/2− +J +N = 1 +QQ11(1) ++ +QR12(1) ++ +FIG. 4. +Zeeman spectroscopy of the ˜ +X(010) state. The main plot shows the transition frequency shift (with subtracted offset) +in a magnetic field, the blue lines are optimized model predictions, and the orange circles are experimental measurements. +Error bars are 1-σ measured peak widths, set by a combination of radiative broadening and unresolved hyperfine structure, +limiting the ability to resolve closely-spaced lines. Lower subplots are slices of the spectra at various magnetic field values, +with experimental data in orange and predicted line locations indicated with vertical dashed blue lines. On the left, we show +the field-free level structure of the transitions studied. +To obtain energy levels and predicted lines, we fix the field-free parameters and diagonalize the combined Stark, +Zeeman, and field-free Hamiltonian. We obtain optimal estimates for free Stark and Zeeman parameters by least- +squares minimization of the residuals between observed and predicted line positions. +Both ground and excited levels are magnetically sensitive. The Zeeman shifts of the ˜A2Π1/2(000) and ˜X2Σ+(000) +states were previously studied at similar magnetic field strengths in Ref. [45], and recently at high fields (∼1 T) in +Ref. [74]. Following these references, we use the following effective Zeeman Hamiltonians for the ground and excited +states: +HZee +X += gSµBSZBZ +(5a) +HZee +A += g′ +SµBSZBZ + gLLZBZ + g′ +lµB +� +e−2iθS+B+ + e2iθS−B− +� +(5b) +Here, Z refers to the lab-frame projection, ± refer to the molecule frame projections, and θ is the electronic azimuthal +coordinate. For the excited state, we use the values from Ref. [74], fixing g′ +S = 1.860, gL = 1.0, and g′ +l = −0.724. For +the ground state, we allow gS to vary in the fits to find an effective value that accurately describes the Zeeman shifts. +While we do not include them here, at higher resolution or at higher field values, additional terms are expected to +contribute in the effective Zeeman Hamiltonian, including terms associated with the bending angular momentum [58]. +The Zeeman fits prefer a value of gS = 2.07(2), deviating from the free electron g-factor of 2.0023. The experimental +Zeeman shifts and the prediction from the optimized model are shown in Fig. 4. Corrections to gS can arise from +mixing involving other states with different Zeeman tuning. For example, the Zeeman shifts of the ˜A(000) state were +fit to g′ +S = 1.860 in a recent high-field study [74], owing to perturbing 4f 136s2 states. Since we observe perturbations +from these 4f states in the field-free structure of the ˜X(010) state, it is natural to also find their effects in the Zeeman +shifts. Furthermore, the 4f states are split into a higher energy spin-orbit anti-aligned manifold and a lower energy +spin-orbit aligned manifold [69]. Due to energy proximity, while ˜A(000) predominantly interacts with the 4f hole +anti-aligned manifold, ˜X(010) will be perturbed more strongly by the aligned manifold. The difference in electron +orientation of the two spin-orbit 4f manifolds can explain the difference between ˜X(010) and ˜A(000) in the sign of +the deviation of gS from its nominal value. + +11 +Increasing Line Strength +1/2+ +3/2+ +3/2− +3/2− +1/2− +J +N = 1 +QQ11(1) ++ +QR12(1) ++ +FIG. 5. +Stark spectroscopy of the ˜ +X(010) state. The main plot shows the transition frequency shift (with subtracted offset) +in an electric field, the blue lines are optimized model predictions, and the orange circles are experimental measurements. +The blue color gradient represents parity forbidden transitions that gain strength at finite electric field. Error bars are 1-σ +peak widths, set by a combination of radiative broadening and unresolved hyperfine structure, limiting the ability to resolve +closely-spaced lines. Lower subplots are slices of the spectra at various electric field values, with experimental data in orange +and predicted line locations indicated with vertical dashed blue lines. On the left, we show the field-free level structure of the +transitions studied. +To describe the Stark shifts, for the both ground and excited states we use the Hamiltonian HE = − ⃗Dmol · ⃗E. The +molecule frame dipole moment Dmol is kept as a free parameter, and obtained from spectra where EZ is scanned with +BZ < 0.5 G. The optimal fit value is Dmol = 2.16(1) D = 1.09 h MHz/(V/cm). This value is in good agreement with +the measured ˜X(000) dipole moment of 1.9(2) D. In Figure 5, we plot the theoretical prediction based on the optimal +fit against the observed line positions. +The Stark shifts confirm the assignment of the ˜X(010) state and demonstrate the orientation control afforded +by parity doublets. In the bending mode, the projection of the molecular axis on the lab-frame Z-axis is given by +ˆn · ˆZ = ( ⃗ +N·⃗Z)( ⃗ +N·ˆn) +N(N+1) +∝ MNℓ. Note we use X, Y, Z to denote lab-frame axes and x, y, z to denote the molecule-frame. +The molecule z axis and dipole moment Dmol both point from O to Yb. For field-free states, ⟨MNℓ⟩ = 0, and the +molecule is unpolarized. In the presence of an electric field fully mixing parity doublets, the Stark shifts are linear, +and the eigenstates are diagonal in the the decoupled basis |ℓ; MN, MS⟩. In this regime, the levels split into 2N + 1 +dipole moment orientations pointing along +MNℓ +N(N+1), and splittings within each orientation manifold are due to the +spin-rotation interaction. +C. +Anomalous intensities and perturbations +Since the ˜A(000) state has been previously fully characterized [45], the assignment of energy levels in ˜X(010) is fairly +straightforward using the effective Hamiltonian approach. However, because this transition is nominally forbidden, +interpreting the line intensities is a challenge. Electric dipole (E1) transitions involving ∆ℓ ̸= 0 are forbidden in +the Condon approximation, which separates electronic and vibrational degrees of freedom [42, 75]. These nominally +forbidden vibronic transitions have been observed spectroscopically in many species of linear triatomic molecules, +including NCO [76], NCS [77], MgNC [78], CaOH [39, 79, 80], SrOH [36, 73, 81], and YbOH [51], though modeling of +the intensities is less common. +These transitions borrow intensity from E1-allowed bands through a combination of vibronic and spin-orbit per- + +12 +turbations [5, 50]. Branching ratios involving forbidden vibronic transitions in YbOH were measured in a previous +study [50] examining dispersed fluorescence from the ˜A(000) state, with resolution at the 10−5 level. The exper- +imentally observed vibrational branching was in good agreement with a theoretical study published in the same +work [50]. While these transitions are of interest as leakage channels for photon cycling, they can also be a resource +for spectroscopy, as we show in the current work. +The observed spectrum exhibits anomalous rotational line intensities, with certain transitions completely miss- +ing at our level of sensitivity. +For example, despite their expected thermal occupation (N ′′ ≤ 3), the P Q+ +12(1), +P P + +11(2), QQ+ +11(2), P P − +11(3), QP − +11(3), and QR− +12(3) lines are missing (see Supplemental Material for a full list of lines). +Anomalous line intensities for forbidden transitions have been previously observed in other molecules with vibronic +mixing [39, 73, 78, 80, 81]. By considering the intensity-borrowing that gives transition strength to these forbidden +transitions, we develop a model that qualitatively explains the observed line strengths. +In an E1 transition, the transition strength is proportional to the square of the transition dipole moment between +the ground and excited state, |⟨ ˜A|T 1 +p (d)| ˜X⟩|2. We are using spherical tensor notation, where p denotes the component +of the spherical tensor in the lab-frame and q in the molecule-frame. Using a Wigner D matrix, we can write the lab +frame dipole moment in terms of its molecule frame projections: T 1 +p (d) = � +q D(1) +p,q(ω)∗T 1 +q (d). In the E1 approximation, +∆Σ = 0, and the molecule-frame projection q of the transition dipole moment determines the selection rule for Λ. The +perpendicular q = ±1 components drive ∆Λ = ±1 transitions, for example the allowed ˜A − ˜X band, while parallel +q = 0 component drives ∆Λ = 0, for example the allowed ˜B − ˜X band. +In the limit of very large vibronic interaction, Λ and ℓ are fully mixed, and one might consider the ˜X(010) → ˜A(000) +transition as a vibronic 2Π − 2Π parallel band, with ∆K = 0. In reality, the vibronic mixing is perturbative in the +ground and excited states, and Λ and ℓ are well-defined. As a result, the observed line intensities are completely +inconsistent with a solely parallel transition model. +Instead, we model the ˜X(010) → ˜A(000) transition as a mixture of perpendicular and parallel bands. We consider +the effects of vibronic perturbations with the selection rule ∆ℓ = ±1, which can result in intensity borrowing. At first +order, we have the dipolar Renner-Teller (RT) Hamiltonian, also referred to as Herzberg-Teller coupling [43, 55, 76], +HRT = V11 +2 +� +L+q−ei(θ−φ) + L−q+e−i(θ−φ)� +. +(6) +This interaction is a form of linear vibronic coupling [82]. Here, V11 parameterizes the interaction strength, θ is the +electronic azimuthal coordinate, φ is the bending azimuthal coordinate as before, L± is a raising/lowering operator +with ∆Λ = ±1, and q± is a dimensionless raising/lowering operator with ∆ℓ = ±1. Physically, this interaction can +be interpreted as the electrostatic interaction between the displaced bending dipole moment and the electron cloud. +The interaction preserves the composite projection number K = Λ + ℓ. +At second order, the dipolar RT Hamiltonian can combine with the perpendicular spin-orbit Hamiltonian, +HSO = A⊥ +2 (L+S− + L−S+) , +(7) +Where L± is defined as before, A⊥ is the off-diagonal spin-orbit coupling, and S± is the raising/lowering operator +with ∆Σ = ±1. The combination of H(1) +RT × H⊥ +SO is an effective interaction with terms q±S∓. This interaction has +∆K = −∆Σ = ±1, but preserves the total angular momentum projection number P = Λ + Σ + ℓ. +Denote the unperturbed excited state as | ˜A2Π1/2(000)⟩0 and the true, perturbed eigenstate as | ˜A2Π1/2(000)⟩. We +can then expand the perturbed eigenstate in terms of dominant ℓ = 1 vibronic contributions [5, 50]: +| ˜A2Π1/2(000)⟩ ∝ | ˜A2Π1/2(000)⟩0 + cµ|µ2Σ(+) +1/2(010)⟩0 + cκ|κ2Σ(−) +1/2(010)⟩0 + cB| ˜B2Π(010)⟩0. +(8) +The perturbative coefficients cµ, cκ, cB represent the relative admixture of the intensity-borrowing states. The relevant +states and perturbations are shown schematically in Fig. 6. The µ2Σ(+) +1/2 state is the P = 1/2 component of the Ω = 1/2, +v2 = 1, ˜A manifold, and the κ2Σ(−) +1/2 state is the P = 1/2 component in the Ω = 3/2, v2 = 1, ˜A manifold. These +two states are connected to ˜A2Π1/2(000) by the second-order perturbation HRT × HSO. The ˜B2Π vibronic state + +13 +FIG. 6. +Level schematic for relevant states and perturbations in YbOH. Levels are labeled by their vibronic term symbol. We +detect the ˜ +X(010) bending state (which is a vibronic 2Π state) by laser excitation (orange line) up to the ˜A2Π1/2(000) state and +observe the fluorescence from decays to the ground ˜ +X(000) state (yellow wavy line). This excitation is a forbidden E1 transition, +however, it acquires intensity by mixing of the excited ˜A2Π1/2(000) state with other |ℓ| = 1 states. Mixing with ˜B(010) occurs +via first-order (blue) Renner-Teller (RT) interactions, and mixing with the µ, κ(010) states occurs via second-order (purple) +cross terms between RT and spin-orbit (SO) (red) interactions. Not shown for simplicity are similar SO interactions between +˜A2Π1/2(000) and ˜B(000) and similar RT interactions between µ, κ(010) and ˜B(000), which also contribute to state mixing. +is the v2 = 1 component of the ˜B2Σ+ +1/2 electronic state, and is connected to ˜A2Π1/2(000) state via the first-order +perturbation HRT . +Each of these perturbing states contribute to different molecule-frame components of the transition dipole moment +(TDM). For example, the transition ˜X2Π → ˜B2Π is generated by the q = 0, z component of the TDM, with +∆K = ∆P = 0. The other transitions to µ and κ have Π → Σ vibronic character, and couple via the q = ±1, x, y +TDM components. The perturbing µ and κ states have opposite spin orientation compared to the original ˜A2Π1/2 +state. This means the intensity-borrowing states have mixed spin projection Σ, and the ∆Σ = 0 selection rule is not +well-defined. +The transition was modeled by first diagonalizing the ˜A2Π1/2(000) and ˜X2Σ(010) states separately to obtain the +level positions of both states. To evaluate the TDM, the excited state vector is then replaced by a linear combination +of the intensity-borrowing state vectors with coefficients cµ, cκ, cB. The change of basis from ˜A to µ, κ, and ˜B uses +appropriate selection rules for vibronic mixing and preserves parity (see supplementary materials for details). The +total TDM is the sum over the individual TDMs evaluated between ˜X(010) and the intensity-borrowing states. To +obtain the transition intensity, the TDM is squared after the sum, allowing TDMs from different states to interfere +with each other. This interference is the source of the anomalous line intensities. +The mixing coefficients, cµ, cκ, cB could not be modeled with a deperturbation Hamiltonian, since neither the µ, κ, +or ˜B state have been extensively studied or modeled, and both states are expected to be affected by perturbations +from nearby states with 4f 136s2 Yb character [69]. Instead, the mixing coefficients are kept as free parameters and +their ratios were fit to the experimentally observed, relative field-free intensities. For the intensity fits, the rotational +temperature is fixed at T = 2 K (the molecule beam is cooled by expansion out of the cell aperture), and since only +relative intensities were fit, the cB parameter is held fixed. The normalized best fit mixing coefficients are found to +be (cµ, cκ, cB) = (0.28, −0.49, 0.83). This implies ∼69% of the ℓ = 1 character in ˜A2Π1/2(000) arises from mixing +with ˜B(010), ∼24% from κ(010), and ∼7% from µ(010). This is in good agreement with recent theory work on +intensity borrowing in YbOH, which attributed 70% of the intensity borrowing to mixing with ˜B(010) [50]. However, +it is important to note that due to interference effects, relative amplitudes of the coefficients, not their squares, are +important for determining rotational line intensities. + +14 +We find that using these parameters to model the transition provides good qualitative understanding of the observed +spectrum, as evidenced by the theory and experiment comparison in Figure 2. Further studies of the excited state +perturbations would be required to improve the fit; however, as the exact intensities are not critical for future +experiments with this molecule, this model is sufficient to provide physical understanding of the intensities and +behavior of this transition. +IV. +CONCLUSION +In this work, we performed high-resolution optical spectroscopy on the rovibrationally forbidden ˜X2Σ+(010) → +˜A2Π1/2(000) transition of 174YbOH. In total, we observed 39 transitions out of low rotational states with N ′′ ≤ 5. +The ˜X(010) structure is well-described by a Hund’s case (b) 2Π effective Hamiltonian, and the ℓ-type parity doubling +is described by two constants, qℓ = −12.0(2) MHz and pℓ = −11(1) MHz. We modeled the anomalous line intensities +of the forbidden band with mixing coefficients representing vibronic perturbations in the excited state. The anomalous +intensities arise from quantum interference between TDMs from the perturbing ˜B(010), µ(010), and κ(010) states. +From the Zeeman spectra, we found the magnetic tuning of ˜X(010) is consistent with an effective isotropic electron +spin g-factor, gS = 2.07(2). From the Stark spectra, we extracted the molecule-frame dipole moment of 2.16(1) D. +These values are in good agreement with the parameters of the ˜X(000) state. +In our study, the hyperfine structure and higher-order Zeeman g-factors were unresolved. Our work provides a basis +for future studies with narrow-linewidth methods, such as RF, microwave, and two-photon spectroscopy, to precisely +determine these properties. +This work is an essential step towards measurements of CP-violating physics in YbOH [12], as well as other +metal hydroxide molecules proposed for CP violation and parity violation searches that utilize the parity doublets +in the bending mode. We showed the ˜X(010) state ℓ-doubling offers spectroscopically resolvable states of molecule +polarization pointing along, against, and perpendicular to the applied electric field, over a wide range of field values. +This orientation control over the dipole moment offers robust systematic error rejection without compromising laser +cooling. The combination of these features make linear polyatomics a promising platform for new physics searches. +With our measured data, we can compute the EDM sensitivity, which is proportional to the electron spin projection on +the internuclear axis, Σ. We find a local maximum value of ⟨Σ⟩ = 0.40 in the N = 1, J = 1 +2 ++ state at E = 101 V/cm, +similar to what was predicted in prior theoretical work [35, 83]. Furthermore, understanding the structure of 174YbOH +is a step toward characterizing the more complicated structure of the odd isotopologues 171YbOH and 173YbOH, which +have sensitivity to parity violation [38] and hadronic CP violation [29], respectively. +Lastly, our determination of the ˜X(010) location and structure is crucial for understanding the complicated excited +state structure in YbOH. For example, with our knowledge of the bending frequency, we can tentatively assign the +unknown [17.33] band in Ref. [51] to the ˜X2Σ+(010) → ˜A2Π1/2(010) band. This would put the excited ˜A2Π1/2(010) +manifold at approximately 17652 cm−1. This state is an excellent candidate for optically pumping population from +˜X(000) into ˜X(010), an important step for signal-to-noise-ratio improvements in precision measurements using the +bending mode. 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Cederbaum, Theory of vibronic coupling in linear molecules, The Journal of Chemical +Physics 74, 2945 (1981). +[83] B. Augenbraun, Methods for Direct Laser Cooling of Polyatomic Molecules, Ph.D. thesis, Harvard University (2021). +[84] R. S. Mulliken and A. Christy, Λ-Type Doubling and Electron Configurations in Diatomic Molecules, Phys. Rev. 38, 87 +(1931). +[85] J. Brown and A. Merer, Lambda-type doubling parameters for molecules in Π electronic states of triplet and higher +multiplicity, J. Mol. Spectrosc. 74, 488 (1979). +[86] C. Di Lauro and I. Mills, Coriolis interactions about XY axes in symmetric tops, Journal of Molecular Spectroscopy 21, +386 (1966). +[87] J. Brown, I. Kopp, C. Malmberg, and B. Rydh, An analysis of hyperfine interactions in the electronic spectrum of AlF, +Physica scripta 17, 55 (1978). +[88] J. M. Brown and B. J. Howard, An approach to the anomalous commutation relations of rotational angular momenta in +molecules, Mol. Phys. 31, 1517 (1976). + +19 +Supplementary Materials: Characterizing the Fundamental Bending Vibration of a +Linear Polyatomic Molecule for Symmetry Violation Searches +I. +TARGET COMPOSITION +The data were obtained from targets of pressed Yb(OH)3 powder in a stoichiometric mixture with Yb powder. +The powders were mixed to have a 1:1 ratio of Yb and OH, ground using a mortar and pestle, passed through a 230 +mesh sieve, and mixed with 4% PEG8000 binder by weight. The powders were pressed in an 8 mm diameter die at +a pressure of ∼1 GPa for ∼ 30 minutes. For some targets, 10-30% water by mass was added to the powder before +pressing, and while pressing the die was heated to ∼150◦C. This was found to improve target density and ablation +yield consistency. +II. +PHASE CONVENTIONS +A. +Λ-Doubling +There is an accepted convention for Λ-doubling, which was laid out by Mulliken and Christy [84]. The convention +is reiterated by Brown in [85] and Brown and Carrington in [54]. This convention is given by +⟨Λ = ±1|e±2iφe|Λ′ = ∓1⟩ = −1 × δΛ,Λ′±2 +(S1) +Here, e±iφe is a raising/lowering operator with φe the azimuthal angle of the electrons. In this convention, a positive +qe electronic Λ-doubling parameter in a 1Π state corresponds to the (−1)J parity level lying above the (−1)J+1 parity +level. In other words, the + parity state is below the − parity state for J = 1. In the YbOH ˜A state, pe + 2qe is +negative, and the − parity state is below the + parity state. This phase choice also manifests in the signs of the +Λ-doubling Hamiltonian. When written in Hund’s case (a), the J±S± terms have a positive prefactor, and the J±J± +terms have a negative prefactor. For this work, we drop the J±J± term in ˜A as its contribution is negligible. +Now we derive the Λ phase convention, following arguments from [55] and [54]. We begin with the definition of the +Lz angular momentum operator in the molecule frame: +Lz|Λ⟩ = −i ∂ +∂φe +|Λ⟩ = Λ|Λ⟩ +(S2) +This means |Λ⟩ ∝ eiΛφe. Since L is not well defined, we expand |Λ⟩ in terms of spherical harmonics: +|Λ⟩ = +� +L +FLYLΛ(θe, φe) = +� +L +FL +√ +2π eiΛφeΘLΛ(θe) +(S3) +Here, � +L |FL|2 = 1, and ΘLΛ(θe) is proportional to the associated Legendre functions P Λ +L (cos θe). +Θl,m(θ) = (−1)m +� +2l + 1 +2 +(l − m)! +(l + m)!P m +l (cos θ) +for m ≥ 0 += (−1)mΘl,−m(θ) +for m < 0 +(S4) +Note the function ΘLΛ satisfies ΘL,−|Λ| = (−1)ΛΘL,|Λ|. This is the origin of this specific phase-convention. +Now we can evaluate the left hand side of eqn. S1 + +20 +⟨Λ|e±2iφe|Λ′⟩ = +� +sin θedθedφe +� +L,L′ +F ∗ +LFL′YLΛ(θe, φe)∗e±2iφeYL′Λ′(θe, φe) += +� +L,L′ +F ∗ +LFL′δΛ,Λ′±2 +� +sin θedθe(−1)Λ′±2ΘL,−Λ′∓2(θe)ΘL′,Λ′(θe) +(S5) +Where we substitute YL,Λ(θe, φe)∗ = (−1)ΛYL,−Λ(θe, φe) and performed the φe integral taking advantage of the +orthogonality of exponential functions. +Now we simplify the integrand by noting we are interested in Λ = ±1, Λ′ = ∓1. This allows us to write −Λ′∓2 = Λ′. +Then the remaining θe integral can be performed by using the orthogonality relations of the associated Legendre +polynomials, which results in +⟨Λ = ±1|e±2iφe|Λ′ = ∓1⟩ = δΛ,Λ′±2 +� +L +|FL|2(−1)Λ′ = −1 × δΛ,Λ′±2 +(S6) +Where we have substituted |Λ| = 1 in the last line and used the fact that |FL|2 is normalized. +We also note that the behavior of YLΛ upon the transformation Λ → −Λ gives the parity properties of |Λ⟩. The +action of space-fixed inversion, i.e. the parity operator P, is equivalent to a reflection σxz of the xz plane of the +molecule. +This can be derived by considering the effect of space-fixed inversion on the Euler angles relating the +molecule and lab frames. Therefore we have: +PYL,Λ(θe, φe) = σxzYL,Λ(θe, φe) += YL,Λ(θe, 2π − φe) += YL,Λ(θe, φe)∗ += (−1)ΛYL,−Λ(θe, φe) +(S7) +This recovers the result P|Λ⟩ = (−1)Λ| − Λ⟩ (note a Σ− state has an extra factor of (−1) that we do not consider). +For the full parity of the rotational wavefunction, the action of P must also be computed on the spin and rotational +wavefunctions, which also reverse the projection quantum numbers and contribute parity phases of (−1)S−Σ and +(−1)J−Ω respectively. The combination of all phase factors gives the complete case (a) parity phase without bending +motion: (−1)Λ+S−Σ+J−Ω = (−1)J−S, where we have used |Σ| = S and Ω = Λ + Σ to simplify the exponent. +B. +ℓ-Doubling +For the derivation of the parity phase and matrix elements involving ℓ, we follow Ref. [55], which uses the vibrational +phase conventions established by by Di Lauro and Mills [86]. The wavefunction for an isotropic 2D harmonic oscillator +may be written as +|v2, ℓ⟩ = +1 +√ +2π eiℓφnΨv2,ℓ(q) +(S8) +Here, q = +� +q2 +1 + q2 +2, where (q1, q2) are the dimensionless, doubly-degenerate normal coordinates of the bending mode, +and φn = tan−1(q2/q1) is the azimuthal angle of the bending nuclear framework. The function Ψv2,ℓ is given by [86]: +Ψv,ℓ(q) = (−1)(v+|ℓ|)/2Nv,ℓq|ℓ|e−q2/2L|ℓ| +(v+|ℓ|)/2(q2) +(S9) +Here, Nv,ℓ is a normalization factor and Lk +n(x) is an associated Laguerre polynomial. This function explicitly satisfies +Ψv2,|ℓ| = Ψv2,−|ℓ|. + +21 +We now consider the matrix elements between ℓ = ±1 states: +⟨ℓ|e±2iφn|ℓ′⟩ = +� +dqdφ 1 +2π e−iℓφnΨv,ℓ(q)e±2iφneiℓ′φnΨv,ℓ′(q) +(S10) +The integration bounds are taken for q ≥ 0 and 2π > q ≥ 0. The φn integral is evaluated with the orthogonality of +complex exponential functions and enforces δℓ,ℓ′+2. +Restricting our attention to ℓ = ±1 states, the Ψv,ℓ(q) functions depend only on |ℓ|, and do not add an additional +phase. As a result we can evaluate the remaining dq integral using the orthogonality relations of the associated +Laguerre polynomials. We are left with +⟨ℓ|e±2iφn|ℓ′⟩ = 1 × δℓ,ℓ′±2 +(S11) +The difference between parity phase factors for ℓ and Λ can be traced to the difference in phase between Ψvℓ(q) and +ΘLΛ(θe) upon space-fixed inversion. By considering the behavior of the wavefunctions under φn → 2π − φn, we see +the radial q part is unaffected, giving us P|v2, ℓ⟩ = |v2, −ℓ⟩. When combined with rotational and spin parity phase +factors, we then obtain the complete parity phase (−1)J−S−ℓ. +III. +EFFECTIVE HAMILTONIANS AND MATRIX ELEMENTS +A. +˜A2Π1/2(000) +For the ˜A2Π1/2(000) state, we follow Ref. [45], which uses the R2 rotational Hamiltonian formalism. Details can +be found in Ref. [54], Ch. 7. The effective Hamiltonian in Hund’s case (a) and in spherical tensor notation is given +by +H ˜ +A = T0 + AT 1 +q=0(L)T 1 +q=0(S) + B(J − L − S)2 − D(J − L − S)4 ++ (pe + 2qe) +� +q=±1 +e−2iqθT 2 +2q(J, S) + 1 +2(peD + 2qeD) +� +q=±1 +[N, e−2iqθT 2 +2q(J, S)]+ +(S12) +Here, T0 is the state origin, A is the spin-orbit constant, B is the rotational constant, D is the centrifugal distortion +term, pe + 2qe represents electronic Λ-doubling, peD + 2qeD is the centrifugal distortion correction to Λ-doubling, +[·, ·]+ is the anti-commutator, J±S± are defined in the molecule frame, and θ is the azimuthal angle of the electronic +wavefunction. Matrix elements of this Hamiltonian can be found in [54, 55, 87]. Note the L2 +x + L2 +y terms that arise +in the R2 formalism are absorbed in the origin. We use the constants determined in Ref. [45]. +To be explicit, using the phase convention from supplementary section II we reproduce our matrix element for the +Λ-doubling term below: +⟨Λ; S, Σ; J, Ω, M|e2iqθT 2 +2q(J, S)|Λ′; S, Σ′; J′, Ω′, M ′⟩ += δJ,J′δM,M ′δΛ+2q,Λ′ +× (−1)J−Ω +� +J +1 +J +−Ω −q Ω′ +� +� +J(J + 1)(2J + 1) +× (−1)S−Σ +� +S +1 S +−Σ q Σ′ +� +� +S(S + 1)(2S + 1) +(S13) + +22 +S +R +N +J +G +ℓ +n +FIG. S1. +A schematic of the coupling scheme in Hund’s case (b), used to describe the ˜ +X(010) state. The vibrational angular +momentum G is projected onto the internuclear axis to form ℓ. The molecule rotation R is coupled to ℓ to form N. Finally the +spin-rotation interaction couples S and N to form J. Coupling to the H nuclear spin is not pictured. +B. +˜ +X2Σ+(010) +We reproduce the ˜X2Σ+(010) Hamiltonian below in spherical tensor notation. +H ˜ +X = T0 + B(N 2 − ℓ2) + γ +� +N · S − T 1 +q=0(N)T 1 +q=0(S) +� ++ γGT 1 +q=0(N)T 1 +q=0(S) + +� +q=±1 +e−2iqφ � +pGT 2 +2q(N, S) − qGT 2 +2q(N, N) +� +(S14) +The bending mode energy levels are well represented by Hund’s case (b) eigenstates. A pictorial representation of +the coupling scheme is given in Figure S1. +As mentioned in the main text, the spin rotation interaction is modified to account for the bending motion. Here we +provide further explanation. In the effective Hamiltonian approach, the spin-rotation parameter receives contributions +from various orders of perturbation theory, γ = γ(1)+γ(2)+· · · [54]. The first order term γ(1) results from the magnetic +interaction between the electron spin and the magnetic dipole moment of the rotating molecule. In heavy molecules, +the first order term is small compared to the dominant second order contribution γ(2), arising from off-diagonal spin- +orbit and rotational perturbations. For linear molecules with Nz = 0, the spin-rotation term N · S implicitly only +contains contributions from NxSx and NySy. However for a bending molecule, since Nz ̸= 0, we explicitly subtract +away NzSz. +Matrix elements for the N 2 and N · S terms can be found in Refs. [54, 55]. Here we reproduce matrix elements for +the terms specific to the bending mode. +⟨ℓ; N, S, J, M|T 1 +q=0(N)T 1 +q=0(S)|ℓ′; N ′, S, J′, M ′⟩ += δJ,J′δN,N ′δM,M ′δℓ,ℓ′ × ℓ +× (−1)J+N ′+S +� +N +S J +S N 1 +� +× (−1)N−ℓ +� +N +1 N +−ℓ 0 +ℓ +� +(2N + 1) +× +� +S(S + 1)(2S + 1) +(S15) + +23 +⟨ℓ; N, S, J, M|T 2 +2q(N, S)e−2iqφ|ℓ′; N ′, S, J′, M ′⟩ += δJ,J′δN,N ′δM,M ′δℓ,ℓ′+2q +× (−1)J+N+S +� +5 +2 +� +N +S J +S N 1 +� +× +� +S(S + 1)(2S + 1) +× +√ +3 +� +2 +1 +1 +N N N +� +� +N(N + 1)(2N + 1) +× (−1)N−ℓ +� +N +2 +N +−ℓ 2q +ℓ +� +(2N + 1) +× +� +S(S + 1)(2S + 1) +(S16) +⟨ℓ; N, S, J, M|T 2 +2q(N, N)e−2iqφ|ℓ′; N ′, S, J′, M ′⟩ += δJ,J′δN,N ′δM,M ′δℓ,ℓ′+2q +× (−1)J+N+S +� +N +J +S +J +N 0 +� +× +√ +5 +� +2 +2 +0 +N N N +� +× +1 +2 +√ +6 +� +(2N − 1)(2N)(2N + 1)(2N + 2)(2N + 3) +× (−1)N−ℓ +� +N +2 +N +−ℓ 2q +ℓ +� +(2N + 1) +(S17) +C. +Stark and Zeeman Matrix Elements +For ˜X(010), the Stark and Zeeman matrix elements are given in Hund’s case (b). For the Stark matrix element, +we only consider the contribution from the dipole component along the molecular z axis. +⟨ℓ; N, S, J, M|T 1 +p (d)|ℓ′; N ′, S, J′, M ′⟩ += (−1)J−M +� +J +1 +J′ +−M p −M +� +× (−1)J′+N+S+1� +(2J + 1)(2J′ + 1) +� +N ′ J′ S +J +N +1 +� +× (−1)N−ℓ� +(2N + 1)(2N ′ + 1) +� +N +1 N ′ +−ℓ 0 +ℓ′ +� +(S18) +⟨ℓ; N, S, J, M|T 1 +p (S)|ℓ′; N ′, S, J′, M ′⟩ += δℓ,ℓ′(−1)J−M +� +J +1 +J′ +−M p −M +� +× (−1)J+N+S+1� +(2J + 1)(2J′ + 1) +� +S J′ N +J +S +1 +� +× +� +S(S + 1)(2S + 1) +(S19) + +24 +IV. +INTENSITY BORROWING AND TRANSITION DIPOLE MOMENTS +A. +Hund’s Case (b) to Case (a) Change of Basis +The eigenstates of ˜X(010) are best described by Hund’s case (b) wavefunctions, while the eigenstates of ˜A(000) +are described by Hund’s case (a) wavefunctions. To calculate transitions, we convert between the two cases using the +following formula from Brown [88]: +|N, K, S, J, M⟩ = +� +Σ,P +(−1)N−S+P √ +2N + 1 +� +J +S +N +P −Σ −K +� +|S, Σ; J, P, M⟩ +(S20) +Here, P = Λ + Σ + ℓ, and K = Λ + ℓ. Note this form is equivalent to that given by Hirota in Ref. [55]. +B. +Transition Dipole Moment +The transition dipole moment (TDM) matrix element is evaluated in Hund’s case (a): +⟨ℓ; Λ; S, Σ; J, P, M|T 1 +p (d)|ℓ′; Λ′; S, Σ′; J′, P ′, M ′⟩ += δΣ,Σ′δℓ,ℓ′ +× (−1)J−M +� +J +1 +J′ +−M p M ′ +� +× +� +(2J + 1)(2J′ + 1)(−1)J−M +× +� +q +� +J +1 J′ +−P q P ′ +� +δΛ,Λ′+q +× ⟨Λ||T 1 +q (d)||Λ′⟩ +(S21) +The last term is the reduced matrix element encoding the transition dipole integral between two electronic states. +The ∆ℓ = 0 selection rule is explicit in the above matrix element. This means we can only drive ˜X(010) to admixtures +in ˜A(000) with |ℓ| = 1. +C. +Mixing with |ℓ| = 1 states +To model the transition intensities, as stated in the main text, we first separately diagonalize the ˜A2Π1/2(000) and +˜X(010) Hamiltonians. We then convert the ˜X(010) eigenvectors from Hund’s case (b) to case (a), using equation S20. +Since we use effective Hamiltonians, the ˜A2Π1/2(000) eigenvectors have ℓ = 0. However, in reality these eigenvectors +are perturbed by other states, and contain admixtures with |ℓ| = 1. These admixed states provide the transition +intensity and non-zero transition dipole moment. +To represent the admixed states, we perform a change of basis to transform the ˜A2Π1/2(000) effective Hamiltonian +eigenvectors into eigenvectors of the admixed states. As states in the main text, the states of interest with ℓ = 1 are +˜Aµ2Σ(+) +1/2(010), ˜Aκ2Σ(−) +1/2(010), and ˜B2Π(010), where we are using vibronic term symbols 2S+1KP . Each eigenvector of +˜A(000) is transformed into a linear combination of eigenvectors from the admixed states, with amplitudes cµ, cκ, cB. +The mixing between ˜A2Π1/2(000) and ˜B2Π(010) occurs at first order due to HRT (see main text). +Since this +interaction preserves K and P, it simply exchanges one quanta between ℓ and Λ. Since ˜A2Π1/2(000) has P = 1/2, we +only consider mixing other P = 1/2 states. We perform the following change of basis: + +25 +⟨ ˜B(010), Λ = 0, ℓ, Σ, P | ˜A(000), Λ′, ℓ′ = 0, Σ′, P ′ = ±1/2⟩ = δℓ,Λ′δP,P ′δΣ,Σ′(−1)P −1/2 +(S22) +Note the phase factor (−1)P −1/2 is explicitly included to preserve parity. This accounts for the extra (−1)ℓ phase +factor in the parity of an ℓ ̸= 0 state compared to an ℓ = 0 state. This factor can arise naturally if HRT is written as +∝ sin (θ − φ) instead of being ∝ cos (θ − φ). While the latter form is most often found in the literature [43, 55], the +former can be found in Ref. [82] in the context of Σ− states. +The admixture of the µ and κ states occurs via a second-order combination of HRT and HSO. These interactions +preserve P but can change K. For µ(010) we obtain the following change of basis: +⟨µ(010), Λ, ℓ, Σ, P | ˜A(000), Λ′, ℓ′ = 0, Σ′, P ′ = ±1/2⟩ = δΛ,−Λ′δℓ,Λ′δΣ,−Σ′(−1)P −1/2 +(S23) +And for κ(010): +⟨κ(010), Λ, ℓ, Σ, P| ˜A(000), Λ′, ℓ′ = 0, Σ′, P ′ = ±1/2⟩ = δΛ,Λ′δℓ,−Λ′δΣ,−Σ′(−1)P −1/2 +(S24) +After changing basis to states with |ℓ| = 1, we compute the transition dipole matrix element using equation S21. +The transition amplitudes for the different state admixtures are added together, and the resulting interference depends +on the mixing coefficients cµ, cκ, cB. Finally, to obtain relative intensities, we square the total transition amplitude. +V. +ASSIGNED LINES +See Table S1. Line notation is described in the main text. + +26 +TABLE S1. +Observed lines, ground states quantum numbers (N ′′, J′′, P′′), excited states quantum numbers (J′, P′), observed +positions, and residuals of ˜ +X2Σ+(010) → ˜A2Π1/2(000) band of YbOH. There are in total 38 lines assigned to 39 transitions as +the QR− +12(1) and P Q− +12(5) lines are overlapped. The fit residual is 6.1 MHz. +Line +N ′′, J′′, P′′ +J′, P′ +Obs. (cm−1) +Obs. - Calc. (MHz) +OP + +12 +2, 3/2, + +1/2, − +17002.4883 +4.4 +3, 5/2, + +3/2, − +17002.4312 +−7.4 +4, 7/2, + +5/2, − +17000.6512 +−2.7 +OP − +12 +2, 3/2, − +1/2, + +17002.9232 +−0.1 +3, 5/2, − +3/2, + +17001.5614 +14.9 +P P + +11 +1, 3/2, + +1/2, − +17003.4683 +−0.2 +3, 7/2, + +5/2, − +17002.6114 +1.7 +5, 11/2, + +9/2, − +17001.8212 +12.2 +P P − +11 +1, 3/2, − +1/2, + +17003.9070 +−2.2 +2, 5/2, − +3/2, + +17003.0314 +−3.6 +4, 9/2, − +7/2, + +17002.2076 +−4.8 +P Q+ +12 +2, 3/2, + +3/2, − +17003.9039 +−8.8 +3, 5/2, + +5/2, − +17002.6012 +−5.8 +5, 9/2, + +9/2, − +17001.8046 +12.7 +P Q− +12 +1, 1/2, − +1/2, + +17003.9053 +−5.8 +2, 3/2, − +3/2, + +17003.0250 +−5.0 +3, 5/2, − +5/2, + +17003.9208 +−5.3 +5, 9/2, − +9/2, + +17004.0076 +13.3 +QQ+ +11 +1, 3/2, + +3/2, − +17004.8846 +3.3 +3, 7/2, + +7/2, − +17005.9150 +−13.0 +5, 11/2, + +11/2, − +17007.0123 +−3.5 +QQ− +11 +1, 3/2, − +3/2, + +17004.0091 +5.5 +2, 5/2, − +5/2, + +17005.3917 +−0.7 +4, 9/2, − +9/2, + +17006.4556 +−1.6 +QR+ +12 +1, 1/2, + +3/2, − +17004.8824 +1.3 +2, 3/2, + +5/2, − +17004.0743 +5.1 +3, 5/2, + +7/2, − +17005.9052 +−5.8 +QR− +12 +1, 1/2, − +3/2, + +17004.0076 +7.3 +2, 3/2, − +5/2, + +17005.3853 +0.9 +4, 7/2, − +9/2, + +17006.4421 +−6.9 +RR+ +11 +1, 3/2, + +5/2, − +17005.0543 +−1.5 +2, 5/2, + +7/2, − +17007.3837 +−0.7 +3, 7/2, + +9/2, − +17006.2215 +2.3 +4, 9/2, + +11/2, − +17009.4646 +−2.9 +RR− +11 +1, 3/2, − +5/2, + +17006.3695 +12.7 +2, 5/2, − +7/2, + +17005.6298 +6.1 +3, 7/2, − +9/2, + +17008.4157 +3.7 +4, 9/2, − +11/2, + +17006.8298 +−0.6 +5, 11/2, − +13/2, + +17010.5312 +−0.9 + diff --git a/ltE2T4oBgHgl3EQfywiY/content/tmp_files/load_file.txt b/ltE2T4oBgHgl3EQfywiY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb2ae3ef5edf5ae6347f033f20872b254d4c1932 --- /dev/null +++ b/ltE2T4oBgHgl3EQfywiY/content/tmp_files/load_file.txt @@ -0,0 +1,1448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf,len=1447 +page_content='Characterizing the Fundamental Bending Vibration of a Linear Polyatomic Molecule for Symmetry Violation Searches Arian Jadbabaie,1, ∗ Yuiki Takahashi,1, ∗ Nickolas H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Pilgram,2, † Chandler J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Conn,1 Yi Zeng,1 Chi Zhang,1 and Nicholas R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Hutzler1 1California Institute of Technology, Division of Physics, Mathematics, and Astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Pasadena, CA 91125 2California Institute of Technology, Division of Engineering and Applied Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Pasadena, CA 91125 (Dated: January 11, 2023) Polyatomic molecules have been identified as sensitive probes of charge-parity violating and parity violating physics beyond the Standard Model (BSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For example, many linear triatomic molecules are both laser-coolable and have parity doublets in the ground electronic ˜ X2Σ+(010) state aris- ing from the bending vibration, both features that can greatly aid BSM searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Understanding the ˜ X2Σ+(010) state is a crucial prerequisite to precision measurements with linear polyatomic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Here, we characterize fundamental bending vibration of 174YbOH using high-resolution optical spectroscopy on the nominally forbidden ˜ X2Σ+(010) → ˜A2Π1/2(000) transition at 588 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We assign 39 transitions originating from the lowest rotational levels of the ˜ X2Σ+(010) state, and accurately model the state’s structure with an effective Hamiltonian using best-fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Addi- tionally, we perform Stark and Zeeman spectroscopy on the ˜ X2Σ+(010) state and fit the molecule- frame dipole moment to Dmol = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='16(1) D and the effective electron g-factor to gS = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='07(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Further, we use an empirical model to explain observed anomalous line intensities in terms of inter- ference from spin-orbit and vibronic perturbations in the excited ˜A2Π1/2(000) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Our work is an essential step toward searches for BSM physics in YbOH and other linear polyatomic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' INTRODUCTION Polyatomic molecules are at the frontier of advanced control over quantum complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Their additional rovibra- tional degrees of freedom provide a large degree of control and tunability of both molecular structure and interactions with a wide range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Rapid progress [1–3] has been made in laser cooling molecules, including polyatomic CaOH [4, 5], CaOCH3 [6], SrOH [7, 8], and YbOH [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Recently, CaOH was optically trapped and laser-cooled to ultracold temperatures [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Quantum control of polyatomic molecules will benefit next-generation searches for new physics beyond the Standard Model [12–15], and will enable advances in quantum computation, simulation, and chemistry [16–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Currently, measurements of diatomic ThO and HfF+ bound charge-parity (CP) violating new physics at TeV energy scales [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' These experiments benefit significantly from parity doubling, the occurrence of nearly-degenerate levels of opposite parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Molecules with parity doublets can be easily aligned in the lab frame with the application of modest electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Furthermore, when polarized, these molecules have both aligned and anti-aligned states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Known as internal co-magnetometers, these states allow for reversal of CP-violating interactions without modifying the external lab field [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This degree of control over molecular alignment is highly advantageous for robust systematic error rejection in searches for CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In diatomic molecules, parity doublets require orbital angular momentum, which conflicts with electronic requirements for efficient laser cooling, especially for heavy molecules with enhanced sensitivity to new physics [12, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Polyatomic molecules offer both generic parity doublets and laser cooling, and therefore provide a route to sig- nificantly improve constraints on new CP-violating physics by multiple orders of magnitude [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' A number of CP ∗ These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' † Current affiliation: NIST Physical Measurement Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Gaithersburg, MD 20899 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='04124v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='atom-ph] 10 Jan 2023 2 violation searches are underway with laser-coolable diatomic molecules, such as BaF [23], YbF [24, 25], TlF [26], and RaF [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Without parity doublets in their ground states, these molecules require large electric fields (>10 kV/cm) for significant polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' By contrast, molecules with parity doublets offer similar polarization in much smaller fields, and the variety of molecular orientations offer richer possibilities for state tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In polyatomic molecules, parity doublets arise from rotation around the inter-nuclear axis and exist independently of the electronic structure used for laser cooling [1, 2, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Examples of polyatomic parity doublets include K doublets in rotations of symmetric molecules, asymmetry doublets in the rotations of asymmetric molecules, and ℓ doublets in bending modes of linear polyatomic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' YbOH molecules in their doubly-degenerate bending mode have been identified as sensitive probes of CP-violating physics [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The Yb-centered, core-penetrating valence electron provides both new physics sensitivity and optical cycling, which was demonstrated with Sisyphus cooling of a YbOH beam to a transverse temperature of <600 µK [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Meanwhile, the vibrational bending motion provides ℓ-type parity doublets that allow polarization control and internal co-magnetometry in modest external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Furthermore, the multiple stable isotopes of Yb provide opportunities for CP violation searches in both the hadronic and leptonic sectors of the Standard Model [12, 29–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Finally, other experiments leveraging the bending motion of linear triatomic molecules, including CP violation searches with SrOH [36] and RaOH [12, 37], and parity-violation searches with linear triatomics [38], warrant further investigation of these states, for which there is no previous, complete study of all molecular properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Here, we present a high-resolution, optical spectroscopy study of the fundamental bending vibration in the electronic ground state of 174YbOH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The spectra are obtained by laser excitation on a rovibrationally forbidden electronic transition in a cryogenic buffer gas beam (CBGB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' By analyzing the field-free, Stark, and Zeeman spectra, we model the rotational structure of the bending molecule, characterize the electric and magnetic tuning of the levels, and extract the molecule-frame dipole moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Our results demonstrate the high level of control available in polyatomic molecules, which will be useful for future symmetry violation searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' First, we provide a brief overview of the overall molecular structure in section I A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The methods are described in section II, with section II A describing the experimental apparatus, and section II B describing the effective Hamiltonians used to model the molecular states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In section III we describe our experimental results and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Section III A discusses the field-free spectrum and optimal state parameters, section III C describes our model for the anomalous line intensities of the forbidden transition, and section III B presents the Stark and Zeeman spectra and their analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We conclude in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Molecular Structure In this section, we briefly review the structure of linear polyatomic molecules, including states with bending vibra- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We label the ground and excited state electronic states as ˜X and ˜A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Electronic states of linear polyatomic molecules are labeled with the term symbol 2S+1ΛΩ(v1 vl 2 v3), where Λ = ⃗L · ˆn is the projection of elec- tronic orbital angular momentum L on the internuclear axis ˆn, Σ = ⃗S · ˆn is the projection of the electron spin S, Ω = Λ + Σ = ⃗J · ˆn is the total projection of the spin and rotational angular momentum J, and vi denotes the number of quanta in the three vibrational modes of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For Λ = 0 states, an additional +/− subscript is used to denote the parity of the electronic configuration, and the Ω subscript is sometimes dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In YbOH [12], the v1 mode is the Yb-O stretch, the v3 mode the O-H stretch, and, due to the Yb mass, the doubly-degenerate vℓ 2 mode can be viewed as the bending of the H atom relative to the Yb-O axis [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The additional ℓ label denotes the number of quanta of vibrational angular momentum G projected on the internuclear axis, ℓ = ⃗G · ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The degeneracy of ±ℓ states are lifted by higher order perturbations, giving rise to parity doublets [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The above electronic labeling scheme treats the vibrational degrees of freedom separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' However, for states with non-zero ℓ and Λ, interactions of the electrons with the bending vibration, known as Renner-Teller couplings [42, 43], will cause rovibrational splittings for different states of K = Λ+ℓ = ⃗N·ˆn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Here is ⃗N = ⃗J−⃗S is the rovibrational angular momentum of the electrons and nuclei, excluding spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Note that N can receive contributions from multiple sources: the end-over-end molecular rotation R, electronic orbital angular momentum L, and vibrational angular momentum 3 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' When both Renner-Teller and spin-orbit couplings are present, neither K nor Ω are completely conserved, and instead the eigenstates have well defined projection quantum number P = ⃗J · ˆn = Λ + ℓ + Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We note that the total angular momentum cannot be less than the projection angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For example, in a state with well-defined N and K, we always have N ≥ |K|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' a consequence relevant for this work is that the lowest rotational level of an ℓ = 1 bending mode has N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We will restrict our discussion to states with v1 = v3 = 0 and v2 ∈ {1, 0}, allowing us to write vibronic term symbols as 2S+1KP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Note that in the term symbols, both Λ and K are denoted as Σ, Π, ∆, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' to indicate 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=', similar to the S, P, D, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' notation in atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This can lead to confusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' for example the (010) vibrational state in the ground electronic state is a 2Σ+ electronic state, but a 2Π vibronic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Whenever we do not include the (v1 v2 v3) label, we are referring to a vibronic term, unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In this work, we study the ˜X2Σ+ 1/2(0110) → ˜A2Π1/2(000) band of 174YbOH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This transition is nominally forbidden in the dipole approximation, which requires ∆ℓ = 0, and it occurs via intensity borrowing in the excited state, as we discuss later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We will neglect the other spin-orbit manifold, ˜A2Π3/2(000), which is located ∼40 THz above ˜A2Π1/2(000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The large spin-orbit coupling in YbOH means Ω is an approximately good quantum number, even in bending states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For simplicity, we will abbreviate the ground state label as ˜X(010) and the excited state label as ˜A(000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In 174YbOH, the 174Yb nucleus has no nuclear spin, and the hyperfine structure from the distant hydrogen nuclear spin I is optically unresolved [44] and only contributes to broadening in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Therefore in this study we neglect I, and label states with well-defined total angular momentum J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Ground state quantum numbers are denoted with a double prime, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N ′′, and excited states with a single prime, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We denote rotational lines with notation similar to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Given the parity doubling in both ˜X(010) and ˜A(000), we add an additional label to denote the parity of the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We label transitions as ∆N∆JP′′ F ′ i ,F ′′ i (N ′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Here, F ′ i = 1 for the excited state, F ′′ i = 1, 2 denotes ground states with J′′ = N ′′ ± S, and P′′ = ± denotes the ground state parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Experiment: Apparatus and Signals The cryogenic buffer gas beam (CBGB) apparatus (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 1a) is similar to that from our previous work [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In summary, the buffer gas cell is formed from a copper block with an interior cylindrical bore 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='5 cm long and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='7 mm in diameter, with windows on the sides for optical access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The cell is surrounded by radiation shields and cooled by a pulse tube refrigerator down to ∼4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Helium buffer gas is introduced in the back of the cell via a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='2 mm gas inlet tube, and passes through a diffuser 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='2 mm downstream in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Typical flow rates are 3 − 6 standard cubic centimeters per minute (SCCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The buffer gas exits the cell via a 5 mm diameter aperture at the front of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Activated charcoal fins on the interior surface of the 4 K radiation shields provide efficient cryo-pumping of the He buffer gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' YbOH molecules are produced by laser ablation of pressed powder targets made from a 1:1 stoichiometric mixture of Yb(OH)3 powder and Yb powder (see supplementary materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Laser ablation is performed by a Nd:YAG laser at 532 nm with ∼10 ns pulse length, 25−40 mJ pulse energy, and ∼9 Hz repetition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The ablation laser is focused with a 300 or 400 mm lens placed approximately one focal length away from the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Hot molecules produced via ablation are subsequently thermalized by collisions with ∼4 K He buffer gas atoms [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We further increase YbOH yield by around an order of magnitude by exciting atomic Yb to the excited 3P1 state [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Specifically, we send ∼300 mW of 556 nm light into the cell to resonantly drive the 1S0 → 3P1 transition of 174Yb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This technique significantly increases the quantity of YbOH in excited vibrational states, including the ˜X(010) state, whose population is increased by a factor of ∼10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' A few milliseconds after ablation, the He gas flow extracts the molecules out of the cell through the aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Molecule density is monitored both in the cell and outside the cell aperture with 577 nm absorption probes resonant 4 Time After Ablation (ms) Ablation Skimmer Collimator Fluorescence probe YbOH 4 K PMT Photodiodes Target Fill line Buffer gas Chemical enhancement Absorption probes ITO-coated electrodes Light pipe Optical filters Magnetic field coils (ii) (iii) (a) in Cell Front of Cell in Beam (b) (i) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (a) Experimental schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' YbOH molecules are produced in the 4 K cryogenic buffer gas cell (brown box) by laser ablation (dark green triangle) of a solid pressed target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The molecules are thermalized by collisions with He buffer gas continuously flowed into the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Chemical production of YbOH is enhanced by exciting Yb atoms using a laser (light green line) resonant with the 1S0 → 3P1 atomic Yb transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Some of the molecules are produced in the ˜ X(010) bending mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The molecules are entrained in the He gas flow and extracted out of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We detect the molecule number density in the ˜ X(000) state via absorption spectroscopy (yellow lines) both in the cell (i) and in front of the cell (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The molecular beam is collimated by a skimmer and collimators before entering the probe region with electric and magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We apply magnetic fields using coils outside the vacuum chamber, and apply electric fields using ITO coated glass electrodes inside the vacuum chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In the center of the fields, molecules in the ˜ X(010) state are excited by a laser (orange line) and their fluorescence is collected through a light pipe to a PMT (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (b) Sample signals from the CBGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (i) In-cell absorption on the RR11(0) line of YbOH ˜ X(000) → ˜A(000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The peak optical depth corresponds to a molecule density of ∼5×109 cm−3 in the ˜ X(000), N = 0 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (ii) Front of cell absorption on the same RR11(0) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The peak optical depth corresponds to a molecule density of ∼2×109 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (iii) Fluorescence after excitation of the bending mode on a strong ˜ X(010) → ˜A(000) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The integrated signal corresponds to ∼8300 photons detected on the PMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' with the RR11(0) line of the ˜X(000) → ˜A(000) transition at 17325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0365 cm−1 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The extracted beam is rotationally and translationally cold, but can have significant excited vibrational population, a result of inefficient vibrational thermalization from buffer gas collisions [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This provides a significant advantage, as we obtain ∼109 molecules exiting the cell in the excited bending mode as a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The molecular beam is collimated by a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4 mm diameter skimmer 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='8 cm downstream from the cell aperture, a 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='5 mm diameter hole 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4 cm downstream from the cell aperture, and a 5 mm diameter hole 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='7 cm downstream from the cell aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The beam travels at 150 − 200 m/s toward the laser-induced fluorescence (LIF) measurement region located ∼60 cm downstream from the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The region is pumped by multiple turbomolecular pumps, and typical pressures when flowing He gas are 1−5×10−7 Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In YbOH, the ˜A(000) → ˜X(010) transition has a vibrational branching ratio of r010 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='054(4)% [50], and the lifetime of the ˜A2Π1/2 state is τ = 20(2) ns [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The excited state population primarily decays to the vibrational ground state, ˜X(000), with r000 = 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='44% branching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Therefore, in our experiment, the fluorescence signal will saturate after roughly one photon scatter as the molecules are optically pumped out of the bending mode and mostly into the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' With a ∼1 mm Gaussian laser beam intersecting a ∼200 m/s molecular beam, we 5 can estimate the saturation parameter required for a single photon scatter as s ≈ 1 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Using the definition of saturation intensity for a transition with branching ratio r as Is = πhc/(λ3τr) [52], we compute an intensity of I ≈ 280 mW/cm2 required to optically pump the forbidden transition ˜X(010) → ˜A(000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For a 1 mm diameter Gaussian laser beam, this requires ≳ 2 mW of optical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' While we have neglected rotational branching and other experimental imperfections in this analysis, we observe the power requirements needed to produce fluorescence on such a forbidden line are feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Downstream in the LIF region, molecules in the ˜X(010) bending mode are excited by a 588 nm laser resonant with the nominally forbidden ˜X(010) → ˜A(000) transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The laser beam, with a ∼1 mm diameter and ∼40 mW of power, is sent perpendicular to the molecular beam (see Fig 1a) through windows at Brewster’s angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The resulting 577 nm fluorescence from decays to the ˜X(000) state is collected with a 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4 mm diameter fused-quartz light pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' A 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4 mm diameter, 19 mm focal length retroreflecting concave mirror opposite the light pipe improves collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We filter out the 588 nm scattered background light using a combination of interference and colored glass filters on the exit of the light pipe, obtaining a signal-to-noise ratio of >10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The fluorescence signal is incident on a photomultiplier tube (PMT) module (Hamamatsu H13543-300), and the resulting photocurrent is amplified with a 10−8 A/V trans-impedance amplifier with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='5 kHz low pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' To obtain the field-free spectrum, we scan the 588 nm probe laser and record its frequency using a wavelength meter (HighFinesse WS7-30) with an absolute accuracy of 30 MHz and a measurement resolution of 1 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' To improve the absolute accuracy, we use the probe light to co-record sub-Doppler I2 spectra, obtained with amplitude modulated saturated absorption spectroscopy [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Calibration of the laser frequency using the I2 spectra results in one standard deviation error of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='35 MHz in absolute frequency accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Figure 1b shows typical absorption and LIF signals obtained in a single shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The LIF signal size typically varies from shot to shot due to ablation yield fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' To construct the field-free spectrum, we scan the laser at approximately 1-2 MHz per shot, average the LIF signal for 4 shots, integrate over the molecule pulse duration, and plot the data against the calibrated probe frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The observed peaks are fit well by a Lorentzian function, with fitting errors < 3 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For the Stark and Zeeman spectra, we step the laser in 3 MHz increments, and average the LIF signal for 10 shots at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For Stark spectroscopy, we use two indium tin oxide (ITO) coated glass plates separated by a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='99(3) mm gap to apply fields up to 265 V/cm in the LIF region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Before entering the field region, the molecular beam is further collimated with a 3 mm hole in a grounded aluminum plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The molecules traveling through the ITO plates are then excited by the 588 nm laser (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The resulting fluorescence is collected through the glass plates with the setup described earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For Zeeman spectroscopy, we generate magnetic fields of 0 − 70 Gauss using two pairs of wire coils outside the vacuum chamber (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The two coil pairs have a diameter of 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4 cm with 500 windings each, and are each symmetrically spaced from the LIF region with distances of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='5(1) cm and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3(1) cm to the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Theory: Effective Hamiltonian The ground and excited states are modeled with an effective Hamiltonian approach [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The ˜A(000) state is well described by a Hund’s case (a) Hamiltonian, using parameters from a previous optical study on a supersonic YbOH beam [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Complete details of the effective Hamiltonian are provided in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In the excited state, strong spin-orbit interactions mean N is not a well-defined quantum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Conversely, the molecule-frame projection quantum numbers Λ, Σ, and Ω are well-defined in Hund’s case (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Cross terms of spin-orbit and rotational perturbations give rise to the Λ-doubling interaction, which mixes the projection quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The resulting Hund’s case (a) ˜A eigenstates are symmetric and anti-symmetric superpositions of projections with well defined parity P: |Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J, Ω, M, P = ±⟩ = 1 √ 2(|Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J, Ω, M⟩ ± (−1)pa| − Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S, −Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J, −Ω, M⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (1) 6 The phase factor pa = J−S−ℓ is connected to the convention for the action of the parity operator, P|Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J, Ω, M⟩ = (−1)pa| − Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S, −Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J, −Ω, M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This phase convention is followed by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [43, 55] (Details in the supplementary materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We model the ground ˜X(010) state using a Hund’s case (b) effective Hamiltonian describing a 2Π vibronic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This approach has provided an accurate description of the vibrational bending modes in other metal hydroxide molecules, such as CaOH and SrOH in optical [39] and millimeter wave [56] studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The lack of first-order spin-orbit interaction means the electron spin S is largely independent of the internuclear axis, and therefore both Σ and P are undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Hund’s case (b) is the natural basis, with N and its projection ℓ as good quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The spin-rotation interaction then couples N with S to form well-defined J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Higher-order perturbations give rise to the ℓ-doubling interaction, and the ˜X eigenstates of good parity are written as: |ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N, S, J, M, P = ±⟩ = 1 √ 2(|ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N, S, J, M⟩ ± (−1)pb| − ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N, S, J, M⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (2) The phase factor in Hund’s case (b) is defined as pb = (−1)N−ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The additional factor of ℓ = 1 means the action of the parity operator on a singly excited bending mode is similar to that of a Σ− electronic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' While this phase convention has physical basis (see supplementary materials) and has been used in literature [43, 55, 57–59], the choice is not universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The parity phase and the sign of the ℓ-doubling Hamiltonian together determine if the lowest energy state is positive or negative parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We use an effective Hamiltonian for the ˜X(010) state given by H ˜ X(010) = B( ⃗N 2 − ℓ2) + γ( ⃗N · ⃗S − NzSz) + γGNzSz + pG 2 � N+S+e−i2φ + N−S−ei2φ� − qG 2 � N 2 +e−i2φ + N 2 −ei2φ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (3) This form was first derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [60] and is presented in detail in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [57, 58, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Here, all subscripts on angular momenta (z, ±) denote molecule-frame quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The azimuthal angle of the bending nuclear framework is given by φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The first term gives the rotational energy of a symmetric top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The next two terms describe the spin-rotation interaction coupling N and S to form J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The last two terms describe ℓ-type parity doubling caused by terms off-diagonal in the vibrational angular momentum G, and cause splittings of opposite parity states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For the spin-rotation interaction we have modified the usual expression, γN · S, by subtracting γNzSz to account for the bending motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This modification is crucial for accurate description of low-N spectra (see supplementary materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Other perturbations can reintroduce this axial spin-rotation term into the Hamiltonian, labeled in the literature with the coefficient γ′ [60] or γG [57, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The first order contribution to γG arises from magnetic dipole interactions [62] and is negligible for the Yb-centered electron in YbOH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' At higher order, a combination of vibronic coupling and spin-orbit interactions can contribute to γG by mixing states with Π electronic character, as observed in NCO [63], CCH [64], and FeCO [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 3, the qG parity-doubling term is standard for a bending molecule in a 2Σ electronic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This term arises from Coriolis effects at second order, similar to the q term in Λ-doubling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The pG term, also in analogy with Λ- doubling, is equivalent to a parity-dependent spin-rotation interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Owing to the weak coupling of the spin to the internuclear axis in Σ electronic states, this term is small and has only been observed in submillimeter spectroscopy of metal hydroxides [56, 66], ZnCN [67], and CrCN [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' As with γG, this term receives higher-order contributions from vibronic mixing with electronic Π states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In spherical harmonic notation [54], the ℓ-type doubling terms may be written in the molecule frame as � q=±1 e−2iqφ � pGT 2 2q(N, S) − qGT 2 2q(N, N) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We are using a sign convention for the ℓ-type doubling Hamiltonian outlined by Brown [59, 61], where the ℓ-type doubling Hamiltonian mirrors that used for Λ-doubling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' However matrix elements of ℓ involve different phases than Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' As a result of the (−1)ℓ factor in our parity phase, we have the matrix elements ⟨ℓ = ±1|e±2iφ|ℓ′ = ∓1⟩ = 1, differing from the azimuthal matrix elements for Λ-doubling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Matrix elements and complete details of the effective Hamiltonian and conventions used are provided in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We construct the predicted spectrum by first separately diagonalizing the effective Hamiltonians for the ground and excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The Hamiltonian basis is truncated at N ′′ = 6 for the ˜X(010) state and J′ = 15/2 for the ˜A(000) 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Field-free spectrum over a ∼9 cm−1 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Orange upper part is experimental observation and blue lower part is theory prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Prediction is using effective model detailed in section III C with coefficients (cµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='28, cκ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='49, cB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='83) and a temperature of T = 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Lines marked with * are unassigned and could arise from other isotopologues or bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [45], we include the P = 3/2 manifold when diagonalizing ˜A(000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' After obtaining eigenvectors and eigenvalues, we convert all eigenvectors to Hund’s case (a) and compute matrix elements of the transition dipole moment (TDM) operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Details of the TDM operator are given in section III C and in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For transitions with non-zero TDM, we compute the line position by taking the difference of excited and ground eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Field-Free Spectrum The observed spectrum (Fig 2) exhibits large splittings that match the excited state Λ-doubling and rotational separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We perform combination-difference tests [54] with these splittings to obtain initial quantum number assignments of transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' With these assignments, we compute initial guesses for the B, γ, and qG Hamiltonian parameters for the ˜X(010) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Using these values and fixing the excited state parameters, we construct a predicted spectrum and perform further line assignments (line notation is described in I A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' With this analysis, we determined the need for additional parameters pG and γG to accurately describe the full spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Without the pG term, various R and P branch features deviate from the prediction by a magnitude >20 MHz, much larger than our frequency error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Specifically, in the region scanned in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 2, without pG, lines with significant residuals are: RR+ 11(2), RR− 11(3), OP + 12(4), P Q+ 12(5), and P P + 11(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The magnitude and parity behavior of these residuals cannot be explained by centrifugal distortion, but can be explained by a parity-dependent spin-rotation interaction, namely pG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' By introducing pG into the prediction, all of these residuals are reduced to values commensurate with the experimental error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Furthermore, using the fit value of pG, we predicted and found the RR+ 11(4) and RR− 11(5) lines (not visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' These additional lines are added to the final fit and confirm the need for a pG term to accurately model the full spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Unlike pG, the γG term does not scale with N ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' However, we find this term necessary to describe the N ′′ = 1 structure, which was crucial for accurate Stark and Zeeman analysis in section III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In particular, we recorded multiple field-free calibration scans of the QQ+ 11(1) and QR+ 12(1) lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Since these lines share the same excited state, their separation is insensitive to error in the ˜A(000) state parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We use the separation of these lines to determine the N ′′ = 1+ spin-rotation splitting to be 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='8(20) MHz, and we add this value as an additional data point for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' By including the γG term in the spectral prediction, were we obtain an accurate prediction of the N ′′ = 1+ splitting commensurate with our measurement error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In total, we assigned 38 of the observed lines to 39 transitions originating from the N ′′ = 1 through N ′′ = 5 levels of the ˜X(010) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Note the QR− 12(1) and P Q− 12(5) lines are overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' To obtain optimal effective Hamiltonian 8 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Spectroscopic parameters for the low-lying vibrational states of the ˜ X2Σ+ manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The ˜ X(010) parameters are obtained from the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Parameter ˜ X(000) [44] ˜ X(010) ˜ X(100) [45] T0/cm−1 0 319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='90901(6) 529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3269(3) B/MHz 7348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4005(3) 7328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='64(15) 7305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='37(24) γ/MHz −81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='15(6) −88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='7(9) −110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='6(21) γG/MHz – 16(2) – qG/MHz – −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0(2) – pG/MHz – −11(1) – parameters, we vary the ˜X(010) state parameters and hold fixed the ˜A(000) state parameters to the values given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We construct predicted spectra and perform nonlinear least-squares minimization of the residuals between the observed and predicted positions of all 39 assigned lines and the N ′′ = 1+ spin-rotation splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' A full list of line assignments is provided in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The best fit parameters are presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The fit residuals have a standard deviation of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='1 MHz, consistent to order unity with the error reported in the previous optical study of the ˜A(000) state [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The rotational and spin rotational ˜X(010) parameters are in good agreement with those for ˜X(000) and ˜X(100), also collected in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The location of the origin T0 is in excellent agreement with previous dispersed fluorescence studies [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The rotational constant B decreases in ˜X(010) as a result of vibrational corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The increasingly negative spin- rotation parameter γ between the three vibrational states is a result of second order spin-orbit perturbations from low-lying electronic states with 4f 136s2 electronic configuration for the Yb centered electron, known as “4f hole” states [44, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Vibronic mixing with electronic 2Π states can also explain the observed γG and pG parameters, which are not typical for the bending mode of an isolated electronic 2Σ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Vibronic mixing exchanges ℓ and Λ while preserving K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' As a result, the ˜X(010) state can acquire some Λ > 0 electronic character, inheriting spin-orbit and Λ-doubling interactions from neighboring 2Π states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Specifically, in the effective Hamiltonian, these interactions can arise at third-order via a combination of linear vibronic coupling and spin-orbit effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This term was first described by Brown in the context of spin-orbit corrections to electronic 2Π states as a result of mixing with other 2Σ or 2∆ states [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Neighboring states that can contribute to γG and pG include both the ˜A manifold and the 4f hole states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The exact nature of the 4f hole states and their vibronic mixing in YbOH is currently unknown and merits further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' However, their proximity to the ground state and their large spin-orbit interactions could explain the significant magnitude of pG and γG in YbOH compared to other metal hydroxides [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The ℓ-type doubling parameter qG is a similar magnitude to that of other metal-hydroxide ˜X(010) states [39, 56], and is in agreement with a recent theoretical calculation [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The parameter qG can be interpreted in terms of the Coriolis coupling constants of a triatomic molecule [39, 41]: qG = −(v2 + 1)B2 ω2 � 1 + � n=1,3 ζ2 2n 4ω2 2 ω2n − ω2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (4) Here, v2 is the number of quanta in the bending vibration ω2, and ζ2n is the Coriolis coupling constant between the bending mode and the vn stretch modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' To estimate ζ21, we can estimate the value of ω3 (O-H stretch) using the CaOH value of 3778 cm−1 [72], and we set v2 = 1, ω2 ≈ T0, and ω1 ≈ 529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3 cm−1 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Furthermore, we can use the relationship ζ2 21 + ζ2 23 = 1 [41] to eliminate ζ2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Using our values of B and qG, we then obtain a value of ζ21 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='137, slightly smaller than in CaOH (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='1969) [39] and SrOH (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='179) [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This is likely due to the break down of the harmonic approximation ω2 ≈ T0 and the approximation of Be ≈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Further work is needed for a complete vibrational characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Using the parameters obtained from our analysis, we construct a field-free level diagram for the N = 1 manifold of the ˜X(010) state, shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' As stated previously, N = 1 is the lowest rotational manifold in the ˜X(010) 9 −pG − 2qG ≈35 MHz −3/4 (γ ＋ γG) ＋ pG/4 ＋ 2qG ≈28 MHz pG/2 − 2qG ≈19 MHz FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Field-free level structure of the N = 1 manifold in the ˜ X(010) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' States are arranged vertically by energy and horizontally by their MF angular momentum projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' States are labeled in the parity basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The hyperfine structure was not resolved in our work, and is instead approximated using parameters from a study of the ˜ X(000) state [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' state, as we always have | ⃗N · ˆn| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Due to their small parity splittings, N = 1 states are easily polarized, making them useful for precision measurements [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The effect of the parity-dependent spin-rotation term, pG, is apparent in the asymmetric parity-doubling of the J = 1/2 and J = 3/2 manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Though we are not sensitive to hyperfine splittings, for completeness we have included the H hyperfine structure using the parameters obtained for the ˜X(000) state in a previous study [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The hyperfine structure is not expected to change significantly in the bending mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The recorded spectrum has lines present that could not be assigned with combination-differences using the ˜A(000) structure, and are not observed in the prediction using the best-fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The lines are marked with * in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We conclude that some of these lines are indeed from 174YbOH by comparing their chemical enhancement [46] when using 1S0 → 3P1 transitions for different Yb isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' These lines could be unthermalized rotational states, or possibly another overlapping ∆ℓ = ±1 band, such as the ˜X2Σ+(020,20) → ˜A2Π1/2(010) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Stark and Zeeman Spectra After fitting the molecular structure with the field-free spectrum, we study the Stark and Zeeman spectra of the molecule in the presence of static (DC) electric and magnetic fields, using the experimental setup described in II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We obtain the spectra by scanning the 588 nm probe laser across two lines corresponding to the field-free N ′′ = 1+ → J′ = 3 2 − transition, QQ+ 11(1) and QR+ 12(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The applied DC fields point along z, while the laser polarization is along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Spectra are taken with the E-field varied from 0 − 264 V/cm and with the applied B-field varied from 0 − 70 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Calibration spectra are taken with EZ = 0 V/cm and BZ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='5 G, and the observed line positions are compared to the I2-corrected field-free positions to calibrate for frequency offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The lines of interest are relatively well-isolated from other features, and the small N ′′ = 1 parity doubling allows us to enter the linear stark regime with modest laboratory fields ≳100 V/cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Since the parity splittings of the excited ˜A2Π1/2 state are >13 GHz, and its molecule frame dipole moment is D˜A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='43(10) D [45], at the fields we consider the excited state Stark shifts are essentially negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Furthermore, given our frequency resolution and the natural linewidth, we are only sensitive to the isotropic interaction of BZ with the electron spin magnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Curl-type relationships [58] estimate anisotropic spin interactions at 6 × 10−3µB, and the nuclear magnetic moment is also suppressed at a similar level, with both effects giving shifts below our resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 10 1/2+ 3/2+ 3/2− 3/2− 1/2− J N = 1 QQ11(1) + QR12(1) + FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Zeeman spectroscopy of the ˜ X(010) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The main plot shows the transition frequency shift (with subtracted offset) in a magnetic field, the blue lines are optimized model predictions, and the orange circles are experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Error bars are 1-σ measured peak widths, set by a combination of radiative broadening and unresolved hyperfine structure, limiting the ability to resolve closely-spaced lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Lower subplots are slices of the spectra at various magnetic field values, with experimental data in orange and predicted line locations indicated with vertical dashed blue lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' On the left, we show the field-free level structure of the transitions studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' To obtain energy levels and predicted lines, we fix the field-free parameters and diagonalize the combined Stark, Zeeman, and field-free Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We obtain optimal estimates for free Stark and Zeeman parameters by least- squares minimization of the residuals between observed and predicted line positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Both ground and excited levels are magnetically sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The Zeeman shifts of the ˜A2Π1/2(000) and ˜X2Σ+(000) states were previously studied at similar magnetic field strengths in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [45], and recently at high fields (∼1 T) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Following these references, we use the following effective Zeeman Hamiltonians for the ground and excited states: HZee X = gSµBSZBZ (5a) HZee A = g′ SµBSZBZ + gLLZBZ + g′ lµB � e−2iθS+B+ + e2iθS−B− � (5b) Here, Z refers to the lab-frame projection, ± refer to the molecule frame projections, and θ is the electronic azimuthal coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For the excited state, we use the values from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [74], fixing g′ S = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='860, gL = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0, and g′ l = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For the ground state, we allow gS to vary in the fits to find an effective value that accurately describes the Zeeman shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' While we do not include them here, at higher resolution or at higher field values, additional terms are expected to contribute in the effective Zeeman Hamiltonian, including terms associated with the bending angular momentum [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The Zeeman fits prefer a value of gS = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='07(2), deviating from the free electron g-factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The experimental Zeeman shifts and the prediction from the optimized model are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Corrections to gS can arise from mixing involving other states with different Zeeman tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For example, the Zeeman shifts of the ˜A(000) state were fit to g′ S = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='860 in a recent high-field study [74], owing to perturbing 4f 136s2 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Since we observe perturbations from these 4f states in the field-free structure of the ˜X(010) state, it is natural to also find their effects in the Zeeman shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Furthermore, the 4f states are split into a higher energy spin-orbit anti-aligned manifold and a lower energy spin-orbit aligned manifold [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Due to energy proximity, while ˜A(000) predominantly interacts with the 4f hole anti-aligned manifold, ˜X(010) will be perturbed more strongly by the aligned manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The difference in electron orientation of the two spin-orbit 4f manifolds can explain the difference between ˜X(010) and ˜A(000) in the sign of the deviation of gS from its nominal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 11 Increasing Line Strength 1/2+ 3/2+ 3/2− 3/2− 1/2− J N = 1 QQ11(1) + QR12(1) + FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Stark spectroscopy of the ˜ X(010) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The main plot shows the transition frequency shift (with subtracted offset) in an electric field, the blue lines are optimized model predictions, and the orange circles are experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The blue color gradient represents parity forbidden transitions that gain strength at finite electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Error bars are 1-σ peak widths, set by a combination of radiative broadening and unresolved hyperfine structure, limiting the ability to resolve closely-spaced lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Lower subplots are slices of the spectra at various electric field values, with experimental data in orange and predicted line locations indicated with vertical dashed blue lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' On the left, we show the field-free level structure of the transitions studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' To describe the Stark shifts, for the both ground and excited states we use the Hamiltonian HE = − ⃗Dmol · ⃗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The molecule frame dipole moment Dmol is kept as a free parameter, and obtained from spectra where EZ is scanned with BZ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='5 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The optimal fit value is Dmol = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='16(1) D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='09 h MHz/(V/cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This value is in good agreement with the measured ˜X(000) dipole moment of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='9(2) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In Figure 5, we plot the theoretical prediction based on the optimal fit against the observed line positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The Stark shifts confirm the assignment of the ˜X(010) state and demonstrate the orientation control afforded by parity doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In the bending mode, the projection of the molecular axis on the lab-frame Z-axis is given by ˆn · ˆZ = ( ⃗ N·⃗Z)( ⃗ N·ˆn) N(N+1) ∝ MNℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Note we use X, Y, Z to denote lab-frame axes and x, y, z to denote the molecule-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The molecule z axis and dipole moment Dmol both point from O to Yb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For field-free states, ⟨MNℓ⟩ = 0, and the molecule is unpolarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In the presence of an electric field fully mixing parity doublets, the Stark shifts are linear, and the eigenstates are diagonal in the the decoupled basis |ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' MN, MS⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In this regime, the levels split into 2N + 1 dipole moment orientations pointing along MNℓ N(N+1), and splittings within each orientation manifold are due to the spin-rotation interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Anomalous intensities and perturbations Since the ˜A(000) state has been previously fully characterized [45], the assignment of energy levels in ˜X(010) is fairly straightforward using the effective Hamiltonian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' However, because this transition is nominally forbidden, interpreting the line intensities is a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Electric dipole (E1) transitions involving ∆ℓ ̸= 0 are forbidden in the Condon approximation, which separates electronic and vibrational degrees of freedom [42, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' These nominally forbidden vibronic transitions have been observed spectroscopically in many species of linear triatomic molecules, including NCO [76], NCS [77], MgNC [78], CaOH [39, 79, 80], SrOH [36, 73, 81], and YbOH [51], though modeling of the intensities is less common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' These transitions borrow intensity from E1-allowed bands through a combination of vibronic and spin-orbit per- 12 turbations [5, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Branching ratios involving forbidden vibronic transitions in YbOH were measured in a previous study [50] examining dispersed fluorescence from the ˜A(000) state, with resolution at the 10−5 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The exper- imentally observed vibrational branching was in good agreement with a theoretical study published in the same work [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' While these transitions are of interest as leakage channels for photon cycling, they can also be a resource for spectroscopy, as we show in the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The observed spectrum exhibits anomalous rotational line intensities, with certain transitions completely miss- ing at our level of sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For example, despite their expected thermal occupation (N ′′ ≤ 3), the P Q+ 12(1), P P + 11(2), QQ+ 11(2), P P − 11(3), QP − 11(3), and QR− 12(3) lines are missing (see Supplemental Material for a full list of lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Anomalous line intensities for forbidden transitions have been previously observed in other molecules with vibronic mixing [39, 73, 78, 80, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' By considering the intensity-borrowing that gives transition strength to these forbidden transitions, we develop a model that qualitatively explains the observed line strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In an E1 transition, the transition strength is proportional to the square of the transition dipole moment between the ground and excited state, |⟨ ˜A|T 1 p (d)| ˜X⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We are using spherical tensor notation, where p denotes the component of the spherical tensor in the lab-frame and q in the molecule-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Using a Wigner D matrix, we can write the lab frame dipole moment in terms of its molecule frame projections: T 1 p (d) = � q D(1) p,q(ω)∗T 1 q (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In the E1 approximation, ∆Σ = 0, and the molecule-frame projection q of the transition dipole moment determines the selection rule for Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The perpendicular q = ±1 components drive ∆Λ = ±1 transitions, for example the allowed ˜A − ˜X band, while parallel q = 0 component drives ∆Λ = 0, for example the allowed ˜B − ˜X band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In the limit of very large vibronic interaction, Λ and ℓ are fully mixed, and one might consider the ˜X(010) → ˜A(000) transition as a vibronic 2Π − 2Π parallel band, with ∆K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In reality, the vibronic mixing is perturbative in the ground and excited states, and Λ and ℓ are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' As a result, the observed line intensities are completely inconsistent with a solely parallel transition model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Instead, we model the ˜X(010) → ˜A(000) transition as a mixture of perpendicular and parallel bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We consider the effects of vibronic perturbations with the selection rule ∆ℓ = ±1, which can result in intensity borrowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' At first order, we have the dipolar Renner-Teller (RT) Hamiltonian, also referred to as Herzberg-Teller coupling [43, 55, 76], HRT = V11 2 � L+q−ei(θ−φ) + L−q+e−i(θ−φ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (6) This interaction is a form of linear vibronic coupling [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Here, V11 parameterizes the interaction strength, θ is the electronic azimuthal coordinate, φ is the bending azimuthal coordinate as before, L± is a raising/lowering operator with ∆Λ = ±1, and q± is a dimensionless raising/lowering operator with ∆ℓ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Physically, this interaction can be interpreted as the electrostatic interaction between the displaced bending dipole moment and the electron cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The interaction preserves the composite projection number K = Λ + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' At second order, the dipolar RT Hamiltonian can combine with the perpendicular spin-orbit Hamiltonian, HSO = A⊥ 2 (L+S− + L−S+) , (7) Where L± is defined as before, A⊥ is the off-diagonal spin-orbit coupling, and S± is the raising/lowering operator with ∆Σ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The combination of H(1) RT × H⊥ SO is an effective interaction with terms q±S∓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This interaction has ∆K = −∆Σ = ±1, but preserves the total angular momentum projection number P = Λ + Σ + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Denote the unperturbed excited state as | ˜A2Π1/2(000)⟩0 and the true, perturbed eigenstate as | ˜A2Π1/2(000)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We can then expand the perturbed eigenstate in terms of dominant ℓ = 1 vibronic contributions [5, 50]: | ˜A2Π1/2(000)⟩ ∝ | ˜A2Π1/2(000)⟩0 + cµ|µ2Σ(+) 1/2(010)⟩0 + cκ|κ2Σ(−) 1/2(010)⟩0 + cB| ˜B2Π(010)⟩0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (8) The perturbative coefficients cµ, cκ, cB represent the relative admixture of the intensity-borrowing states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The relevant states and perturbations are shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The µ2Σ(+) 1/2 state is the P = 1/2 component of the Ω = 1/2, v2 = 1, ˜A manifold, and the κ2Σ(−) 1/2 state is the P = 1/2 component in the Ω = 3/2, v2 = 1, ˜A manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' These two states are connected to ˜A2Π1/2(000) by the second-order perturbation HRT × HSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The ˜B2Π vibronic state 13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Level schematic for relevant states and perturbations in YbOH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Levels are labeled by their vibronic term symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We detect the ˜ X(010) bending state (which is a vibronic 2Π state) by laser excitation (orange line) up to the ˜A2Π1/2(000) state and observe the fluorescence from decays to the ground ˜ X(000) state (yellow wavy line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This excitation is a forbidden E1 transition, however, it acquires intensity by mixing of the excited ˜A2Π1/2(000) state with other |ℓ| = 1 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Mixing with ˜B(010) occurs via first-order (blue) Renner-Teller (RT) interactions, and mixing with the µ, κ(010) states occurs via second-order (purple) cross terms between RT and spin-orbit (SO) (red) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Not shown for simplicity are similar SO interactions between ˜A2Π1/2(000) and ˜B(000) and similar RT interactions between µ, κ(010) and ˜B(000), which also contribute to state mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' is the v2 = 1 component of the ˜B2Σ+ 1/2 electronic state, and is connected to ˜A2Π1/2(000) state via the first-order perturbation HRT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Each of these perturbing states contribute to different molecule-frame components of the transition dipole moment (TDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For example, the transition ˜X2Π → ˜B2Π is generated by the q = 0, z component of the TDM, with ∆K = ∆P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The other transitions to µ and κ have Π → Σ vibronic character, and couple via the q = ±1, x, y TDM components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The perturbing µ and κ states have opposite spin orientation compared to the original ˜A2Π1/2 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This means the intensity-borrowing states have mixed spin projection Σ, and the ∆Σ = 0 selection rule is not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The transition was modeled by first diagonalizing the ˜A2Π1/2(000) and ˜X2Σ(010) states separately to obtain the level positions of both states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' To evaluate the TDM, the excited state vector is then replaced by a linear combination of the intensity-borrowing state vectors with coefficients cµ, cκ, cB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The change of basis from ˜A to µ, κ, and ˜B uses appropriate selection rules for vibronic mixing and preserves parity (see supplementary materials for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The total TDM is the sum over the individual TDMs evaluated between ˜X(010) and the intensity-borrowing states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' To obtain the transition intensity, the TDM is squared after the sum, allowing TDMs from different states to interfere with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This interference is the source of the anomalous line intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The mixing coefficients, cµ, cκ, cB could not be modeled with a deperturbation Hamiltonian, since neither the µ, κ, or ˜B state have been extensively studied or modeled, and both states are expected to be affected by perturbations from nearby states with 4f 136s2 Yb character [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Instead, the mixing coefficients are kept as free parameters and their ratios were fit to the experimentally observed, relative field-free intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For the intensity fits, the rotational temperature is fixed at T = 2 K (the molecule beam is cooled by expansion out of the cell aperture), and since only relative intensities were fit, the cB parameter is held fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The normalized best fit mixing coefficients are found to be (cµ, cκ, cB) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='28, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='49, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This implies ∼69% of the ℓ = 1 character in ˜A2Π1/2(000) arises from mixing with ˜B(010), ∼24% from κ(010), and ∼7% from µ(010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This is in good agreement with recent theory work on intensity borrowing in YbOH, which attributed 70% of the intensity borrowing to mixing with ˜B(010) [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' However, it is important to note that due to interference effects, relative amplitudes of the coefficients, not their squares, are important for determining rotational line intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 14 We find that using these parameters to model the transition provides good qualitative understanding of the observed spectrum, as evidenced by the theory and experiment comparison in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Further studies of the excited state perturbations would be required to improve the fit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' however, as the exact intensities are not critical for future experiments with this molecule, this model is sufficient to provide physical understanding of the intensities and behavior of this transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' CONCLUSION In this work, we performed high-resolution optical spectroscopy on the rovibrationally forbidden ˜X2Σ+(010) → ˜A2Π1/2(000) transition of 174YbOH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In total, we observed 39 transitions out of low rotational states with N ′′ ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The ˜X(010) structure is well-described by a Hund’s case (b) 2Π effective Hamiltonian, and the ℓ-type parity doubling is described by two constants, qℓ = −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0(2) MHz and pℓ = −11(1) MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We modeled the anomalous line intensities of the forbidden band with mixing coefficients representing vibronic perturbations in the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The anomalous intensities arise from quantum interference between TDMs from the perturbing ˜B(010), µ(010), and κ(010) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' From the Zeeman spectra, we found the magnetic tuning of ˜X(010) is consistent with an effective isotropic electron spin g-factor, gS = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='07(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' From the Stark spectra, we extracted the molecule-frame dipole moment of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='16(1) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' These values are in good agreement with the parameters of the ˜X(000) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In our study, the hyperfine structure and higher-order Zeeman g-factors were unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Our work provides a basis for future studies with narrow-linewidth methods, such as RF, microwave, and two-photon spectroscopy, to precisely determine these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This work is an essential step towards measurements of CP-violating physics in YbOH [12], as well as other metal hydroxide molecules proposed for CP violation and parity violation searches that utilize the parity doublets in the bending mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We showed the ˜X(010) state ℓ-doubling offers spectroscopically resolvable states of molecule polarization pointing along, against, and perpendicular to the applied electric field, over a wide range of field values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This orientation control over the dipole moment offers robust systematic error rejection without compromising laser cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The combination of these features make linear polyatomics a promising platform for new physics searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' With our measured data, we can compute the EDM sensitivity, which is proportional to the electron spin projection on the internuclear axis, Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We find a local maximum value of ⟨Σ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='40 in the N = 1, J = 1 2 + state at E = 101 V/cm, similar to what was predicted in prior theoretical work [35, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Furthermore, understanding the structure of 174YbOH is a step toward characterizing the more complicated structure of the odd isotopologues 171YbOH and 173YbOH, which have sensitivity to parity violation [38] and hadronic CP violation [29], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Lastly, our determination of the ˜X(010) location and structure is crucial for understanding the complicated excited state structure in YbOH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For example, with our knowledge of the bending frequency, we can tentatively assign the unknown [17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='33] band in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [51] to the ˜X2Σ+(010) → ˜A2Π1/2(010) band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This would put the excited ˜A2Π1/2(010) manifold at approximately 17652 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This state is an excellent candidate for optically pumping population from ˜X(000) into ˜X(010), an important step for signal-to-noise-ratio improvements in precision measurements using the bending mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Furthermore, the location of ˜X(010) is necessary for the determination of repumping pathways for laser cooling, slowing, and trapping of YbOH, toward next-generation CP violation searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge many helpful discussions with the PolyEDM collaboration and the Doyle group at Harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We thank Tim Steimle, Phelan Yu, and Ashay Patel for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This work was supported by a NIST Precision Measurement Grant (60NANB18D253), an NSF CAREER Award (PHY-1847550), the Heising-Simons Foundation (2022-3361), the Gordon and Betty Moore Foundation (GBMF7947), and the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Sloan Foundation (G-2019- 12502).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' YT was supported by the Masason Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 15 [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Doyle, Radiation pressure force from optical cycling on a polyatomic molecule, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 134002 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [8] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Kozyryev, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Baum, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Matsuda, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Augenbraun, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Anderegg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Sedlack, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} 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+page_content=' thesis, Harvard University (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [84] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Mulliken and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Christy, Λ-Type Doubling and Electron Configurations in Diatomic Molecules, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 38, 87 (1931).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [85] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Brown and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Merer, Lambda-type doubling parameters for molecules in Π electronic states of triplet and higher multiplicity, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Spectrosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 74, 488 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [86] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Di Lauro and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Mills, Coriolis interactions about XY axes in symmetric tops, Journal of Molecular Spectroscopy 21, 386 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [87] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Brown, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Kopp, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Malmberg, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Rydh, An analysis of hyperfine interactions in the electronic spectrum of AlF, Physica scripta 17, 55 (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [88] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Brown and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Howard, An approach to the anomalous commutation relations of rotational angular momenta in molecules, Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 31, 1517 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 19 Supplementary Materials: Characterizing the Fundamental Bending Vibration of a Linear Polyatomic Molecule for Symmetry Violation Searches I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' TARGET COMPOSITION The data were obtained from targets of pressed Yb(OH)3 powder in a stoichiometric mixture with Yb powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The powders were mixed to have a 1:1 ratio of Yb and OH, ground using a mortar and pestle, passed through a 230 mesh sieve, and mixed with 4% PEG8000 binder by weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The powders were pressed in an 8 mm diameter die at a pressure of ∼1 GPa for ∼ 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For some targets, 10-30% water by mass was added to the powder before pressing, and while pressing the die was heated to ∼150◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This was found to improve target density and ablation yield consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' PHASE CONVENTIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Λ-Doubling There is an accepted convention for Λ-doubling, which was laid out by Mulliken and Christy [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The convention is reiterated by Brown in [85] and Brown and Carrington in [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This convention is given by ⟨Λ = ±1|e±2iφe|Λ′ = ∓1⟩ = −1 × δΛ,Λ′±2 (S1) Here, e±iφe is a raising/lowering operator with φe the azimuthal angle of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In this convention, a positive qe electronic Λ-doubling parameter in a 1Π state corresponds to the (−1)J parity level lying above the (−1)J+1 parity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In other words, the + parity state is below the − parity state for J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In the YbOH ˜A state, pe + 2qe is negative, and the − parity state is below the + parity state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This phase choice also manifests in the signs of the Λ-doubling Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' When written in Hund’s case (a), the J±S± terms have a positive prefactor, and the J±J± terms have a negative prefactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For this work, we drop the J±J± term in ˜A as its contribution is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Now we derive the Λ phase convention, following arguments from [55] and [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We begin with the definition of the Lz angular momentum operator in the molecule frame: Lz|Λ⟩ = −i ∂ ∂φe |Λ⟩ = Λ|Λ⟩ (S2) This means |Λ⟩ ∝ eiΛφe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Since L is not well defined, we expand |Λ⟩ in terms of spherical harmonics: |Λ⟩ = � L FLYLΛ(θe, φe) = � L FL √ 2π eiΛφeΘLΛ(θe) (S3) Here, � L |FL|2 = 1, and ΘLΛ(θe) is proportional to the associated Legendre functions P Λ L (cos θe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Θl,m(θ) = (−1)m � 2l + 1 2 (l − m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (l + m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='P m l (cos θ) for m ≥ 0 = (−1)mΘl,−m(θ) for m < 0 (S4) Note the function ΘLΛ satisfies ΘL,−|Λ| = (−1)ΛΘL,|Λ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This is the origin of this specific phase-convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Now we can evaluate the left hand side of eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S1 20 ⟨Λ|e±2iφe|Λ′⟩ = � sin θedθedφe � L,L′ F ∗ LFL′YLΛ(θe, φe)∗e±2iφeYL′Λ′(θe, φe) = � L,L′ F ∗ LFL′δΛ,Λ′±2 � sin θedθe(−1)Λ′±2ΘL,−Λ′∓2(θe)ΘL′,Λ′(θe) (S5) Where we substitute YL,Λ(θe, φe)∗ = (−1)ΛYL,−Λ(θe, φe) and performed the φe integral taking advantage of the orthogonality of exponential functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Now we simplify the integrand by noting we are interested in Λ = ±1, Λ′ = ∓1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This allows us to write −Λ′∓2 = Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Then the remaining θe integral can be performed by using the orthogonality relations of the associated Legendre polynomials, which results in ⟨Λ = ±1|e±2iφe|Λ′ = ∓1⟩ = δΛ,Λ′±2 � L |FL|2(−1)Λ′ = −1 × δΛ,Λ′±2 (S6) Where we have substituted |Λ| = 1 in the last line and used the fact that |FL|2 is normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We also note that the behavior of YLΛ upon the transformation Λ → −Λ gives the parity properties of |Λ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The action of space-fixed inversion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' the parity operator P, is equivalent to a reflection σxz of the xz plane of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This can be derived by considering the effect of space-fixed inversion on the Euler angles relating the molecule and lab frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Therefore we have: PYL,Λ(θe, φe) = σxzYL,Λ(θe, φe) = YL,Λ(θe, 2π − φe) = YL,Λ(θe, φe)∗ = (−1)ΛYL,−Λ(θe, φe) (S7) This recovers the result P|Λ⟩ = (−1)Λ| − Λ⟩ (note a Σ− state has an extra factor of (−1) that we do not consider).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For the full parity of the rotational wavefunction, the action of P must also be computed on the spin and rotational wavefunctions, which also reverse the projection quantum numbers and contribute parity phases of (−1)S−Σ and (−1)J−Ω respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The combination of all phase factors gives the complete case (a) parity phase without bending motion: (−1)Λ+S−Σ+J−Ω = (−1)J−S, where we have used |Σ| = S and Ω = Λ + Σ to simplify the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' ℓ-Doubling For the derivation of the parity phase and matrix elements involving ℓ, we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [55], which uses the vibrational phase conventions established by by Di Lauro and Mills [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The wavefunction for an isotropic 2D harmonic oscillator may be written as |v2, ℓ⟩ = 1 √ 2π eiℓφnΨv2,ℓ(q) (S8) Here, q = � q2 1 + q2 2, where (q1, q2) are the dimensionless, doubly-degenerate normal coordinates of the bending mode, and φn = tan−1(q2/q1) is the azimuthal angle of the bending nuclear framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The function Ψv2,ℓ is given by [86]: Ψv,ℓ(q) = (−1)(v+|ℓ|)/2Nv,ℓq|ℓ|e−q2/2L|ℓ| (v+|ℓ|)/2(q2) (S9) Here, Nv,ℓ is a normalization factor and Lk n(x) is an associated Laguerre polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This function explicitly satisfies Ψv2,|ℓ| = Ψv2,−|ℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 21 We now consider the matrix elements between ℓ = ±1 states: ⟨ℓ|e±2iφn|ℓ′⟩ = � dqdφ 1 2π e−iℓφnΨv,ℓ(q)e±2iφneiℓ′φnΨv,ℓ′(q) (S10) The integration bounds are taken for q ≥ 0 and 2π > q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The φn integral is evaluated with the orthogonality of complex exponential functions and enforces δℓ,ℓ′+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Restricting our attention to ℓ = ±1 states, the Ψv,ℓ(q) functions depend only on |ℓ|, and do not add an additional phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' As a result we can evaluate the remaining dq integral using the orthogonality relations of the associated Laguerre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We are left with ⟨ℓ|e±2iφn|ℓ′⟩ = 1 × δℓ,ℓ′±2 (S11) The difference between parity phase factors for ℓ and Λ can be traced to the difference in phase between Ψvℓ(q) and ΘLΛ(θe) upon space-fixed inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' By considering the behavior of the wavefunctions under φn → 2π − φn, we see the radial q part is unaffected, giving us P|v2, ℓ⟩ = |v2, −ℓ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' When combined with rotational and spin parity phase factors, we then obtain the complete parity phase (−1)J−S−ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' EFFECTIVE HAMILTONIANS AND MATRIX ELEMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' ˜A2Π1/2(000) For the ˜A2Π1/2(000) state, we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [45], which uses the R2 rotational Hamiltonian formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Details can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [54], Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The effective Hamiltonian in Hund’s case (a) and in spherical tensor notation is given by H ˜ A = T0 + AT 1 q=0(L)T 1 q=0(S) + B(J − L − S)2 − D(J − L − S)4 + (pe + 2qe) � q=±1 e−2iqθT 2 2q(J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S) + 1 2(peD + 2qeD) � q=±1 [N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' e−2iqθT 2 2q(J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S)]+ (S12) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' T0 is the state origin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' A is the spin-orbit constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' B is the rotational constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' D is the centrifugal distortion term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' pe + 2qe represents electronic Λ-doubling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' peD + 2qeD is the centrifugal distortion correction to Λ-doubling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' ·]+ is the anti-commutator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J±S± are defined in the molecule frame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' and θ is the azimuthal angle of the electronic wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Matrix elements of this Hamiltonian can be found in [54, 55, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Note the L2 x + L2 y terms that arise in the R2 formalism are absorbed in the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We use the constants determined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' To be explicit, using the phase convention from supplementary section II we reproduce our matrix element for the Λ-doubling term below: ⟨Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J, Ω, M|e2iqθT 2 2q(J, S)|Λ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S, Σ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J′, Ω′, M ′⟩ = δJ,J′δM,M ′δΛ+2q,Λ′ × (−1)J−Ω � J 1 J −Ω −q Ω′ � � J(J + 1)(2J + 1) × (−1)S−Σ � S 1 S −Σ q Σ′ � � S(S + 1)(2S + 1) (S13) 22 S R N J G ℓ n FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' A schematic of the coupling scheme in Hund’s case (b), used to describe the ˜ X(010) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The vibrational angular momentum G is projected onto the internuclear axis to form ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The molecule rotation R is coupled to ℓ to form N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Finally the spin-rotation interaction couples S and N to form J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Coupling to the H nuclear spin is not pictured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' ˜ X2Σ+(010) We reproduce the ˜X2Σ+(010) Hamiltonian below in spherical tensor notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' H ˜ X = T0 + B(N 2 − ℓ2) + γ � N · S − T 1 q=0(N)T 1 q=0(S) � + γGT 1 q=0(N)T 1 q=0(S) + � q=±1 e−2iqφ � pGT 2 2q(N, S) − qGT 2 2q(N, N) � (S14) The bending mode energy levels are well represented by Hund’s case (b) eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' A pictorial representation of the coupling scheme is given in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' As mentioned in the main text, the spin rotation interaction is modified to account for the bending motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Here we provide further explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In the effective Hamiltonian approach, the spin-rotation parameter receives contributions from various orders of perturbation theory, γ = γ(1)+γ(2)+· · · [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The first order term γ(1) results from the magnetic interaction between the electron spin and the magnetic dipole moment of the rotating molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' In heavy molecules, the first order term is small compared to the dominant second order contribution γ(2), arising from off-diagonal spin- orbit and rotational perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For linear molecules with Nz = 0, the spin-rotation term N · S implicitly only contains contributions from NxSx and NySy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' However for a bending molecule, since Nz ̸= 0, we explicitly subtract away NzSz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Matrix elements for the N 2 and N · S terms can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Here we reproduce matrix elements for the terms specific to the bending mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' ⟨ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N, S, J, M|T 1 q=0(N)T 1 q=0(S)|ℓ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N ′, S, J′, M ′⟩ = δJ,J′δN,N ′δM,M ′δℓ,ℓ′ × ℓ × (−1)J+N ′+S � N S J S N 1 � × (−1)N−ℓ � N 1 N −ℓ 0 ℓ � (2N + 1) × � S(S + 1)(2S + 1) (S15) 23 ⟨ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N, S, J, M|T 2 2q(N, S)e−2iqφ|ℓ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N ′, S, J′, M ′⟩ = δJ,J′δN,N ′δM,M ′δℓ,ℓ′+2q × (−1)J+N+S � 5 2 � N S J S N 1 � × � S(S + 1)(2S + 1) × √ 3 � 2 1 1 N N N � � N(N + 1)(2N + 1) × (−1)N−ℓ � N 2 N −ℓ 2q ℓ � (2N + 1) × � S(S + 1)(2S + 1) (S16) ⟨ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N, S, J, M|T 2 2q(N, N)e−2iqφ|ℓ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N ′, S, J′, M ′⟩ = δJ,J′δN,N ′δM,M ′δℓ,ℓ′+2q × (−1)J+N+S � N J S J N 0 � × √ 5 � 2 2 0 N N N � × 1 2 √ 6 � (2N − 1)(2N)(2N + 1)(2N + 2)(2N + 3) × (−1)N−ℓ � N 2 N −ℓ 2q ℓ � (2N + 1) (S17) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Stark and Zeeman Matrix Elements For ˜X(010), the Stark and Zeeman matrix elements are given in Hund’s case (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For the Stark matrix element, we only consider the contribution from the dipole component along the molecular z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' ⟨ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N, S, J, M|T 1 p (d)|ℓ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N ′, S, J′, M ′⟩ = (−1)J−M � J 1 J′ −M p −M � × (−1)J′+N+S+1� (2J + 1)(2J′ + 1) � N ′ J′ S J N 1 � × (−1)N−ℓ� (2N + 1)(2N ′ + 1) � N 1 N ′ −ℓ 0 ℓ′ � (S18) ⟨ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N, S, J, M|T 1 p (S)|ℓ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' N ′, S, J′, M ′⟩ = δℓ,ℓ′(−1)J−M � J 1 J′ −M p −M � × (−1)J+N+S+1� (2J + 1)(2J′ + 1) � S J′ N J S 1 � × � S(S + 1)(2S + 1) (S19) 24 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' INTENSITY BORROWING AND TRANSITION DIPOLE MOMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Hund’s Case (b) to Case (a) Change of Basis The eigenstates of ˜X(010) are best described by Hund’s case (b) wavefunctions, while the eigenstates of ˜A(000) are described by Hund’s case (a) wavefunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' To calculate transitions, we convert between the two cases using the following formula from Brown [88]: |N, K, S, J, M⟩ = � Σ,P (−1)N−S+P √ 2N + 1 � J S N P −Σ −K � |S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J, P, M⟩ (S20) Here, P = Λ + Σ + ℓ, and K = Λ + ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Note this form is equivalent to that given by Hirota in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Transition Dipole Moment The transition dipole moment (TDM) matrix element is evaluated in Hund’s case (a): ⟨ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S, Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J, P, M|T 1 p (d)|ℓ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Λ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' S, Σ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' J′, P ′, M ′⟩ = δΣ,Σ′δℓ,ℓ′ × (−1)J−M � J 1 J′ −M p M ′ � × � (2J + 1)(2J′ + 1)(−1)J−M × � q � J 1 J′ −P q P ′ � δΛ,Λ′+q × ⟨Λ||T 1 q (d)||Λ′⟩ (S21) The last term is the reduced matrix element encoding the transition dipole integral between two electronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The ∆ℓ = 0 selection rule is explicit in the above matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This means we can only drive ˜X(010) to admixtures in ˜A(000) with |ℓ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Mixing with |ℓ| = 1 states To model the transition intensities, as stated in the main text, we first separately diagonalize the ˜A2Π1/2(000) and ˜X(010) Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We then convert the ˜X(010) eigenvectors from Hund’s case (b) to case (a), using equation S20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Since we use effective Hamiltonians, the ˜A2Π1/2(000) eigenvectors have ℓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' However, in reality these eigenvectors are perturbed by other states, and contain admixtures with |ℓ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' These admixed states provide the transition intensity and non-zero transition dipole moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' To represent the admixed states, we perform a change of basis to transform the ˜A2Π1/2(000) effective Hamiltonian eigenvectors into eigenvectors of the admixed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' As states in the main text, the states of interest with ℓ = 1 are ˜Aµ2Σ(+) 1/2(010), ˜Aκ2Σ(−) 1/2(010), and ˜B2Π(010), where we are using vibronic term symbols 2S+1KP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Each eigenvector of ˜A(000) is transformed into a linear combination of eigenvectors from the admixed states, with amplitudes cµ, cκ, cB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The mixing between ˜A2Π1/2(000) and ˜B2Π(010) occurs at first order due to HRT (see main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Since this interaction preserves K and P, it simply exchanges one quanta between ℓ and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Since ˜A2Π1/2(000) has P = 1/2, we only consider mixing other P = 1/2 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' We perform the following change of basis: 25 ⟨ ˜B(010), Λ = 0, ℓ, Σ, P | ˜A(000), Λ′, ℓ′ = 0, Σ′, P ′ = ±1/2⟩ = δℓ,Λ′δP,P ′δΣ,Σ′(−1)P −1/2 (S22) Note the phase factor (−1)P −1/2 is explicitly included to preserve parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This accounts for the extra (−1)ℓ phase factor in the parity of an ℓ ̸= 0 state compared to an ℓ = 0 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' This factor can arise naturally if HRT is written as ∝ sin (θ − φ) instead of being ∝ cos (θ − φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' While the latter form is most often found in the literature [43, 55], the former can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' [82] in the context of Σ− states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The admixture of the µ and κ states occurs via a second-order combination of HRT and HSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' These interactions preserve P but can change K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' For µ(010) we obtain the following change of basis: ⟨µ(010), Λ, ℓ, Σ, P | ˜A(000), Λ′, ℓ′ = 0, Σ′, P ′ = ±1/2⟩ = δΛ,−Λ′δℓ,Λ′δΣ,−Σ′(−1)P −1/2 (S23) And for κ(010): ⟨κ(010), Λ, ℓ, Σ, P| ˜A(000), Λ′, ℓ′ = 0, Σ′, P ′ = ±1/2⟩ = δΛ,Λ′δℓ,−Λ′δΣ,−Σ′(−1)P −1/2 (S24) After changing basis to states with |ℓ| = 1, we compute the transition dipole matrix element using equation S21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The transition amplitudes for the different state admixtures are added together, and the resulting interference depends on the mixing coefficients cµ, cκ, cB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Finally, to obtain relative intensities, we square the total transition amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' ASSIGNED LINES See Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Line notation is described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' 26 TABLE S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Observed lines, ground states quantum numbers (N ′′, J′′, P′′), excited states quantum numbers (J′, P′), observed positions, and residuals of ˜ X2Σ+(010) → ˜A2Π1/2(000) band of YbOH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' There are in total 38 lines assigned to 39 transitions as the QR− 12(1) and P Q− 12(5) lines are overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' The fit residual is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='1 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' Line N ′′, J′′, P′′ J′, P′ Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (cm−1) Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' - Calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content=' (MHz) OP + 12 2, 3/2, + 1/2, − 17002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4883 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4 3, 5/2, + 3/2, − 17002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4312 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4 4, 7/2, + 5/2, − 17000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='6512 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='7 OP − 12 2, 3/2, − 1/2, + 17002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='9232 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='1 3, 5/2, − 3/2, + 17001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='5614 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='9 P P + 11 1, 3/2, + 1/2, − 17003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4683 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='2 3, 7/2, + 5/2, − 17002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='6114 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='7 5, 11/2, + 9/2, − 17001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='8212 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='2 P P − 11 1, 3/2, − 1/2, + 17003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='9070 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='2 2, 5/2, − 3/2, + 17003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0314 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='6 4, 9/2, − 7/2, + 17002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='2076 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='8 P Q+ 12 2, 3/2, + 3/2, − 17003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='9039 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='8 3, 5/2, + 5/2, − 17002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='6012 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='8 5, 9/2, + 9/2, − 17001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='8046 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='7 P Q− 12 1, 1/2, − 1/2, + 17003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='9053 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='8 2, 3/2, − 3/2, + 17003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0250 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0 3, 5/2, − 5/2, + 17003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='9208 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3 5, 9/2, − 9/2, + 17004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0076 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3 QQ+ 11 1, 3/2, + 3/2, − 17004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='8846 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3 3, 7/2, + 7/2, − 17005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='9150 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0 5, 11/2, + 11/2, − 17007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0123 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='5 QQ− 11 1, 3/2, − 3/2, + 17004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0091 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='5 2, 5/2, − 5/2, + 17005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3917 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='7 4, 9/2, − 9/2, + 17006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4556 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='6 QR+ 12 1, 1/2, + 3/2, − 17004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='8824 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3 2, 3/2, + 5/2, − 17004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0743 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='1 3, 5/2, + 7/2, − 17005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='9052 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='8 QR− 12 1, 1/2, − 3/2, + 17004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0076 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3 2, 3/2, − 5/2, + 17005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='9 4, 7/2, − 9/2, + 17006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4421 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='9 RR+ 11 1, 3/2, + 5/2, − 17005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='0543 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='5 2, 5/2, + 7/2, − 17007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3837 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='7 3, 7/2, + 9/2, − 17006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='2215 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3 4, 9/2, + 11/2, − 17009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4646 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='9 RR− 11 1, 3/2, − 5/2, + 17006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='3695 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='7 2, 5/2, − 7/2, + 17005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='6298 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='1 3, 7/2, − 9/2, + 17008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='4157 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} +page_content='7 4, 9/2, − 11/2, + 17006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfywiY/content/2301.04124v1.pdf'} 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0000000000000000000000000000000000000000..86c70437b919f9559dcbdd1b2024258cdb1c194d --- /dev/null +++ b/pingpong/content/tmp_files/load_file.txt @@ -0,0 +1,69 @@ +filepath=D:\projects\langchain-ChatGLM-master\knowledge_base\pingpong\content\之江实验室2023年度乒乓球团体赛方案.pdf,len=68 +page_content='2023年之江实验室乒乓球团体赛活动方案 之江乒乓，健健康康。为迎接祖国亚运会的召开，乒乓球协会围 绕“立足之江、增强体质、团结拼搏”宗旨，活跃科研人员业余文化生 活，拟于 9 月组织之江实验室 2023 年度乒乓球团体赛。具体活动方 案如下： 一、比赛主题 迎亚运，乒乓激荡在之江。 二、比赛目的 促进各部门各中心间交流，丰富室友业余生活，培育合作精神和 拼搏精神，提升室友荣誉感、归属感和责任感。 三、策划组织 本次由之江实验室工会举办，之江实验室乒乓球社团具体策划组 织实施。 四、参加人员 之江实验室全体乒乓球爱好者。 五、时间地点 时间：2023 年 9 月 19 日和 9 月 21 日（一周内比完） 地点：之江实验室南湖总部主楼 A 层乒乓球室 六、活动内容 （一）比赛项目 实验室乒乓球团队赛，出场顺序为男双、混双、男单、女单、男 单；男女运动员均不得兼项（每名队员在每轮比赛中仅限出场一次）。 比赛采用 5 场 3 胜制， （每局 11 分制，10 平后先得二分者比赛结束）。 小组赛每场 3 局 2 胜制，淘汰赛每场 5 局 3 胜制。小组赛 5 场打满， 淘汰赛先胜 3 场即结束。 （二）参赛要求 1、参赛运动员须为实验室在职人员，' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='各自部门（研究院）项目 聘用和实习生也可报名参加。 2、比赛报名总人数 9 人，其中男选手不少于 5 名，女选手不少 于 2 名（报名表见附件 1）。 （三）比赛规则 1.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='比赛分小组赛和淘汰赛，共 10 支队伍，分组情况详见附件 2， 无需再自行组队。 2.' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='小组赛阶段 10 个队分为两组进行循环赛，每组取前 4 名进入 淘汰赛；淘汰赛采用双方 4 支队伍直接交叉比赛的方式决出前 4（如 A1-B4,A2-B3），负者争夺 5-8 名。 之江实验室工会 2023 年 8 月 21 日 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='附件 1：报名表 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='参赛球队报名表 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='球队名称（如职能队）： ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='序号 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='姓名 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='性别 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='部门（中心） ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='人员性质 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='1 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='2 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='3 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='4 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='5 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='6 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='7 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='8 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='9 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='10 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='附件 2：分组情况 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='乒乓球比赛分组 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='部门、研究中心、直属单位 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='队伍名称 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='报名联系人 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='所有职能部门 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='职能队 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='陈均杰（000557） ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='基础理论研究院+ ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='北京研究中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='基础理论&北研 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='中心队 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='郭浩（001255） ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='人工智能研究院 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='人工智能队 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='冯琳清（000974） ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智能网络研究院 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智能网络队 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='程小峰（002803） ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智能感知研究院 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智能感知队 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='庄逸洋（001276） ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智能计算研究院 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智能计算队 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='王军（000150） ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智能装备研究院 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智能装备队 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='安学良（002015） ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='之江实验室科技控股有限公司 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='之科控股队 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='董雯歌（ZJTH2057） ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='健康医疗大数据研究中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='科艺融合研究中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智能芯片与器件研究中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智能科技标准化研究中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智能社会治理研究中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='交叉创新一队 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='王昱(000143) ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='之江/燧原联合创新研究中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智能机器人研究中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='金融科技研究中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='杭钢/之江数字经济联合创新 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='研究中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='智慧交通研究中心 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='交叉创新二队 ' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} +page_content='王艺涵(000404)' metadata={'source': 'D:\\projects\\langchain-ChatGLM-master\\knowledge_base\\pingpong\\content\\之江实验室2023年度乒乓球团体赛方案.pdf'} diff --git "a/pingpong/content/tmp_files/\344\271\213\346\261\237\345\256\236\351\252\214\345\256\2442023\345\271\264\345\272\246\344\271\222\344\271\223\347\220\203\345\233\242\344\275\223\350\265\233\346\226\271\346\241\210.pdf.txt" "b/pingpong/content/tmp_files/\344\271\213\346\261\237\345\256\236\351\252\214\345\256\2442023\345\271\264\345\272\246\344\271\222\344\271\223\347\220\203\345\233\242\344\275\223\350\265\233\346\226\271\346\241\210.pdf.txt" new file mode 100644 index 0000000000000000000000000000000000000000..b5cdae7c297a9cfe14194128045eb3f3c5a6b910 --- /dev/null +++ "b/pingpong/content/tmp_files/\344\271\213\346\261\237\345\256\236\351\252\214\345\256\2442023\345\271\264\345\272\246\344\271\222\344\271\223\347\220\203\345\233\242\344\275\223\350\265\233\346\226\271\346\241\210.pdf.txt" @@ -0,0 +1,107 @@ +2023年之江实验室乒乓球团体赛活动方案 +之江乒乓，健健康康。为迎接祖国亚运会的召开，乒乓球协会围 +绕“立足之江、增强体质、团结拼搏”宗旨，活跃科研人员业余文化生 +活，拟于 9 月组织之江实验室 2023 年度乒乓球团体赛。具体活动方 +案如下： +一、比赛主题 +迎亚运，乒乓激荡在之江。 +二、比赛目的 +促进各部门各中心间交流，丰富室友业余生活，培育合作精神和 +拼搏精神，提升室友荣誉感、归属感和责任感。 +三、策划组织 +本次由之江实验室工会举办，之江实验室乒乓球社团具体策划组 +织实施。 +四、参加人员 +之江实验室全体乒乓球爱好者。 +五、时间地点 +时间：2023 年 9 月 19 日和 9 月 21 日（一周内比完） +地点：之江实验室南湖总部主楼 A 层乒乓球室 +六、活动内容 +（一）比赛项目 +实验室乒乓球团队赛，出场顺序为男双、混双、男单、女单、男 +单；男女运动员均不得兼项（每名队员在每轮比赛中仅限出场一次）。 + +比赛采用 5 场 3 胜制， +（每局 11 分制，10 平后先得二分者比赛结束）。 +小组赛每场 3 局 2 胜制，淘汰赛每场 5 局 3 胜制。小组赛 5 场打满， +淘汰赛先胜 3 场即结束。 +（二）参赛要求 +1、参赛运动员须为实验室在职人员，各自部门（研究院）项目 +聘用和实习生也可报名参加。 +2、比赛报名总人数 9 人，其中男选手不少于 5 名，女选手不少 +于 2 名（报名表见附件 1）。 +（三）比赛规则 +1.比赛分小组赛和淘汰赛，共 10 支队伍，分组情况详见附件 2， +无需再自行组队。 +2.小组赛阶段 10 个队分为两组进行循环赛，每组取前 4 名进入 +淘汰赛；淘汰赛采用双方 4 支队伍直接交叉比赛的方式决出前 4（如 +A1-B4,A2-B3），负者争夺 5-8 名。 +之江实验室工会 +2023 年 8 月 21 日 + +附件 1：报名表 +参赛球队报名表 +球队名称（如职能队）： +序号 +姓名 +性别 +部门（中心） +人员性质 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 + +附件 2：分组情况 +乒乓球比赛分组 +部门、研究中心、直属单位 +队伍名称 +报名联系人 +所有职能部门 +职能队 +陈均杰（000557） +基础理论研究院+ +北京研究中心 +基础理论&北研 +中心队 +郭浩（001255） +人工智能研究院 +人工智能队 +冯琳清（000974） +智能网络研究院 +智能网络队 +程小峰（002803） +智能感知研究院 +智能感知队 +庄逸洋（001276） +智能计算研究院 +智能计算队 +王军（000150） +智能装备研究院 +智能装备队 +安学良（002015） +之江实验室科技控股有限公司 +之科控股队 +董雯歌（ZJTH2057） +健康医疗大数据研究中心 +科艺融合研究中心 +智能芯片与器件研究中心 +智能科技标准化研究中心 +智能社会治理研究中心 +交叉创新一队 +王昱(000143) +之江/燧原联合创新研究中心 +智能机器人研究中心 +金融科技研究中心 +杭钢/之江数字经济联合创新 +研究中心 +智慧交通研究中心 +交叉创新二队 +王艺涵(000404) + diff --git a/ptE4T4oBgHgl3EQfvQ3k/content/tmp_files/2301.05241v1.pdf.txt b/ptE4T4oBgHgl3EQfvQ3k/content/tmp_files/2301.05241v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e1a46f0cd144461dea385bc8eaee3767713952b0 --- /dev/null +++ b/ptE4T4oBgHgl3EQfvQ3k/content/tmp_files/2301.05241v1.pdf.txt @@ -0,0 +1,1599 @@ +MNRAS 000, 1–14 (2022) +Preprint 16 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Debiasing Standard Siren Inference of the Hubble Constant with +Marginal Neural Ratio Estimation +Samuel Gagnon-Hartman,1,2,3★ John Ruan1, Daryl Haggard2 +1Department of Physics and Astronomy, Bishop’s University, 2600 College Street, Sherbrooke J1M 1Z7, Canada +2Department of Physics and McGill Space Institute, McGill University, Montreal, QC, Canada H3A 2T8 +3Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Gravitational wave (GW) standard sirens may resolve the Hubble tension, provided that stan- +dard siren inference of 𝐻0 is free from systematic biases. However, standard sirens from +binary neutron star (BNS) mergers suffer from two sources of systematic bias, one arising +from the anisotropy of GW emission, and the other from the anisotropy of electromagnetic +(EM) emission from the kilonova. For an observed sample of BNS mergers, the traditional +Bayesian approach to debiasing involves the direct computation of the detection likelihood. +This is infeasible for large samples of detected BNS merger due to the high dimensionality +of the parameter space governing merger detection. In this study, we bypass this computation +by fitting the Hubble constant to forward simulations of the observed GW and EM data under +a simulation-based inference (SBI) framework using marginal neural ratio estimation. A key +innovation of our method is the inclusion of BNS mergers which were only detected in GW, +which allows for estimation of the bias introduced by EM anisotropy. Our method corrects +for ∼90% of the bias in the inferred value of 𝐻0 when telescope follow-up observations of +BNS mergers have extensive tiling of the merger localization region, using known telescope +sensitivities and assuming a model of kilonova emission. Our SBI-based method thus enables +a debiased inference of the Hubble constant of BNS mergers, including both mergers with +detected EM counterparts and those without. +Key words: transients: neutron star mergers – gravitational waves – methods: data analysis – +cosmology: observations +1 +INTRODUCTION +The value of the Hubble constant, 𝐻0, is currently the subject of +dispute as a ∼5𝜎 tension exists between the latest late-time mea- +surement using the cosmic distance ladder by the SH0ES Team +(Riess et al. 2022) and the early-time value inferred from cosmic +microwave background (CMB) anisotropies by the Planck satellite +(Aghanim et al. 2020). Gravitational wave (GW) standard sirens +provide an independent way to measure 𝐻0, and thus have the po- +tential to resolve this dispute (Abbott et al. 2017b). +GW observations of binary neutron star (BNS) mergers pro- +vide an estimate of the luminosity distance (𝐷𝐿) of each merger +through modeling of its GW waveform. If the electromagnetic (EM) +emission from the kilonova counterparts of the mergers are also de- +tected, then the BNS mergers can be precisely localized, thus provid- +ing their cosmological redshifts through the host galaxy spectrum. +Given a sample of BNS mergers with known redshifts and luminos- +ity distances, 𝐻0 can be inferred. This approach is known as the stan- +dard siren method of inferring the Hubble constant (Schutz 1986; +★ samuel.gagnonhartman@sns.it +Holz & Hughes 2005). The binary neutron star merger GW170817 +was the first standard siren, providing a ∼10% measurement of +𝐻0 (Abbott et al. 2017a,c; Soares-Santos et al. 2017; Abbott et al. +2017b; Hotokezaka et al. 2019). Forecasts predict that a 2% mea- +surement of 𝐻0 can be achieved by combining a future sample of +∼50 standard sirens (e.g., Dalal et al. 2006; Nissanke et al. 2010, +2013; Chen et al. 2018; Feeney et al. 2019). +In order for a standard siren measurement to resolve the tension +in 𝐻0, it must be free of systematic biases. Here, we seek to address +two major sources of bias in standard sirens: GW anisotropy bias and +EM anisotropy bias. The bias from anisotropic GW emission has +been shown to inflate the value of 𝐻0 inferred from standard sirens +(Talbot & Thrane 2020), and a method to mitigate these biases +was presented by Gerardi et al. (2021) using a simulation-based +inference (SBI) approach. An additional source of bias is intro- +duced by observational selection effects owing to the anisotropic +EM emission from the kilonova, as detailed in Chen (2020), and +this additional bias was not addressed by Gerardi et al. (2021). In +both sources of bias, the anisotropy of BNS merger emission pro- +duces a selection effect where mergers consistent with a high value +of 𝐻0 are preferentially observed. +© 2022 The Authors +arXiv:2301.05241v1 [astro-ph.CO] 12 Jan 2023 + +2 +Gagnon-Hartman et al. +Correcting for bias introduced by a selection effect requires a +careful understanding of the selection criterion itself (Vitale et al. +2021). Gerardi et al. (2021) corrected for GW anisotropy bias +through modelling the dependence of successful GW detection on +both BNS merger parameters and cosmological parameters, using +general relativity (GR) and the GW detector’s configuration. GR +directly allows calculation of the strain and polarization breakdown +of a GW produced by a BNS merger from that merger’s measured +parameters. Making a determination of detection on this GW fur- +ther requires knowledge of the polarization sensitivity and strain +detection threshold of the GW detector. +We extend this logic to EM anisotropy bias, using simulation- +based inference (SBI) to characterize the EM selection criterion in +a sample of BNS mergers. For an observed sample of BNS mergers +detected in GWs, we expect to have both mergers with identified EM +counterparts and those without. Mergers without EM counterparts +in the sample allow us to infer the probability of EM detection for a +BNS merger, given its measured parameters and assuming a model +for the EM emission. This allows us to characterize the dependence +of EM selection on BNS and cosmological parameters, and thereby +correct for the EM anisotropy bias in standard siren measurements +of 𝐻0. The main innovation of our method is this inclusion of BNS +mergers detected in GWs without detected EM counterparts. +In the absence of an analytic likelihood that includes both +GW and EM selection effects, we intead opt for a SBI approach, +where a BNS merger forward simulator enables emulation of both +selection effects. SBI refers to a class of inference methods that +rely on surrogate likelihood functions from a simulator forward +model, enabling Bayesian inference even in situations where the +likelihood function is intractable, such as ours (Cranmer et al. 2020). +SBI methods typically function by simulating a data set from a +set of input parameters, and then producing a summary statistic +which describes the difference between the simulated data and the +observed data. Through repeated sampling of the parameter space, +these summary statistics are used to construct a surrogate likelihood +function, and thus enable posterior inference. +Here, we present an SBI-based method which corrects for +both GW and EM anisotropy biases in standard siren measure- +ments of 𝐻0. Our approach exploits our singular goal of inferring +the marginal posterior distribution of 𝐻0, allowing us to treat all +BNS merger parameters as nuisance parameters. Specifically, we +use marginal neural ratio estimation (MNRE), in which a summary +statistic is generated for each set of input parameters. The summary +statistic is a quantity which encapsulates the consistency of the pa- +rameters drawn with the observed data. These summary statistics +are then used as a training set for a neural ratio estimator network, +which learns the marginal posterior distribution for 𝐻0, our param- +eter of interest (Miller et al. 2021). +MNRE is a recently-developed SBI method which requires far +fewer samples than competing methods to produce an informative +posterior. It accomplishes this by estimating only marginal poste- +riors rather than joint probability distributions (Miller et al. 2021). +MNRE is thus appropriate in situations where only marginal dis- +tributions are of interest, and the computational expense of each +sample is high. MNRE has recently been applied by Karchev et al. +(2022) in the context of Type Ia supernova cosmology, where it was +used to marginalize over a large number of of supernova parame- +ters. Similarly, our study is only interested in the marginal posterior +distribution for 𝐻0, and thus MNRE is an ideal tool. +To test and validate our SBI-based approach on a mock data +set, we assume a set of true BNS mergers with associated GW +strains. Of these BNS mergers, only a subset have associated EM +detections. The expected kilonova light curves from the full merger +sample is repeatedly simulated with randomly-sampled kilonova +and cosmological parameter configurations, assuming a model for +the kilonova emission, and the summary statistics from each round +are used to train an MNRE. We demonstrate that by including GW- +only mergers in a sample of BNS mergers, we can correct for the +EM anisotropy bias latent within the sample. Our method provides +a GW anisotropy bias correction level comparable to that in Gerardi +et al. (2021), and further addresses EM anisotropy bias for standard +siren cosmology. +The structure of this paper is as follows. Section 2 explains the +physical cause of each source of bias. Section 3 discusses the mock +data sets used in this study and the simulator. Section 4 provides +a detailed discussion of our SBI-based method. Section 5 reviews +results from each validation test performed on the mock data sets. +We summarize and conclude in Section 6. +2 +SOURCES OF BIAS +2.1 +GW anisotropy and bias +GW detections of BNS mergers suffer from GW anisotropy bias, +which results in standard siren measurements overestimating 𝐻0 +(Malmquist 1922). This stems from the luminosity distance – in- +clination angle degeneracy of the GW strain produced from a +BNS merger, which skews the inferred luminosity distance of a +BNS merger in a way that is dependent on its inclination angle 𝜄 +(Wahlquist 1987). We illustrate this effect in Figure 1, where a se- +ries of BNS mergers evenly spaced in luminosity distance have their +inferred luminosity distances and associated uncertainties shown +for two possible inclination angles, one face-on and one oblique. +Uncertainty in luminosity distance estimates from GW detections +arises from the detector’s polarization sensitivity (Cutler & Flana- +gan 1994). In Figure 1, we assume the characteristics of the Ad- +vanced LIGO/Virgo experiment as expected in O4 (Yi et al. 2022). +Mergers placed at oblique inclinations have an inferred luminosity +distance greater than their true value, while the opposite is true of +mergers placed at face-on inclinations. This is because GW strain is +strongest along the angular momentum axis of a BNS merger, and +weakest perpendicular to this axis. A weak GW strain may be weak +either because the merger is in fact very distant and face-on, or be- +cause the merger is nearby but at an oblique inclination, thus giving +rise to the aforementioned degeneracy. An example of this lumi- +nosity distance-inclination angle degeneracy is shown in Figure 2, +which displays the overlaid 𝑃(𝐷𝐿, 𝜄) joint posterior distributions for +10 otherwise identical BNS mergers at different inclination angles. +This luminosity distance – inclination angle degeneracy would +be unproblematic in the absence of selection effects. However, the +amplitude of a GW strain curve is related to the probability that the +merger’s GW is detected at all. In order for a BNS merger to be +detected, its strain must exceed some critical signal-to-noise ratio. +Naturally, this is less likely for weak-strain mergers than for strong- +strain mergers. The nature of this relationship is shown in Figure 3, +where we show that face-on mergers are likely to be detected even +at large distances, while oblique mergers are unlikely to be detected +even at small distances. As a result, GWs from face-on mergers +are preferentially detected in a sample of BNS mergers, biasing the +luminosity distances to be nearer than their true values, and thus +inflating the value of 𝐻0 as inferred from standard sirens. +MNRAS 000, 1–14 (2022) + +Debiasing Standard Siren Cosmology +3 +Figure 1. Hubble diagram illustrating the source of gravitational wave GW +anisotropy bias. Face-on mergers (𝑣 = cos(𝜄) = 0.9) have inferred luminos- +ity distances (⟨𝐷𝐿 ⟩) skewed nearer to the observer than their actual value, +while the opposite is true for oblique mergers (𝑣 = 0.1). The tendency to +preferentially detect face-on mergers therefore results in an inflated value of +the Hubble constant. +Figure 2. The joint 𝑃(𝐷𝐿, 𝜄) posterior distributions for ten simulated BNS +mergers. Each merger is at exactly the same distance (𝐷𝐿 = 100 Mpc) with +the same NS masses (both 1.4 M⊙). Each merger varies only in its inclination +angle, 𝜄. Mergers at low 𝜄 have highly degenerate posteriors, with significant +probability density assigned to higher-than-true distances. These posteriors, +when marginalized over 𝜄, are the source of the bias illustrated in Figure 1. +2.2 +EM anisotropy and bias +A second source of bias affecting standard siren measurements +arises from anisotropy of the kilonova EM emission and its asso- +ciated selection effect. We provide a brief explanation for this bias +below, and its expected effect on 𝐻0. For a more in-depth discussion +of the origin and nature of this bias, we refer to Chen (2020). +BNS mergers produce an associated kilonova, whose EM emis- +sion enables multi-messenger standard siren cosmology. This emis- +sion primarily arises from 𝑟-process nucleosynthesis in the neutron- +rich ejecta of the BNS merger. Simple kilonova models often invoke +two ejecta components, a ‘blue’ polar component, and a ‘red’ equa- +torial component. The geometry of the emission from these compo- +nents is schematically depicted in Figure 4. Following Nicholl et al. +(2021) and Bulla (2019), we characterize this geometry using the +Figure 3. Probability of detecting GWs from a BNS merger given its lumi- +nosity distance, 𝐷𝐿, and inclination angle 𝜄. Probabilities were generated us- +ing GWToolbox assuming its parameterization of the Advanced LIGO/Virgo +experiment expected in O4 (Yi et al. 2022). Face-on mergers (𝜄 ≈ 0◦) are +in general more likely to be detected than oblique mergers (𝜄 ≈ 90◦). This +effect is especially significant for distant mergers. +blue component half-opening angle, 𝜃. The exact value of this angle +is uncertain, with Bulla (2019) estimating 𝜃 = 30◦ for GW170817 +and Nicholl et al. (2021) suggesting that 𝜃 = 45◦ is an appropriate +guess for all kilonovae. +As a result of the anisotropic EM emission from the kilonova, +the inclination angle of a BNS merger influences whether or not the +kilonova can be detected in EM telescope follow-up observations. +When a BNS merger is detected in GWs but its kilonova is not +discovered, the observer does not know whether the kilonova was +missed due to excessive distance or inclination. This issue is illus- +trated in Figure 5, where kilonova light curves from a merger at a +fixed redshift are produced for various assumed 𝐻0 and inclination +angles. This figure demonstrates that identical BNS mergers can +fail to produce a detectable EM signal due to either the assumed +value of 𝐻0, or its inclination. As a result, this effect can cause +the observer to preferentially discover face-on kilonovae in optical +imaging follow-up searches. Since this EM anisotropy bias further +enforces the discovery of standard sirens whose inferred luminosity +distances are nearer than their true values, this again inflates the +inferred value of 𝐻0. Thus, EM selection acts to exacerbate the +already extant GW anisotropy bias in a sample of BNS mergers. +3 +MOCK DATA SETS +To develop and validate our approach, we use a mock GW data +set of sample of BNS mergers, as well as accompanying simu- +lated EM light curves for a subset of these mergers for which the +kilonova is detected. A GW detection provides three relevant BNS +parameter distributions: (1) the joint luminosity distance – inclina- +tion angle distribution 𝑃(𝐷𝐿, 𝜄), (2) the joint NS mass distribution +𝑃(𝑀1, 𝑀2), and (3) the GW sky localization 𝑃(RA, DEC). An EM +detection of the kilonova counterpart provides a redshift 𝑧, and +precise sky location (RA, DEC). +For each merger, the joint probability distribution for the incli- +nation angle and luminosity distance is the related to the merger’s +true values by the relation +MNRAS 000, 1–14 (2022) + +0.07 +0.06 +redshift +0.05 +0.04 +0.03 +True Du +(DL) (v= 0.1) +0.02 +(D) (v =0.9) +3 +0.01 +100 +200 +300 +400 +500 +-50 +0 +50 +D, [Mpc] +ResidualsTrue Event +140 +0.0020 +120 +0.0015 +DL [Mpc] +P(DL, t) +100 ++ +0.0010 +80 +0.0005 +60 +0.0000 +0 +20 +40 +60 +80 +[。]]500 +400 +0.8 +D, [Mpc] +300 +0.6 +detect +200 +100 +0.2 +0.0 +0 +20 +40 +60 +80 +[.114 +Gagnon-Hartman et al. +Figure 4. Geometry of a binary neutron star (BNS) merger and its associated kilonova. Left panel shows the geometry of the ejecta and emission, including the +red and blue ejecta components, and the gamma-ray burst (GRB) shock cocoon. 𝜃 labels the half-opening angle of the blue component, and 𝜃𝑐 labels that of +the GRB shock cocoon. Right panel demonstrates how the inclination angle, 𝜄, is defined with respect to the angualar momentum axis of the BNS. 𝑣, defined +as the cosine of the inclination angle, is often used in the literature in lieu of 𝜄. +0.4 +0.6 +0.8 +1.0 +Time [days] +22.8 +23.0 +23.2 +23.4 +23.6 +23.8 +24.0 +z-magnitude +H0 = 60 km s +1 Mpc +1 +v = 0.1 +v = 0.5 +v = 0.9 +0.4 +0.6 +0.8 +1.0 +Time [days] +22.8 +23.0 +23.2 +23.4 +23.6 +23.8 +24.0 +H0 = 70 km s +1 Mpc +1 +0.4 +0.6 +0.8 +1.0 +Time [days] +22.8 +23.0 +23.2 +23.4 +23.6 +23.8 +24.0 +H0 = 80 km s +1 Mpc +1 +Figure 5. Simulated kilonova light curves, observed at various inclination angles, and in various cosmologies. Each light curve was generated using MOSFiT +using the same parameter values, except for 𝐻0 and inclination angle. When a kilonova is detected, it can be uncertain whether its detection was due to a high +value of the 𝐻0, or a low inclination angle. +𝑑𝑝(𝑣, 𝐷𝐿) = N +� 𝐷𝐿 +𝐷0 +�2 +× exp +������ +− 1 +2Δ2 +1 +� 𝑣𝐷0 +𝐷𝐿 +− 𝑣0 +�2 +− +1 +2Δ2 +2 +� +𝐷0(1 + 𝑣2) +2𝐷𝐿 +− +1 + 𝑣2 +0 +2 +�2������ +× Θ(𝐷𝐿/𝐷0)Θ +� 𝐷max +𝐷0 +− 𝐷𝐿 +𝐷0 +� +Θ(1 − 𝑣2)𝑑𝑣𝑑𝐷𝐿, +(1) +where 𝑣 = cos(𝜄) is the inferred cosine of the inclination angle, +𝐷𝐿 is the inferred luminosity distance, 𝐷0 is the true luminosity +distance, 𝑣0 = cos(𝜄0) is the true cosine of the inclination angle, +Δ1 and Δ2 are functions which encode the polarization sensitivity +functions of the GW detector, Θ is the Heaviside step function, and +𝐷max is an arbitrarily high distance which is treated as the cutoff for +the probability distribution (Cutler & Flanagan 1994). In this study, +we use a fiducial 𝐷max = 6.5 Gpc. While the inclination angle is +often marginalized over to convert this into a luminosity distance +posterior distribution, it is important for us to leave them separate, +as our goal is to discern the bias produced by the inclination angle. +We perform two kinds of validation tests, each depending on a +different underlying mock sample of BNS mergers. The first class of +tests are contrived tests, where the parameters of each BNS merger +are conspicuously chosen to highlight a certain effect. For example, +a sample of 100 mergers may be placed at the same true distance and +redshift, but each with different inclination angles, to exaggerate and +test the effect of EM inclination angle selection bias. The second +class of tests are cosmological tests, where the BNS parameters are +simulated using GWToolbox, a toolkit for generating realistic GW +source populations and their probability of detection with various +GW instruments (Yi et al. 2022). In our cosmological tests, we treat +all mergers as having been detected by LIGO/Virgo during O4. +GWToolbox generates a set of true parameters for each merger, +produces a strain curve from those parameters, and then verifies that +the strain’s signal-to-noise ratio (SNR) is high enough for the merger +to be detected. If detected, the strain curve is then interpreted to +produce BNS parameter estimates. In this study, we set the detection +SNR to 8. Mergers generated using GWToolbox will thus suffer from +GW anisotropy bias due to the dependence of the probability of +detection on luminosity distance and inclination angle, as illustrated +MNRAS 000, 1–14 (2022) + +(a) +(b) +axis of rotation +0 +0. +c +L +Lanthanide +rich region +line of sight +Lanthanide. +GRB shock +poor region +)so = +gamma ray burst (GRB)Debiasing Standard Siren Cosmology +5 +in Figure 3. This stands in contrast with the contrived mergers, +where a sample of mergers is assumed to be detected regardless +of their parameters, and is therefore not subject to GW anisotropy +bias. Contrived mergers are therefore only useful in tests where +EM-selection bias is considered independently of GW anisotropy +bias. +Once a sample of GW mergers is produced through either +method, the telescope follow-up imaging must be specified for each +merger. In our simplest case, it is assumed that a telescope is always +available for EM follow-up and it is always pointed in the right +location in the sky to detect the kilonova. We also assume that the +exposure depths of the images are the same for each merger. In this +case, EM follow-up can only fail if the kilonova is too dim to be +detected in the exposure depths of the images. A merger may be too +dim either because it is too distant, or because its inclination angle +is too oblique, thus giving rise to EM-selection bias. This method +is employed for the contrived tests in this study. +In the more complex but realistic case, the precise sky local- +ization of the BNS merger is considered. Given the true location of +a BNS merger on the sky and the telescope imaging pointings, there +exists some probability that the merger lies outside all pointings +and could not have been observed. Furthermore, even if the merger +indeed lies within a pointing, the depth, time post-merger, and filter +of that image, as well as the Galactic dust extinction at that sky +location, must all be considered. This method is employed for the +cosmological tests in this study. +In our validation tests including sky localization, we assume +the same GW sky localization and EM follow-up for each merger. +One test uses the GOTO-4 follow-up for GW190425 (Steeghs et al. +2022), which is a very incomplete follow-up tiling (having only 29% +probability coverage of the GW localization region), while another +uses the CFHT follow-up for GW 190814 to contrast with a deeper +and more complete tiling with 64% probability coverage (Abbott +et al. 2020a,b; Vieira et al. 2020). +The Galactic dust extinction at a BNS merger’s sky location +will also affect the probability of EM detection of the kilonova, as +a kilonova in a sky region with high dust extinction is less likely to +be detected than one in a region with low dust extinction. We use +the Galactic dust maps of Schlafly & Finkbeiner (2011) through the +dustmaps package. In contrived tests, wherein sky plane sampling +is not considered, dust extinction is set to a fiducial value of 𝐸𝐵−𝑉 = +2.2. +When evaluating whether or not a kilonova should be detected, +its light curve must be generated. We generate kilonova light curves +for each simulated merger from its BNS parameters using MOSFiT, +a software package for astrophysical transient simulation (Guillo- +chon et al. 2018; Nicholl et al. 2021). If the kilonova’s magnitude +is brighter than the exposure depth of the relevant telescope point- +ing, then the EM counterpart is detected. In this way, each BNS +merger in the merger sample is evaluated and classified either as +an ‘EM+GW’ merger (EM counterpart detected) or a ‘GW-only’ +merger (EM counterpart not detected). Leveraging information from +both EM+GW mergers and GW-only mergers to correct for EM se- +lection bias is a core feature of this study. +4 +METHOD +4.1 +The Case for SBI +In this section we discuss the insufficiency of traditional inference +methods in accounting for EM and GW anisotropy biases. We begin +by laying out the traditional approach to inferring parameters from +sets of BNS mergers. To simplify this discussion, we neglect the +inference of BNS parameters to focus solely on 𝐻0. Given a fixed +catalogue of mergers, x, with estimates on 𝐷𝐿, 𝑧, and 𝜄, we may +write the posterior distribution for 𝐻0 as +𝑃(𝑧, 𝐷𝐿, 𝜄, 𝐻0|x) ∝ +𝑃(𝐻0) +[ ¯𝑁(𝐻0)]𝑁 × +𝑁 +� +𝑖=1 +𝑃(ˆ𝑧𝑖|𝑧𝑖, 𝐻0, 𝐷𝐿)𝑃( ˆ𝐷𝐿,𝑖, ˆ𝜄𝑖|𝐷𝐿,𝑖, 𝜄𝑖), +(2) +where ˆ𝑧𝑖, ˆ𝐷𝐿,𝑖, and ˆ𝜄𝑖 are the estimates of 𝑧𝑖, 𝐷𝐿,𝑖, and 𝜄𝑖 for +each merger, ¯𝑁 is the mean number of detected EM+GW mergers +as a function of 𝐻0, and 𝑁 is the number of EM+GW mergers in +the catalogue (Mortlock et al. 2019). No analytic formula exists to +produce ¯𝑁 as a function of 𝐻0; for any given catalogue of mergers, +each merger’s luminosity distance and inclination angle influence +both the probability of GW detection and the probability of EM +detection, as discussed in Section 2. +A brute force estimation of ¯𝑁 for a given 𝐻0 requires gener- +ating the total number of mergers within a volume of space over +some duration of time, and then determining which among them +would be detected as a standard siren. The determination of de- +tection requires simulating each merger in the catalogue and then +comparing the expected GW and EM emission to both the response +functions of the GW detector and the telescope tiling of the localiza- +tion region. Since each merger has a unique luminosity distance and +inclination angle, the number of parameters influencing the number +of detected mergers in a given sample, 𝑁, scales with the number of +mergers included within that sample. Estimating ¯𝑁 at a particular +𝐻0 furthermore requires a dense sampling of this parameter space, +and an estimation of ¯𝑁(𝐻0) requires dense sampling assuming var- +ious values of 𝐻0. Computational expenses mount as the required +number of samples increases, rendering this approach prohibitively +computationally expensive beyond a merger sample of more than a +few mergers. +We therefore adopt an SBI approach to estimating ¯𝑁(𝐻0). +Within the SBI framework, we repeatedly sample the parameter +space and simulate a sample of BNS mergers at each point. The +mock observables from these ‘forward simulations’ enable infer- +ence of the parameter of interest, in this study, 𝐻0. We train a +neural network to compare the mock observables to real data and +thereby learn the parameter values underlying that data. Our method +employs marginal posterior inference, completely bypassing calcu- +lation of the likelihood function and thus removing the need for +explicit knowledge of ¯𝑁(𝐻0) (Talbot & Thrane 2020). +4.2 +Layout of Approach +Our SBI method follows a six-step approach: +(i) Draw parameter sample Θ +(ii) Generate realistic observables, x|x0, Θ +(iii) Summarize consistency of generated observables with data +using a summary statistic S = 𝑓 (x, x0) +(iv) Repeat steps i-iii to produce data set (Θ, S) +(v) Intermediate processing to prepare data set for training +(vi) Train neural network to infer the marginal posterior of 𝐻0 +from (Θ, S). +Below, we discuss each step in greater detail, proceeding in order +from i to vi. Figure 6 displays a schematic of our approach. +MNRAS 000, 1–14 (2022) + +6 +Gagnon-Hartman et al. +Figure 6. Overview of our approach to inference. EM and GW data supplied to the simulator informs the parameter prior distributions set by the model. In +practice, this should be real data, but in this study we use a simulated mock data set. At each step of simulation, BNS and cosmological parameters are sampled +from these prior distributions and passed to a forward model. The forward model simulates the BNS mergers assuming these parameters and determines +whether their GW and EM emission should be detected. This is done repeatedly to build a set of training data. The data then undergoes intermediate processing +to allow for the correction of EM anisotropy before being passed into the MNRE for training. A fully-trained MNRE outputs a marginal posterior distribution +for 𝐻0. +Figure 7. Process overview of the forward model developed for this study. In the first step, the parameters necessary for BNS merger generation are sampled +and computed. The inclination angles, 𝜄𝐺 and 𝜄𝑆, as well as the Hubble constant, 𝐻0, are sampled from a prior distribution. x represents the GW strain +data, which is used to generate a random luminosity distance for GW-only mergers, 𝐷𝐿,𝐺. Meanwhile, the luminosity distances of standard siren mergers, +𝐷𝐿,𝑆, is constrained by the sampled 𝐻0 and the measured redshifts, 𝑧𝑆. The redshifts of GW-only mergers are similarly computed from 𝐷𝐿,𝐺 and 𝐻0. These +parameters are summarized as Θ, which is passed to MOSFiT for light curve generation. Each light curve is either observed or not observed according to some +magnitude criterion. If all EM+GW mergers are confirmed to have been EM-observed and all GW-only mergers are not EM-observed, then the simulated +observations are said match the data, and a summary statistic is computed. Otherwise, the minimum, or ‘null’, summary statistic is returned. +In step (i), the forward model gathers a sample of parameters +from their relevant prior space. These parameters include the lumi- +nosity distances and inclination angles of each BNS merger, as well +as 𝐻0. During tests considering the impact of telescope follow-up +we also include the sky location, (RA, DEC). Table 1 summarizes +the appropriate minimally-informative prior distributions. +After gathering the parameter sample Θ, the forward model +generates the GW and EM observables for each event (step (ii)). +Since training a neural network directly on kilonova light curves and +merger GWs is difficult, we introduced a data compression scheme +to aid training convergence (for a similar example in cosmology, +see Alsing et al. 2018). We perform this compression in steps (iii) +and (v). Step (iii) computes the similarity between the simulated +and actual observables, producing a quantity called the summary +statistic S. +In step (iv), the forward model repeatedly samples parameter +space and generates observables, thus producing a data set of input +parameters, Θ, and their associated summary statistics, S. This data +MNRAS 000, 1–14 (2022) + +sample from prior +forward model +build training data +amplitude +S +GW +EM +xnl! +time +marginal posterior +train MNRE +intermediate +inference +processing +(0',s) +(0'.S) +(0, S) +(°H) +P +Ho +P(O'is)x +light curve generation +parameter generation +kilonova +(tg,..., ) +simulator +(us,... s)* +determination +of observation +Ho- +compute summary +(2s,.·, 2g)* +statistic +Di +IF simulated +ELSE +observations +match data +return +- measured * - included inO +S +s() +mir + sampledDebiasing Standard Siren Cosmology +7 +Parameter +Prior +Motivation +𝐻0 +Uniform[60, 80] +Treat as unconstrained +𝐷𝐿 +𝑃(𝐷𝐿, 𝜄|x𝐺𝑊 ) +Inferred from GW strain +𝜄 +𝑃(𝐷𝐿, 𝜄|xGW) +Inferred from GW strain +(RA, DEC) +𝑃(RA, DEC|x𝐺𝑊 ) +From GW LAL Inference +Table 1. The four free parameters considered in this study and their +prior distributions. A GW detection providing strain xGW produces a joint +posterior distribution for the luminosity distance and inclination angle, +𝑃(𝐷𝐿, 𝜄|xGW). This distribution is treated as the credible region for an +merger’s 𝐷𝐿 and 𝜄. Similarly, a GW detection has an associated sky local- +ization, 𝑃(RA, DEC|xGW), produced by LAL inference of the GW strain. +set, (Θ, S), contains information on the posterior distributions of +the input parameters, including 𝐻0. In this work, we train a neural +network to reconstruct the marginal posterior distribution 𝑃(𝐻0) us- +ing the data set produced by the forward model. However, we found +that the noise inherent to our choice of summary statistic hindered +training convergence. We therefore apply intermediate processing +in step (v), wherein we recast the data to a basis more amenable +to training convergence. This transformation varies with the global +quantities of the dataset (e.g., maximum and minimum), so we ap- +ply it after the completion of the sampling and forward modelling +phase. Then, in step (vi), a marginal neural ratio estimator (MNRE) +trained on these transformed data produces the marginal posterior +distribution for 𝐻0. +4.3 +Forward Model +4.3.1 +Observable Generation +The forward model generates observable data, x, given a parameter +vector, Θ. Our input parameters consist of 𝐻0, an inclination angle +𝜄 for each merger, and a luminosity distance 𝐷𝐿 for each GW-only +merger. The observables in this study consist of two parts: GW +emission and EM emission. +The GW strain for a BNS merger provides a joint estimate +for its 𝐷𝐿 and 𝜄. Example joint (𝐷𝐿, 𝜄) distributions are shown in +Figure 2. For an EM+GW merger, the measured redshift 𝑧 and +sampled 𝐻0 specify a 𝐷𝐿. We then sample an inclination an- +gle 𝜄 from the corresponding conditional posterior distribution, +𝑃(𝜄|𝐷𝐿, 𝑥GW), where 𝑥GW represents the GW strain for that merger. +Meanwhile, a GW-only merger lacks a measured redshift, leaving +both 𝐷𝐿 and 𝜄 as free parameters to sample from the joint poste- +rior distribution 𝑃(𝐷𝐿, 𝜄|𝑥GW). Following these rules, the forward +model fixes the parameters of all mergers in the sample prior to light +curve generation. For a graphical representation, see the parameter +generation panel of Figure 7. +Before generating light curves, we produce GW observables +for each merger. The forward model uses GWToolbox to produce a +GW strain and detection signal-to-noise ratio for each merger in the +sample. We count the GW emission of a BNS merger as detected if +the merger’s SNR exceeds 8. Only GW-detected events contribute +to parameter inference. Such events supply the GW-inferred joint +posterior distribution for 𝐷𝐿 and 𝜄. +The forward model then generates realistic EM emission (kilo- +novae) for each GW-detected merger using the MOSFiT software +package. MOSFiT uses a number of parameters to specify the fea- +tures of a kilonova, for a detailed description of these parameters +see Nicholl et al. (2021). While we sample 𝐷𝐿 and 𝜄 from their prior +distributions and pass those samples to the light curve simulator, +we fix all other relevant parameters to fiducial values. Appendix +A1 discusses our assumed values for the following kilonova pa- +rameters: the kilonova ejecta component opacities, 𝜅red and 𝜅blue, +the neutron star masses, the disk ejection fraction 𝜖disk, the blue +ejecta enhancement factor 𝛼, and the geometric parameters 𝜃 and +𝜃𝑐, which respectively describe the half-opening angles of the blue +ejecta component and gamma ray burst-shocked region. +4.3.2 +Summary Statistic +The summary statistic S represents the similarity of the simulated +observables with the observed data, x. Since the simulated observ- +ables are generated from the parameter sample Θ, the summary +statistic captures the consistency of Θ with x. +For samples where the simulated and observed data do not +meet a minimum consistency threshold, we set the summary statistic +to a minimum or ‘null’ value. We refer to such samples as ‘non- +informative’ since they do not contribute positively to the inferred +posterior density. Our minimum consistency check ensures that only +the correct mergers within a sample have detected EM counterparts. +To illustrate this ‘minimum consistency check’ with a basic +example, consider a real set of 2 BNS mergers where 1 has a +detected EM counterpart. Following our procedure, we sample a +parameter sample from prior space and produce mock GW and EM +observables from that sample. Should zero or both BNS mergers +have detected EM counterparts, then we say that the simulated ob- +servables are completely inconsistent with the real data, and a null +return is produced. Furthermore, if the wrong merger has a detected +EM counterpart in the simulated data, then that is also completely +inconsistent, producing a null return. +For samples where the minimum consistency threshold is met, +we then assess the degree of consistency. We do this by comparing +the sampled 𝐷𝐿 and 𝜄 to the (𝐷𝐿, 𝜄) joint posterior distribution +inferred from each merger’s GW signal. Appendix A2 discusses the +exact form of this comparison, as well as our choice of minimum +return for non-informative samples. +4.3.3 +EM Follow-Up +We investigate how imperfect EM follow-up observation attempts +influence our ability to correct for EM anisotropy bias. To motivate +this, consider a scenario where we can image the full sky to some +exposure depth in some filter for several days after a BNS merger is +detected in GWs. In this scenario, the only cause for non-detection of +the kilonova is the flux from the kilonova does not meet the detection +thresholds of the telescope images. Insufficient flux can only be +explained by either the distance of the merger or its inclination, +both features of the underlying BNS merger. Therefore, the EM non- +detection places some constraints on these BNS merger parameters. +The existence of these constraints allows for the correction of EM +anisotropy bias. +Let us now consider the case where a realistic telescope follow- +up is performed. When a BNS merger is detected in GWs, LAL +inference is used to localize its origin on the sky with a probability +density map (Veitch et al. 2015). Telescope follow-up tiling of the +localization region is not exhaustive, as they do not cover 100% +of the sky, nor are the exposures always deep enough to guarantee +kilonova detection. This introduces new failure modes for EM de- +tection: the possibility that an merger was not within the field of +MNRAS 000, 1–14 (2022) + +8 +Gagnon-Hartman et al. +view of any images during the follow-up campaign, or the images +were not of sufficient depth. For any given EM non-detection, it is +therefore unclear whether the non-detected was due to factors in- +trinsic to the BNS merger (𝐷𝐿, 𝜄) or factors relating to the telescope +follow-up. +4.4 +Intermediate Processing +We refer to the process of repeatedly simulating observables from +parameter samples as a ‘run’ of the forward model. A run of the +forward model produces a table of input parameter vectors, Θ, and +their associated summary statistics, S. These tables contain the +information necessary to produce a marginal posterior distribution +for 𝐻0. +To correct for EM anisotropy bias, we use information from +GW-only mergers to estimate the bias implicit in EM+GW merg- +ers. We do this by inspecting the shift in summary statistics when +we include all mergers in the sample versus when the sample only +includes EM+GW mergers. Our approach to EM anisotropy correc- +tion requires two full runs of the forward model. We refer to the +process whereby we graft the output of one run onto the other as +‘intermediate processing’. +In the first run, the data set includes only EM+GW mergers, +and the forward model generates a sample of EM+GW summary +statistics called SS. In the second run, the data set includes GW- +only mergers along with EM+GW mergers, and the forward model +generates a second set of summary statistics called SGW. The null +statistics in SS indicate Θ for which one or more mergers lack +a detected EM counterpart. Meanwhile, the null statistics in SGW +may also indicate Θ for which one or more mergers erroneously have +a detected EM counterpart. This additional constraint marginalizes +over the EM selection effect, debiasing the 𝐻0 posterior prescribed +by the summary statistics. However, the uncertainties in the GW- +only mergers’ redshifts produce greater scatter in SGW than in S𝑆, +effectively prescribing a much broader posterior distribution for 𝐻0. +We therefore process both sets of summary statistics into a modified +summary statistic set S′, which possesses the unbiasedness of SGW +and a precision nearing that of SS. We discuss the details of this +‘intermediate processing’ step in Appendix A3. +4.5 +Inference Model +The inference model learns the marginal posterior distribution of +𝐻0 from the set of input parameter vectors and their associated +summary statistics produced by the forward model. We perform +this inference using Marginal Neural Ratio Estimation (MNRE), a +simulation-based inference approach (Miller et al. 2021). MNRE +learns the marginal likelihood-to-evidence ratio for a parameter of +interest from a set of training data consisting of input parameters +and their associated model outputs. Appendix A4 describes in more +detail neural ratio estimation, the operating principle behind MNRE. +We train the neural network using Adam (Kingma & Ba 2014) +to minimize binary cross-entropy loss, as defined in Miller et al. +(2021). Although MNRE tends to produce conservative posterior +estimates once it converges, inappropriate selection of training hy- +perparameters often leads to convergence issues (e.g., Cranmer et al. +2015; Brehmer et al. 2018, 2019). To produce reliable posteriors in +each training, we adjusted the learning rate schedule and other hy- +perparameters, as shown in Table 2. We implement MNRE using +the swyft software package (Miller et al. 2020, 2022). +Parameter +EM-Only +GW-Only +Combined +𝑙𝑖 +5 · 10−4 +5 · 10−5 +5 · 10−4 +𝑓 +0.1 +0.1 +0.5 +𝑝 +5 +10 +20 +𝑒𝑚 +25 +50 +100 +Table 2. The training hyperparameters used in neural ratio estimation (NRE) +for each test. These are the initial learning rate 𝑙𝑖, the learning rate adjustment +factor, 𝑓 , the patience, 𝑝, and the maximum number of epochs 𝑒𝑚. The +NRE begins training at the initial learning rate, which is then adjusted by +the learning rate scheduler as training progresses. At each step in training, a +validation loss is calculated and passed to the learning rate scheduler. If 𝑝 +rounds pass without any decrease in validation loss, then the learning rate is +multiplied by the factor 𝑓 . This continues for 𝑒𝑚 epochs. +5 +RESULTS ON MOCK MERGER DATA SETS +5.1 +GW-only Correction +In this test, 100 simulated BNS mergers were generated using +GWToolbox assuming the realistic ‘cosmological’ parameter dis- +tribution discussed in Section 3. This sample of mergers suffers +from GW anisotropy bias, as distant mergers are only detected if +their inclination angles are nearly face-on. This is illustrated in Fig- +ure 8. For this test, the forward model was modified to not evaluate +light curves at all, and to always assume that a GW-detected merger +is also an EM+GW merger regardless of its luminosity distance or +inclination angle. This is equivalent to assuming that all mergers lie +within the field of view of a telescope with infinite exposure depth. +The purpose of this modification is to ensure that GW anisotropy +bias is corrected in the absence of EM anisotropy bias. In essence, +this test is equivalent to the GW anisotropy bias correction test +discussed in Gerardi et al. (2021). +Two sets of summary statistics are generated using this forward +model. In one set, called Sb, GW anisotropy is not considered, and +the full sample of 100 mergers are considered in each summary +statistic. In the other set, called Sc, GW anisotropy is considered, +so only a subset of the 100 mergers are considered in each summary +statistic, and this subset varies in size and composition from one +draw to the next. The first of these sets is used to produce a biased +𝐻0 posterior, while the second produces the corrected 𝐻0 posterior. +These posteriors are shown in Figure 8. The value of 𝐻0 pre- +dicted by the biased distribution is 70.54+0.73 +−0.69 km s−1 Mpc−1, which +indicates a GW anisotropy bias of 0.54+0.73 +−0.69 km s−1 Mpc−1. The cor- +rected distribution yeilds 𝐻0 = 70.09+0.69 +−0.76 km s−1 Mpc−1, which +accounts for 83.33% of the bias. The bias in our distribution is +within two standard deviations of the mean bias computed by Ger- +ardi et al. (2021), indicating that our method for merger generation +and bias measurement is consistent with their results. The degree +to which our method corrects for the bias in 𝐻0 is also comparable +to their results, which produce corrected distributions with biases +consistent with 0 to the 1-𝜎 level. +5.2 +EM-only Correction +In this test, 10 BNS mergers occur at the same luminosity distance +and redshift. This distance is set to 135.9 Mpc, which has a cor- +responding redshift of 0.031 assuming 𝐻0 = 70 km s−1 Mpc−1. +Each merger is identical except for its inclination angle, 𝜄. Every +neutron star is assumed to have a mass of 1.4𝑀⊙. The inclination +MNRAS 000, 1–14 (2022) + +Debiasing Standard Siren Cosmology +9 +Biased +Corrected +Biased +Corrected +Biased +GW- + Corrected +Fully- + Corrected +66 +68 +70 +72 +74 +76 +78 +80 +H0 [km s +1Mpc +1] +EM Correction Only +GW Correction Only +Combined Correction +True H0 +Figure 8. The 𝐻0 posteriors produced by each of the first three tests in this paper. The posteriors are coloured according to their test, and labelled according +to their status as biased, corrected, or partially-corrected. In the EM-correction only and GW-correction only tests, two posteriors are produced, one biased +and one corrected. In the combined tests, one with and one without sky-sampling considerations, three posteriors are produced, one biased, one with GW +anisotropy bias corrected, and one with all biases corrected. It is meaningless to correct for EM anisotropy bias without correcting for GW anisotropy bias if +the data suffers from GW anisotropy bias, so those posteriors were not produced in this study. The combined test results pictured here correspond to the C1 test +in Table 3. +Biased +GW-Corrected Fully-Corrected +Biased +GW-Corrected Fully-Corrected +66 +68 +70 +72 +74 +76 +78 +80 +H0 [km s +1Mpc +1] +Realistic Follow-Up +Unrealistic Follow-Up +True H0 +Figure 9. The 𝐻0 posteriors produced by increasing the EM detection sensitivity on the combined EM+GW correction test. The posteriors are coloured +according to their test, and labelled according to their status as biased, corrected, or partially-corrected. Even when realistic follow-up is considered, EM and +GW anisotropy bias can be corrected for using our method, given that the EM detection sensitivity is sufficiently high. With reference to Table 3, the unrealistic +follow-up test corresponds to C3 and the realistic follow-up test corresponds to C5. +angles of the mergers are evenly spaced in cos(𝜄), taking values from +cos(𝜄) = 0.0 to cos(𝜄) = 0.9. Of these, only the three mergers with +the lowest 𝜄 are EM+GW mergers. By placing these mergers at the +same distance and assuming their GW emission are all detected, we +ensure that the only parameters which determine an merger’s status +as a standard siren are its inclination angle and 𝐻0. In this way, we +probe EM anisotropy bias in the absence of GW anisotropy bias, +and can thus investigate the degree to which our method uniquely +corrects for EM anisotropy bias. +In determining the value of the bias, we must compare the mean +of the inferred 𝐻0 posterior to some true value. While in strict terms +the true value of 𝐻0 is that assumed in the generation of the BNS +mergers, it is inappropriate to use this as the baseline for the bias +since it is not necessarily the value which would be inferred if all 10 +BNS mergers were EM+GW mergers. Recall that EM anisotropy +bias is the difference between the value of 𝐻0 inferred from a sample +of EM+GW mergers and the value inferred a subset of those same +mergers. +The biased and corrected posteriors produced in this test are +shown in Figure 8. Both the biased and corrected distributions +exhibit a long tail to high 𝐻0, which is due to the low inclinations +of the three EM+GW mergers. Even after our correction scheme +is applied, some small probability density is still applied to these +high 𝐻0. The mean of the biased distribution is 70.17+1.31 +−1.36 km s−1 +Mpc−1 while that of the corrected distribution is 70.01+1.18 +−0.75 km +s−1 Mpc−1. The corrected distribution accounts for 94.12% of EM +anisotropy bias. This is greater than the correction level achieved +for GW anisotropy bias in both Gerardi et al. (2021) and the GW +MNRAS 000, 1–14 (2022) + +10 +Gagnon-Hartman et al. +Figure 10. The luminosity distances (𝐷𝐿) and inclination angles of the +100 GW-detected BNS mergers generated using GWToolbox. Mergers at +large distances are unlikely to be detected unless they are nearly face-on +(see Figure 3), thus resulting in GW anisotropy bias. Mergers which would +be detected via EM follow-up with a 𝑔-band magnitude cutoff of 23.3 are +marked in red, and mergers which would be detected with a cutoff of 25.0 +are marked in green. +Test Name +Exposure Depth (𝑚AB) +Follow-Up Scheme +C1 +23.3 +Perfect +C2 +23.3 +GW190425+GOTO +C3 +25 +Perfect +C4 +25 +GW190425+GOTO +C5 +25 +GW190814+CFHT +Table 3. The five validation tests performed to evaluate our method’s ability +to correct for both EM and GW anisotropy biases. These tests vary the +exposure depth of the assumed EM follow-up campaign as well as the +coverage of the campaign. The form of the follow-up schemes listed is +(GW localization)+(telescope), except when the follow-up is assumed to be +perfect. A ‘perfect’ follow-up campaign assumes that the targeted merger +lies within the imaged region of the sky. The 𝐻0 posteriors inferred in C1 +are shown in Figure 8 and those inferred in C3 and C5 are shown in Figure +9. +anisotropy-only test presented in this work. The correction itself +manifests as a suppression of high 𝐻0 values, as is expected from +our approach to EM anisotropy bias correction (see Section 4.4). +5.3 +Combined Correction Tests +We performed five tests to gauge the ability of our method to si- +multaneously correct for EM and GW anisotropy bias. Each test +considers the same sample of mergers used in the GW-only cor- +rection test, shown in Figure 10. In these tests, we vary the EM +follow-up campaign specifications for the same 100 mergers. In two +of these tests, the EM follow-up campaign is assumed to have total +sky coverage in the 𝑧 band. The tests differ by the exposure depth +assumed, which is 23.3 𝑚AB in one and 25.0 𝑚AB in the other. +For each exposure depth, we also test how our method can correct +for bias if a specific GW localization and follow-up campaign is +assumed. We perform this using the GW localization of GW190425 +and its GOTO follow-up campaign (Steeghs et al. 2022). Each test +and its specifications is laid out in Table 3. +5.3.1 +C1 and C2 +In C1 and C2, we test the ability of our method to correct for EM and +GW anisotropy bias when the detection threshold for EM follow-up +is set to 23.3 𝑚AB in the 𝑧-band. Three posteriors are produced in C1. +First, the EM+GW mergers are used to perform a biased inference +on 𝐻0. Then an intermediate correction is performed by activating +the GW anisotropy correction method in the forward model. The +final posterior is fully-corrected, accounting for both GW and EM +selection effects in the forward model. These posteriors are shown +in purple in Figure 8. +We found that for this sample of EM+GW mergers, the bias was +negative, with an inferred 𝐻0 of 69.58+0.54 +−0.61 km s−1 Mpc−1. This +is not surprising, and it is fairly common when the EM detection +threshold is low compared to the magnitudes of BNS mergers at +large distances (≳ 200 Mpc) where GW anisotropy bias becomes +important. Gerardi et al. (2021) demonstrates that negatively-biased +EM+GW merger posteriors are common in 100-merger sample. +Applying GW anisotropy bias correction raises the inferred +value of 𝐻0 to 69.89+0.65 +−0.47 km s−1 Mpc−1, correcting for 50.00% of +the bias. This demonstrates that our method for GW anisotropy bias +correction works whether the bias is positive or negative. Including +EM anisotropy changes the inferred value of 𝐻0 to 69.93+0.59 +−0.51 km +s−1 Mpc−1, increasing the level of bias correction to 68.18%. +In C2, the GW localization of GW190425 and the GOTO +follow-up campaign for that merger is assumed. Due to the poor +localization coverage of the follow-up, this resulted in our method +being unable to correct for EM anisotropy bias, although the de- +gree of GW anisotropy bias correction remained the same. This is +sensible, since any given GW-only merger is far more likely to lie +outside the EM follow-up coverage than to be too dim to be seen by +the detector. +5.3.2 +C3 and C4 +C3 and C4 consider the same sample of 100 mergers as the other +combined tests, while assuming that the EM follow-up detection +sensitivity reaches 25 𝑚AB. C3 assumes that all mergers lie within +the follow-up campaign’s sky coverage, while C4 assumes the sky +localization of GW190425 and that merger’s GOTO EM follow- +up campaign for all mergers. The posterior distributions for 𝐻0 +produced in C3 is shown in Figure 9, referred to therein as the +‘unrealistic follow-up’ case. +When perfect localization coverage is assumed (C3), the biased +posterior of 𝐻0 is 69.49+0.69 +−0.51 km s−1 Mpc−1. Once GW anisotropy +correction is applied, this becomes 70.47+1.50 +−1.48 km s−1 Mpc−1. GW +anisotropy correction in this sample of mergers acts to broaden the +posterior significantly due uncertainty in the level of GW anisotropy +bias itself. Once EM anisotropy bias is also corrected for, the inferred +value of 𝐻0 becomes 69.96+1.16 +−1.06 km s−1 Mpc−1, constituting a bias +correction level of 92.12% at the cost of broadening the posterior +by roughly a factor of 2. +Assuming realistic localization coverage (C4), the biased pos- +terior of 𝐻0 is 70.61+1.06 +−0.90 km s−1 Mpc−1. Applying GW anisotropy +bias correction changes this to 70.38+1.01 +−0.96, constituting a 37.71% +correction in the bias level. Applying the EM anisotropy bias cor- +rection in addition to this changes the inferred value of 𝐻0 to +69.81+0.99 +−1.09, constituting a 68.85% reduction in the bias level. This +MNRAS 000, 1–14 (2022) + +90 +GW-detected +80 +EM+GW-detected +70 +(23.3 mAB) +Angle [ +EM+GW-detected +60 +(25.0 mAB) +50 +Inclination +米米 +40 +30 +20 +10 +米 +100 +200 +300 +400 +500 +600 +D, [Mpc]Debiasing Standard Siren Cosmology +11 +demonstrates that increasing the depth of follow-up imaging can +allow for EM anisotropy bias correction even when the localization +coverage is poor. +5.3.3 +C5 +C5 is similar to C4, except that the sky localization of GW190814 +is used along with that merger’s Canada-France-Hawaii Telescope +follow-up campaign. The purpose of this test is to illustrate the qual- +ity with which EM and GW anisotropy biases may be corrected for +when exposures are deep and the EM follow-up campaign coverage +is extensive. In this test, the biased posterior of 𝐻0 is 69.40+1.22 +−1.14 +km s−1 Mpc−1. Once GW anisotropy correction is applied, this be- +comes 70.23+1.38 +−0.96 km s−1 Mpc−1. Once EM anisotropy bias is also +corrected for, the inferred value of 𝐻0 becomes 70.14+1.20 +−1.34 km s−1 +Mpc−1, constituting a bias correction level of 76.67%. The poste- +rior distributions for 𝐻0 produced in this test are displayed in Figure +9, labelled therein as the ‘realistic follow-up’ case. Due to the in- +completeness of the follow-up campaign, all posteriors are broader +than those produced in the case where perfect follow-up is assumed +(C3). We find that bias correction in a regime where EM follow-up +is nearly complete achieves a similar level of bias correction to C3. +6 +CONCLUSIONS +In this study, we have demonstrated how simulation-based inference +can be used to produce an unbiased measurement of 𝐻0 using +mergers detected with EM+GW in addition to GW-only mergers. In +doing so, we account for both GW and EM selection effects. Our +GW anisotropy bias correction method matches the performance +of the SBI method presented by Gerardi et al. (2021) and further +generalizes its inference to account for EM anistropy bias in standard +siren measurements of 𝐻0. +The key to EM anisotropy correction is the inclusion of GW- +only events – mergers whose GW signal is detected in the absence +of a detected EM counterpart. For many such mergers, the fact +of EM non-detection places a constraint on the merger’s apparent +magnitude, which in turn constrains the range of possible inclination +angles and luminosity distances for the merger. These constraints in +turn act to temper, and in many cases fully remove, EM anisotropy +bias in the inferred value of 𝐻0. +We found that the inclusion of EM anisotropy correction in +scenarios where EM anistropy bias is negligible can reduce the +bias correction level of our method by a few percent. However, this +slight reduction in correction efficacy in low-EM anisotropy bias +scenarios is outweighed by the high efficacy of EM anisotropy bias +correction in high-EM anisotropy bias scenarios. Without an ab +initio method of determining whether a population of BNS mergers +is significantly EM anisotropy biased, it is safest to include EM +anisotropy correction in the analysis of a merger sample alongside +GW anisotropy correction. +The tests which consider the GW footprint of GW190425 and +its GOTO EM follow-up campaign (C3 and C4) illustrate the impor- +tance of a comprehensive EM follow-up campaign for each merger. +In these tests, the probability of a merger not lying within the EM +follow-up region is considered. When the probability that a GW- +only merger only lacks a detected EM counterpart because it was +not localized within any telescope pointings is high, only weak +constraints can be placed on that merger’s inclination angle and +luminosity distance. Such was the case for GW190425, where the +localization was poor and the GOTO campaign only had ∼25% +localization coverage. This in turn weakens the correction level on +the EM anisotropy bias. Thus, a maximally-informative EM follow- +up campaign should seek to maximize its coverage of the BNS +merger’s sky localization. We make this point by including a test +where we consider a more complete EM follow-up scenario, the +Canada-France-Hawaii Telescope follow-up of GW190814 (C5), +wherein the bias correction level is comparable to that of perfect +localization coverage. +Future extensions of this work should focus on more realis- +tic inference scenarios. For example, in this work we assumed the +neutron star masses of each merger to be precisely measured from +the GWs, when in reality a GW measurement provides a joint prob- +ability distribution for the merger’s neutron star masses. We also +assumed the same EM follow-up routine for each merger with the +same underlying sky localization. A more realistic treatment of EM +follow-up should consider the real sky localizations of each merger +with merger-specific EM follow-up routines. Future work should +also address the uncertainty in the kilonova parameters themselves, +as incorrectly assuming their values can lead to either insufficient +or overzealous bias correction. +The significance of the biases addressed in this study will only +increase as more BNS mergers are detected in the coming years, +underscoring the importance of a reliable bias correction method. +SBI-based approaches such as ours enable cosmologists to take +full advantage of the incoming deluge of BNS merger detections +to produce unbiased measurements of the Hubble constant, and +possibly resolve the Hubble tension. +ACKNOWLEDGMENTS +The authors thank Sabrina Berger, Michael Matesic, Carter Rhea, +Jason Rowe, Nicholas Vieira, and Clovis Vinant-Tang for help- +ful discussions. S.G.H. acknowledges support from the Natural +Sciences and Engineering Research Council of Canada (NSERC) +through their Canada Graduate Scholarships - Master’s programme, +as well as from the Bishop’s University Foundation through their +Graduate Entrance Scholarship. J.J.R. and D.H. acknowledge sup- +port from the Canada Research Chairs (CRC) program, the NSERC +Discovery Grant program, the FRQNT Nouveaux Chercheurs Grant +program, and the Canadian Institute for Advanced Research (CI- +FAR). J.J.R. acknowledges support from the Canada Foundation +for Innovation (CFI), and the Québec Ministère de l’Économie et +de l’Innovation. Computations were performed on the Cedar and +Béluga supercomputing clusters managed by Compute Canada. +DATA AVAILABILITY +The data underlying this article is available upon request. The in- +ference code is available on the author’s GitHub page: https: +//github.com/samgagnon/kilonova_sim. +REFERENCES +Abbott B. P., et al., 2017a, Phys. Rev. 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P., 2022, The Gravita- +tional Wave Universe Toolbox: A software package to simulate ob- +servation of the Gravitational Wave Universe with different detectors +(arXiv:2106.13662) +APPENDIX A: METHODS SUPPLEMENTAL MATERIAL +A1 +Kilonova Parameters +We generate kilonova light curves using the MOSFiT software. We +assume that kilonovae have two merger ejecta components, one +red and one blue (Metzger 2019), with opacities 𝜅red and 𝜅blue +respectively. We use fiducial values of 𝜅red = 10 cm2 g−1 and +𝜅blue = 0.5 cm2 g−1, following the results of Radice et al. (2018). +A kilonova’s peak luminosity scales with the quantity of ejecta +produced in the NS merger, and thus to the masses of the NS +themselves. Our light curve simulation requires as inputs the BNS +merger chirp mass, +M = +� +(𝑚1𝑚2)3/5 +(𝑚1 + 𝑚2)1/5 +� +, +(A1) +and NS mass ratio, 𝑞 = 𝑚1/𝑚2, where 𝑚1 < 𝑚2. We compute these +quantities from the true NS masses provided by GWToolbox. +The luminosities of red and blue components also depend on +their temperatures, which are influenced by the disk ejection frac- +tion, 𝜖disk, and the blue component enhancement factor 𝛼. 𝜖disk +governs the fraction of the BNS remnant accretion disk ejected +post-merger. We use a fiducial value of 𝜖disk = 0.15, while Nicholl +et al. (2021) state that the true value may range anywhere from 0.05 +to 0.5, following Metzger (2019). Meanwhile, surface winds from +the merger remnant enhance the temperature of the blue ejecta, en- +capsulated by 𝛼. Nicholl et al. (2021) assign 𝛼 a flat prior from 0.1 +to 1.0, while we set it to 1.0 (maximum enhancement). +Finally, the geometry of the kilonova influences which com- +ponent we actually observe. In MOSFiT, kilonova geometry is sim- +plified into two distinct angular regions defined by the half-opening +angle of the blue component, 𝜃. We follow Nicholl et al. (2021) by +setting 𝜃 = 45◦. BNS mergers also produce an associated gamma- +ray burst (GRB), which shocks ejecta material in its vicinity. For the +purposes of this study, we assume that the GRB shock negligibly +effects the blue component, thus setting the half-opening angle of +the GRB shock to 𝜃𝑐 = 0. We do this since the half-opening angle of +the shocked cone is uncertain and its physics are poorly understood. +A2 +Summary Statistic +We measure the consistency of Θ with x explicitly using the (𝐷𝐿, 𝜄) +distributions of each GW-detected merger as measured from its +gravitational wave emission. The likelihood of guessed parameters +𝐷𝐿 and 𝜄 for any particular merger are taken as L = 𝑃(𝐷𝐿, 𝜄|𝑥GW) +MNRAS 000, 1–14 (2022) + +Debiasing Standard Siren Cosmology +13 +where 𝑥GW is the gravitational wave strain for that merger. Each +merger’s likelihood contributes to the informative summary statistic +S, defined as +S = +𝑁 +∑︁ +𝑖 +log(L(𝐷𝐿,𝑖, 𝜄𝑖|𝑥GW,𝑖)), +(A2) +or the sum of log-likelihoods of each merger. We use this definition +of S since it is based on an analytic likelihood, allowing the neural +ratio estimator to easily learn the true likelihood-to-evidence ratio. +Furthermore, variance in S originates entirely from uncertainty in +the proposed parameters 𝐷𝐿,𝑖 and 𝜄𝑖, meaning that our selected +summary statistic does not introduce any new biases that are not +already present in the source data. This follows the principles of +summary statistic selection laid out by Raynal & Onnela (2021). +Our choice of the minimum value of S for null returns follows +from the form of Equation A2. Since the neural ratio estimator is +trained to maximize the log-likelihood of the target parameter, a +null return must be lower than the lowest informative return. To +this end, we set a minimum allowable likelihood for each merger +within a sample, Lmin = 10−6. We choose this as the minimum +likelihood since it is equivalent to the likelihood of sampling 𝐷𝐿 +and 𝜄 from a 2D uniform distribution, given that our likelihood +distributions have a resolution of 1000×1000. As such, any sampled +(𝐷𝐿, 𝜄) with this likelihood or below is no more informative than a +sample drawn from a uniform distribution, and we therefore deem it +‘non-informative’. If a parameter sample is non-informative for all +mergers in a data set (i.e., if S < −6 · 𝑁) then the summary statistic +is automatically returned as null, or Snull = −6 · 𝑁. +A careful inspection of Equation A2 reveals that its maximum +and minimum vary with the number of mergers in a sample. Fur- +thermore, only a fraction of all BNS mergers within a sample have +detected GWs in any given sample. To maintain consistent bounds +for the summary statistic, we treat each merger not detected in +GWs as contributing their maximum log-likelihood to the summary +statistic. Thus, the summary statistic for each sample is computed +as +S = +𝐷 +∑︁ +𝑖 +𝑙𝑖(Θ|𝐷𝐿,𝑖, 𝜄𝑖) + +𝑁 +∑︁ +𝑖 +max(𝑙𝑖(Θ)), +(A3) +where 𝐷 is the number of GW-detected mergers, 𝑁 is the number +of mergers not detected in GWs, 𝑙𝑖(Θ) is the log-likelihood of an +merger indexed 𝑖 with the parameter sample Θ, and (𝐷𝐿,𝑖, 𝜄𝑖) are +the luminosity distance and inclination angle sampled for the merger +𝑖. +A3 +Intermediate Processing +Producing an unbiased 𝐻0 posterior from SGW is not as simple +as training another MNRE. This is because the nonzero returns in +SGW are noise-dominated. Each summary statistic S is the sum of +log-likelihoods for each merger given the sampled parameter vector. +EM+GW mergers are governed by only two free variables: 𝐻0 and 𝜄, +the former of the two is shared for all mergers within a sample. This +means that at a particular 𝐻0, the variance in log-likelihoods is en- +tirely determined by the uncertainty in 𝜄 for each merger. The redshift +of a EM+GW merger is known from EM observations, fixing the lu- +minosity distance when a cosmology is assumed ((𝐻′ +0, 𝑧) −→ 𝐷′ +𝐿). +Meanwhile, GW-only mergers are governed by an additional pa- +rameter, 𝐷𝐿. Without a known redshift, the luminosity distance of +a GW-only merger is free to be sampled independently of 𝐻0. This +means that for a particular 𝐻0, the variance in the log-likelihood is +driven by both the uncertainty in 𝐷𝐿 and the uncertainty in 𝜄. We +find that this drives the variance in nonzero returns to a point where +an MNRE is no longer capable of recovering the underlying pattern, +rendering direct posterior inference impossible. +With nonzero returns rendered useless, we are forced to look to +the zero returns for answers. Zero returns are not non-informative: +they show the values of 𝐻0 which are forbidden given a sample of +mergers. Taking the ratio between the number of zero returns and +the number of total samples as a function of 𝐻0, the ‘rejection’ rate, +or 𝑅(𝐻0), may be constructed. Any set of summary statistics has +an associated rejection rate function, and we denote the rejection +rate produced from SS as 𝑅S(𝐻0), and that produced from SGW as +𝑅GW(𝐻0). +These rejection rates differ from one another, as including +GW-only mergers increases the selectiveness of the forward model. +Crucially, the selectiveness does not change uniformly with 𝐻0, +but follows some functional relationship encoded within 𝑅GW(𝐻0). +The objective in correcting for EM anisotropy bias is thus to modify +the set SS such that it accounts for the information in 𝑅GW(𝐻0). +A responsible modification to SS is one that does not introduce +any information beyond what is encoded in GW-only mergers. The +most obvious modification is to apply 𝑅GW(𝐻0) to SS by changing +some portion of the informative summary statistics to zero returns +according to the target rejection rate. This does not change the +posterior learned from the distribution since the MNRE explicitly +ignores null returns in favour of informative samples. Therefore, the +solution must modify the informative samples in SS without setting +them to zero. We define the modified set of summary statistics as +the product of SS and 𝑅GW(𝐻0), i.e. +S′ +S = SS|SGW = 𝑅GW(𝐻0) · SS. +(A4) +This modified set of summary statistics, S′ +S, includes all in- +formation granted by the EM+GW mergers, while also incorporat- +ing the rejection rate required by GW-only mergers. 𝐻0 posteriors +learned from this modified set correct for the bias introduced by EM +anisotropy without broadening the distribution by including noisy +GW-only mergers. +A4 +Neural Ratio Estimation +Neural ratio estimation (NRE) involves training a classifier network +𝑑𝜙 : Θ × 𝑋 −→ [0, 1] to discriminate between pairs (𝜃, 𝑥) sampled +from the joint posterior distribution 𝑝(𝜃, 𝑥) and the product of the +marginals 𝑝(𝜃)𝑝(𝑥). By the Neyman-Pearson lemma, the proba- +bility that a data pair belongs to the joint posterior is proportional +to the posterior density (Neyman & Pearson 1933). Formally, the +neural network optimizes +𝜙∗ = arg min +𝜙 +E +𝑝(𝜃,𝑥) 𝑝(𝜃′)[L(𝑑𝜙(𝜃, 𝑥)) + L(1 − 𝑑𝜙(𝜃′, 𝑥))], (A5) +where L(𝑝) = − log 𝑝 is the negative log-likelihood. For this task, +the Bayes optimal classifier uses the decision function +𝑑(𝜃, 𝑥) = +𝑝(𝜃, 𝑥) +𝑝(𝜃, 𝑥) + 𝑝(𝜃)𝑝(𝑥) , +(A6) +MNRAS 000, 1–14 (2022) + +14 +Gagnon-Hartman et al. +which defines the likelihood-to-evidence (LTE) ratio +𝑟(𝜃, 𝑥) = +𝑑(𝜃, 𝑥) +1 − 𝑑(𝜃, 𝑥) = +𝑝(𝜃, 𝑥) +𝑝(𝜃)𝑝(𝑥) = 𝑝(𝑥|𝜃) +𝑝(𝑥) += 𝑝(𝜃|𝑥) +𝑝(𝜃) . +(A7) +Thus, NRE grants us an estimator log 𝑟𝜙(𝜃, 𝑥) = logit(𝑑𝜙(𝜃, 𝑥)) +of the LTE log-ratio and a surrogate ˆ𝑝(𝜃|𝑥) = 𝑟𝜙(𝜃, 𝑥)𝑝(𝜃) for the +posterior density. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–14 (2022) + diff --git a/ptE4T4oBgHgl3EQfvQ3k/content/tmp_files/load_file.txt b/ptE4T4oBgHgl3EQfvQ3k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..261b64f9a2ae6f9ef0c9e755c42d66acdc16491c --- /dev/null +++ b/ptE4T4oBgHgl3EQfvQ3k/content/tmp_files/load_file.txt @@ -0,0 +1,1002 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf,len=1001 +page_content='MNRAS 000, 1–14 (2022) Preprint 16 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 Debiasing Standard Siren Inference of the Hubble Constant with Marginal Neural Ratio Estimation Samuel Gagnon-Hartman,1,2,3★ John Ruan1, Daryl Haggard2 1Department of Physics and Astronomy, Bishop’s University, 2600 College Street, Sherbrooke J1M 1Z7, Canada 2Department of Physics and McGill Space Institute, McGill University, Montreal, QC, Canada H3A 2T8 3Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Gravitational wave (GW) standard sirens may resolve the Hubble tension, provided that stan- dard siren inference of 𝐻0 is free from systematic biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' However, standard sirens from binary neutron star (BNS) mergers suffer from two sources of systematic bias, one arising from the anisotropy of GW emission, and the other from the anisotropy of electromagnetic (EM) emission from the kilonova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For an observed sample of BNS mergers, the traditional Bayesian approach to debiasing involves the direct computation of the detection likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This is infeasible for large samples of detected BNS merger due to the high dimensionality of the parameter space governing merger detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In this study, we bypass this computation by fitting the Hubble constant to forward simulations of the observed GW and EM data under a simulation-based inference (SBI) framework using marginal neural ratio estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A key innovation of our method is the inclusion of BNS mergers which were only detected in GW, which allows for estimation of the bias introduced by EM anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Our method corrects for ∼90% of the bias in the inferred value of 𝐻0 when telescope follow-up observations of BNS mergers have extensive tiling of the merger localization region, using known telescope sensitivities and assuming a model of kilonova emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Our SBI-based method thus enables a debiased inference of the Hubble constant of BNS mergers, including both mergers with detected EM counterparts and those without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Key words: transients: neutron star mergers – gravitational waves – methods: data analysis – cosmology: observations 1 INTRODUCTION The value of the Hubble constant, 𝐻0, is currently the subject of dispute as a ∼5𝜎 tension exists between the latest late-time mea- surement using the cosmic distance ladder by the SH0ES Team (Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2022) and the early-time value inferred from cosmic microwave background (CMB) anisotropies by the Planck satellite (Aghanim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Gravitational wave (GW) standard sirens provide an independent way to measure 𝐻0, and thus have the po- tential to resolve this dispute (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' GW observations of binary neutron star (BNS) mergers pro- vide an estimate of the luminosity distance (𝐷𝐿) of each merger through modeling of its GW waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' If the electromagnetic (EM) emission from the kilonova counterparts of the mergers are also de- tected, then the BNS mergers can be precisely localized, thus provid- ing their cosmological redshifts through the host galaxy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Given a sample of BNS mergers with known redshifts and luminos- ity distances, 𝐻0 can be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This approach is known as the stan- dard siren method of inferring the Hubble constant (Schutz 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' ★ samuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='gagnonhartman@sns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='it Holz & Hughes 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The binary neutron star merger GW170817 was the first standard siren, providing a ∼10% measurement of 𝐻0 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2017a,c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Soares-Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Hotokezaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Forecasts predict that a 2% mea- surement of 𝐻0 can be achieved by combining a future sample of ∼50 standard sirens (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=', Dalal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Nissanke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2010, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Feeney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In order for a standard siren measurement to resolve the tension in 𝐻0, it must be free of systematic biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Here, we seek to address two major sources of bias in standard sirens: GW anisotropy bias and EM anisotropy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The bias from anisotropic GW emission has been shown to inflate the value of 𝐻0 inferred from standard sirens (Talbot & Thrane 2020), and a method to mitigate these biases was presented by Gerardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021) using a simulation-based inference (SBI) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' An additional source of bias is intro- duced by observational selection effects owing to the anisotropic EM emission from the kilonova, as detailed in Chen (2020), and this additional bias was not addressed by Gerardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In both sources of bias, the anisotropy of BNS merger emission pro- duces a selection effect where mergers consistent with a high value of 𝐻0 are preferentially observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='05241v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='CO] 12 Jan 2023 2 Gagnon-Hartman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Correcting for bias introduced by a selection effect requires a careful understanding of the selection criterion itself (Vitale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Gerardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021) corrected for GW anisotropy bias through modelling the dependence of successful GW detection on both BNS merger parameters and cosmological parameters, using general relativity (GR) and the GW detector’s configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' GR directly allows calculation of the strain and polarization breakdown of a GW produced by a BNS merger from that merger’s measured parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Making a determination of detection on this GW fur- ther requires knowledge of the polarization sensitivity and strain detection threshold of the GW detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We extend this logic to EM anisotropy bias, using simulation- based inference (SBI) to characterize the EM selection criterion in a sample of BNS mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For an observed sample of BNS mergers detected in GWs, we expect to have both mergers with identified EM counterparts and those without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Mergers without EM counterparts in the sample allow us to infer the probability of EM detection for a BNS merger, given its measured parameters and assuming a model for the EM emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This allows us to characterize the dependence of EM selection on BNS and cosmological parameters, and thereby correct for the EM anisotropy bias in standard siren measurements of 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The main innovation of our method is this inclusion of BNS mergers detected in GWs without detected EM counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In the absence of an analytic likelihood that includes both GW and EM selection effects, we intead opt for a SBI approach, where a BNS merger forward simulator enables emulation of both selection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' SBI refers to a class of inference methods that rely on surrogate likelihood functions from a simulator forward model, enabling Bayesian inference even in situations where the likelihood function is intractable, such as ours (Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' SBI methods typically function by simulating a data set from a set of input parameters, and then producing a summary statistic which describes the difference between the simulated data and the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Through repeated sampling of the parameter space, these summary statistics are used to construct a surrogate likelihood function, and thus enable posterior inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Here, we present an SBI-based method which corrects for both GW and EM anisotropy biases in standard siren measure- ments of 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Our approach exploits our singular goal of inferring the marginal posterior distribution of 𝐻0, allowing us to treat all BNS merger parameters as nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Specifically, we use marginal neural ratio estimation (MNRE), in which a summary statistic is generated for each set of input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The summary statistic is a quantity which encapsulates the consistency of the pa- rameters drawn with the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' These summary statistics are then used as a training set for a neural ratio estimator network, which learns the marginal posterior distribution for 𝐻0, our param- eter of interest (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' MNRE is a recently-developed SBI method which requires far fewer samples than competing methods to produce an informative posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' It accomplishes this by estimating only marginal poste- riors rather than joint probability distributions (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' MNRE is thus appropriate in situations where only marginal dis- tributions are of interest, and the computational expense of each sample is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' MNRE has recently been applied by Karchev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2022) in the context of Type Ia supernova cosmology, where it was used to marginalize over a large number of of supernova parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Similarly, our study is only interested in the marginal posterior distribution for 𝐻0, and thus MNRE is an ideal tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' To test and validate our SBI-based approach on a mock data set, we assume a set of true BNS mergers with associated GW strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Of these BNS mergers, only a subset have associated EM detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The expected kilonova light curves from the full merger sample is repeatedly simulated with randomly-sampled kilonova and cosmological parameter configurations, assuming a model for the kilonova emission, and the summary statistics from each round are used to train an MNRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We demonstrate that by including GW- only mergers in a sample of BNS mergers, we can correct for the EM anisotropy bias latent within the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Our method provides a GW anisotropy bias correction level comparable to that in Gerardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021), and further addresses EM anisotropy bias for standard siren cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The structure of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Section 2 explains the physical cause of each source of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Section 3 discusses the mock data sets used in this study and the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Section 4 provides a detailed discussion of our SBI-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Section 5 reviews results from each validation test performed on the mock data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We summarize and conclude in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2 SOURCES OF BIAS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='1 GW anisotropy and bias GW detections of BNS mergers suffer from GW anisotropy bias, which results in standard siren measurements overestimating 𝐻0 (Malmquist 1922).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This stems from the luminosity distance – in- clination angle degeneracy of the GW strain produced from a BNS merger, which skews the inferred luminosity distance of a BNS merger in a way that is dependent on its inclination angle 𝜄 (Wahlquist 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We illustrate this effect in Figure 1, where a se- ries of BNS mergers evenly spaced in luminosity distance have their inferred luminosity distances and associated uncertainties shown for two possible inclination angles, one face-on and one oblique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Uncertainty in luminosity distance estimates from GW detections arises from the detector’s polarization sensitivity (Cutler & Flana- gan 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In Figure 1, we assume the characteristics of the Ad- vanced LIGO/Virgo experiment as expected in O4 (Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Mergers placed at oblique inclinations have an inferred luminosity distance greater than their true value, while the opposite is true of mergers placed at face-on inclinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This is because GW strain is strongest along the angular momentum axis of a BNS merger, and weakest perpendicular to this axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A weak GW strain may be weak either because the merger is in fact very distant and face-on, or be- cause the merger is nearby but at an oblique inclination, thus giving rise to the aforementioned degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' An example of this lumi- nosity distance-inclination angle degeneracy is shown in Figure 2, which displays the overlaid 𝑃(𝐷𝐿, 𝜄) joint posterior distributions for 10 otherwise identical BNS mergers at different inclination angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This luminosity distance – inclination angle degeneracy would be unproblematic in the absence of selection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' However, the amplitude of a GW strain curve is related to the probability that the merger’s GW is detected at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In order for a BNS merger to be detected, its strain must exceed some critical signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Naturally, this is less likely for weak-strain mergers than for strong- strain mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The nature of this relationship is shown in Figure 3, where we show that face-on mergers are likely to be detected even at large distances, while oblique mergers are unlikely to be detected even at small distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' As a result, GWs from face-on mergers are preferentially detected in a sample of BNS mergers, biasing the luminosity distances to be nearer than their true values, and thus inflating the value of 𝐻0 as inferred from standard sirens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) Debiasing Standard Siren Cosmology 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Hubble diagram illustrating the source of gravitational wave GW anisotropy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Face-on mergers (𝑣 = cos(𝜄) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='9) have inferred luminos- ity distances (⟨𝐷𝐿 ⟩) skewed nearer to the observer than their actual value, while the opposite is true for oblique mergers (𝑣 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The tendency to preferentially detect face-on mergers therefore results in an inflated value of the Hubble constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The joint 𝑃(𝐷𝐿, 𝜄) posterior distributions for ten simulated BNS mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Each merger is at exactly the same distance (𝐷𝐿 = 100 Mpc) with the same NS masses (both 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='4 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Each merger varies only in its inclination angle, 𝜄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Mergers at low 𝜄 have highly degenerate posteriors, with significant probability density assigned to higher-than-true distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' These posteriors, when marginalized over 𝜄, are the source of the bias illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='2 EM anisotropy and bias A second source of bias affecting standard siren measurements arises from anisotropy of the kilonova EM emission and its asso- ciated selection effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We provide a brief explanation for this bias below, and its expected effect on 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For a more in-depth discussion of the origin and nature of this bias, we refer to Chen (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' BNS mergers produce an associated kilonova, whose EM emis- sion enables multi-messenger standard siren cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This emis- sion primarily arises from 𝑟-process nucleosynthesis in the neutron- rich ejecta of the BNS merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Simple kilonova models often invoke two ejecta components, a ‘blue’ polar component, and a ‘red’ equa- torial component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The geometry of the emission from these compo- nents is schematically depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Following Nicholl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021) and Bulla (2019), we characterize this geometry using the Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Probability of detecting GWs from a BNS merger given its lumi- nosity distance, 𝐷𝐿, and inclination angle 𝜄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Probabilities were generated us- ing GWToolbox assuming its parameterization of the Advanced LIGO/Virgo experiment expected in O4 (Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Face-on mergers (𝜄 ≈ 0◦) are in general more likely to be detected than oblique mergers (𝜄 ≈ 90◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This effect is especially significant for distant mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' blue component half-opening angle, 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The exact value of this angle is uncertain, with Bulla (2019) estimating 𝜃 = 30◦ for GW170817 and Nicholl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021) suggesting that 𝜃 = 45◦ is an appropriate guess for all kilonovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' As a result of the anisotropic EM emission from the kilonova, the inclination angle of a BNS merger influences whether or not the kilonova can be detected in EM telescope follow-up observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' When a BNS merger is detected in GWs but its kilonova is not discovered, the observer does not know whether the kilonova was missed due to excessive distance or inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This issue is illus- trated in Figure 5, where kilonova light curves from a merger at a fixed redshift are produced for various assumed 𝐻0 and inclination angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This figure demonstrates that identical BNS mergers can fail to produce a detectable EM signal due to either the assumed value of 𝐻0, or its inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' As a result, this effect can cause the observer to preferentially discover face-on kilonovae in optical imaging follow-up searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Since this EM anisotropy bias further enforces the discovery of standard sirens whose inferred luminosity distances are nearer than their true values, this again inflates the inferred value of 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Thus, EM selection acts to exacerbate the already extant GW anisotropy bias in a sample of BNS mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 3 MOCK DATA SETS To develop and validate our approach, we use a mock GW data set of sample of BNS mergers, as well as accompanying simu- lated EM light curves for a subset of these mergers for which the kilonova is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A GW detection provides three relevant BNS parameter distributions: (1) the joint luminosity distance – inclina- tion angle distribution 𝑃(𝐷𝐿, 𝜄), (2) the joint NS mass distribution 𝑃(𝑀1, 𝑀2), and (3) the GW sky localization 𝑃(RA, DEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' An EM detection of the kilonova counterpart provides a redshift 𝑧, and precise sky location (RA, DEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For each merger, the joint probability distribution for the incli- nation angle and luminosity distance is the related to the merger’s true values by the relation MNRAS 000, 1–14 (2022) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='06 redshift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='03 True Du (DL) (v= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='02 (D) (v =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='9) 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='01 100 200 300 400 500 50 0 50 D, [Mpc] ResidualsTrue Event 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0020 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0015 DL [Mpc] P(DL, t) 100 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0010 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0005 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0000 0 20 40 60 80 [。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=']]500 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='8 D, [Mpc] 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='6 detect 200 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 0 20 40 60 80 [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='114 Gagnon-Hartman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Geometry of a binary neutron star (BNS) merger and its associated kilonova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Left panel shows the geometry of the ejecta and emission, including the red and blue ejecta components, and the gamma-ray burst (GRB) shock cocoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 𝜃 labels the half-opening angle of the blue component, and 𝜃𝑐 labels that of the GRB shock cocoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Right panel demonstrates how the inclination angle, 𝜄, is defined with respect to the angualar momentum axis of the BNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 𝑣, defined as the cosine of the inclination angle, is often used in the literature in lieu of 𝜄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 Time [days] 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 z-magnitude H0 = 60 km s 1 Mpc 1 v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='1 v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='5 v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 Time [days] 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 H0 = 70 km s 1 Mpc 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 Time [days] 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 H0 = 80 km s 1 Mpc 1 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Simulated kilonova light curves, observed at various inclination angles, and in various cosmologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Each light curve was generated using MOSFiT using the same parameter values, except for 𝐻0 and inclination angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' When a kilonova is detected, it can be uncertain whether its detection was due to a high value of the 𝐻0, or a low inclination angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 𝑑𝑝(𝑣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 𝐷𝐿) = N � 𝐷𝐿 𝐷0 �2 × exp ������ − 1 2Δ2 1 � 𝑣𝐷0 𝐷𝐿 − 𝑣0 �2 − 1 2Δ2 2 � 𝐷0(1 + 𝑣2) 2𝐷𝐿 − 1 + 𝑣2 0 2 �2������ × Θ(𝐷𝐿/𝐷0)Θ � 𝐷max 𝐷0 − 𝐷𝐿 𝐷0 � Θ(1 − 𝑣2)𝑑𝑣𝑑𝐷𝐿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (1) where 𝑣 = cos(𝜄) is the inferred cosine of the inclination angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 𝐷𝐿 is the inferred luminosity distance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 𝐷0 is the true luminosity distance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 𝑣0 = cos(𝜄0) is the true cosine of the inclination angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Δ1 and Δ2 are functions which encode the polarization sensitivity functions of the GW detector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Θ is the Heaviside step function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' and 𝐷max is an arbitrarily high distance which is treated as the cutoff for the probability distribution (Cutler & Flanagan 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In this study, we use a fiducial 𝐷max = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='5 Gpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' While the inclination angle is often marginalized over to convert this into a luminosity distance posterior distribution, it is important for us to leave them separate, as our goal is to discern the bias produced by the inclination angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We perform two kinds of validation tests, each depending on a different underlying mock sample of BNS mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The first class of tests are contrived tests, where the parameters of each BNS merger are conspicuously chosen to highlight a certain effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For example, a sample of 100 mergers may be placed at the same true distance and redshift, but each with different inclination angles, to exaggerate and test the effect of EM inclination angle selection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The second class of tests are cosmological tests, where the BNS parameters are simulated using GWToolbox, a toolkit for generating realistic GW source populations and their probability of detection with various GW instruments (Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In our cosmological tests, we treat all mergers as having been detected by LIGO/Virgo during O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' GWToolbox generates a set of true parameters for each merger, produces a strain curve from those parameters, and then verifies that the strain’s signal-to-noise ratio (SNR) is high enough for the merger to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' If detected, the strain curve is then interpreted to produce BNS parameter estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In this study, we set the detection SNR to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Mergers generated using GWToolbox will thus suffer from GW anisotropy bias due to the dependence of the probability of detection on luminosity distance and inclination angle, as illustrated MNRAS 000, 1–14 (2022) (a) (b) axis of rotation 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' c L Lanthanide rich region line of sight Lanthanide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' GRB shock poor region )so = gamma ray burst (GRB)Debiasing Standard Siren Cosmology 5 in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This stands in contrast with the contrived mergers, where a sample of mergers is assumed to be detected regardless of their parameters, and is therefore not subject to GW anisotropy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Contrived mergers are therefore only useful in tests where EM-selection bias is considered independently of GW anisotropy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Once a sample of GW mergers is produced through either method, the telescope follow-up imaging must be specified for each merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In our simplest case, it is assumed that a telescope is always available for EM follow-up and it is always pointed in the right location in the sky to detect the kilonova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We also assume that the exposure depths of the images are the same for each merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In this case, EM follow-up can only fail if the kilonova is too dim to be detected in the exposure depths of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A merger may be too dim either because it is too distant, or because its inclination angle is too oblique, thus giving rise to EM-selection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This method is employed for the contrived tests in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In the more complex but realistic case, the precise sky local- ization of the BNS merger is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Given the true location of a BNS merger on the sky and the telescope imaging pointings, there exists some probability that the merger lies outside all pointings and could not have been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Furthermore, even if the merger indeed lies within a pointing, the depth, time post-merger, and filter of that image, as well as the Galactic dust extinction at that sky location, must all be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This method is employed for the cosmological tests in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In our validation tests including sky localization, we assume the same GW sky localization and EM follow-up for each merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' One test uses the GOTO-4 follow-up for GW190425 (Steeghs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2022), which is a very incomplete follow-up tiling (having only 29% probability coverage of the GW localization region), while another uses the CFHT follow-up for GW 190814 to contrast with a deeper and more complete tiling with 64% probability coverage (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2020a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Vieira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The Galactic dust extinction at a BNS merger’s sky location will also affect the probability of EM detection of the kilonova, as a kilonova in a sky region with high dust extinction is less likely to be detected than one in a region with low dust extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We use the Galactic dust maps of Schlafly & Finkbeiner (2011) through the dustmaps package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In contrived tests, wherein sky plane sampling is not considered, dust extinction is set to a fiducial value of 𝐸𝐵−𝑉 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' When evaluating whether or not a kilonova should be detected, its light curve must be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We generate kilonova light curves for each simulated merger from its BNS parameters using MOSFiT, a software package for astrophysical transient simulation (Guillo- chon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Nicholl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' If the kilonova’s magnitude is brighter than the exposure depth of the relevant telescope point- ing, then the EM counterpart is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In this way, each BNS merger in the merger sample is evaluated and classified either as an ‘EM+GW’ merger (EM counterpart detected) or a ‘GW-only’ merger (EM counterpart not detected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Leveraging information from both EM+GW mergers and GW-only mergers to correct for EM se- lection bias is a core feature of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 4 METHOD 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='1 The Case for SBI In this section we discuss the insufficiency of traditional inference methods in accounting for EM and GW anisotropy biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We begin by laying out the traditional approach to inferring parameters from sets of BNS mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' To simplify this discussion, we neglect the inference of BNS parameters to focus solely on 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Given a fixed catalogue of mergers, x, with estimates on 𝐷𝐿, 𝑧, and 𝜄, we may write the posterior distribution for 𝐻0 as 𝑃(𝑧, 𝐷𝐿, 𝜄, 𝐻0|x) ∝ 𝑃(𝐻0) [ ¯𝑁(𝐻0)]𝑁 × 𝑁 � 𝑖=1 𝑃(ˆ𝑧𝑖|𝑧𝑖, 𝐻0, 𝐷𝐿)𝑃( ˆ𝐷𝐿,𝑖, ˆ𝜄𝑖|𝐷𝐿,𝑖, 𝜄𝑖), (2) where ˆ𝑧𝑖, ˆ𝐷𝐿,𝑖, and ˆ𝜄𝑖 are the estimates of 𝑧𝑖, 𝐷𝐿,𝑖, and 𝜄𝑖 for each merger, ¯𝑁 is the mean number of detected EM+GW mergers as a function of 𝐻0, and 𝑁 is the number of EM+GW mergers in the catalogue (Mortlock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' No analytic formula exists to produce ¯𝑁 as a function of 𝐻0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' for any given catalogue of mergers, each merger’s luminosity distance and inclination angle influence both the probability of GW detection and the probability of EM detection, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A brute force estimation of ¯𝑁 for a given 𝐻0 requires gener- ating the total number of mergers within a volume of space over some duration of time, and then determining which among them would be detected as a standard siren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The determination of de- tection requires simulating each merger in the catalogue and then comparing the expected GW and EM emission to both the response functions of the GW detector and the telescope tiling of the localiza- tion region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Since each merger has a unique luminosity distance and inclination angle, the number of parameters influencing the number of detected mergers in a given sample, 𝑁, scales with the number of mergers included within that sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Estimating ¯𝑁 at a particular 𝐻0 furthermore requires a dense sampling of this parameter space, and an estimation of ¯𝑁(𝐻0) requires dense sampling assuming var- ious values of 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Computational expenses mount as the required number of samples increases, rendering this approach prohibitively computationally expensive beyond a merger sample of more than a few mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We therefore adopt an SBI approach to estimating ¯𝑁(𝐻0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Within the SBI framework, we repeatedly sample the parameter space and simulate a sample of BNS mergers at each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The mock observables from these ‘forward simulations’ enable infer- ence of the parameter of interest, in this study, 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We train a neural network to compare the mock observables to real data and thereby learn the parameter values underlying that data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Our method employs marginal posterior inference, completely bypassing calcu- lation of the likelihood function and thus removing the need for explicit knowledge of ¯𝑁(𝐻0) (Talbot & Thrane 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='2 Layout of Approach Our SBI method follows a six-step approach: (i) Draw parameter sample Θ (ii) Generate realistic observables, x|x0, Θ (iii) Summarize consistency of generated observables with data using a summary statistic S = 𝑓 (x, x0) (iv) Repeat steps i-iii to produce data set (Θ, S) (v) Intermediate processing to prepare data set for training (vi) Train neural network to infer the marginal posterior of 𝐻0 from (Θ, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Below, we discuss each step in greater detail, proceeding in order from i to vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Figure 6 displays a schematic of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) 6 Gagnon-Hartman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Overview of our approach to inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' EM and GW data supplied to the simulator informs the parameter prior distributions set by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In practice, this should be real data, but in this study we use a simulated mock data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' At each step of simulation, BNS and cosmological parameters are sampled from these prior distributions and passed to a forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The forward model simulates the BNS mergers assuming these parameters and determines whether their GW and EM emission should be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This is done repeatedly to build a set of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The data then undergoes intermediate processing to allow for the correction of EM anisotropy before being passed into the MNRE for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A fully-trained MNRE outputs a marginal posterior distribution for 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Process overview of the forward model developed for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In the first step, the parameters necessary for BNS merger generation are sampled and computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The inclination angles, 𝜄𝐺 and 𝜄𝑆, as well as the Hubble constant, 𝐻0, are sampled from a prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' x represents the GW strain data, which is used to generate a random luminosity distance for GW-only mergers, 𝐷𝐿,𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Meanwhile, the luminosity distances of standard siren mergers, 𝐷𝐿,𝑆, is constrained by the sampled 𝐻0 and the measured redshifts, 𝑧𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The redshifts of GW-only mergers are similarly computed from 𝐷𝐿,𝐺 and 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' These parameters are summarized as Θ, which is passed to MOSFiT for light curve generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Each light curve is either observed or not observed according to some magnitude criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' If all EM+GW mergers are confirmed to have been EM-observed and all GW-only mergers are not EM-observed, then the simulated observations are said match the data, and a summary statistic is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Otherwise, the minimum, or ‘null’, summary statistic is returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In step (i), the forward model gathers a sample of parameters from their relevant prior space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' These parameters include the lumi- nosity distances and inclination angles of each BNS merger, as well as 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' During tests considering the impact of telescope follow-up we also include the sky location, (RA, DEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Table 1 summarizes the appropriate minimally-informative prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' After gathering the parameter sample Θ, the forward model generates the GW and EM observables for each event (step (ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Since training a neural network directly on kilonova light curves and merger GWs is difficult, we introduced a data compression scheme to aid training convergence (for a similar example in cosmology, see Alsing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We perform this compression in steps (iii) and (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Step (iii) computes the similarity between the simulated and actual observables, producing a quantity called the summary statistic S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In step (iv), the forward model repeatedly samples parameter space and generates observables, thus producing a data set of input parameters, Θ, and their associated summary statistics, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This data MNRAS 000, 1–14 (2022) sample from prior forward model build training data amplitude S GW EM xnl!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=" time marginal posterior train MNRE intermediate inference processing (0',s) (0'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content="S) (0, S) (°H) P Ho P(O'is)x light curve generation parameter generation kilonova (tg,." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=', ) simulator (us,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' s)* determination of observation Ho- compute summary (2s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='·, 2g)* statistic Di IF simulated ELSE observations match data return measured * - included inO S s() mir sampledDebiasing Standard Siren Cosmology 7 Parameter Prior Motivation 𝐻0 Uniform[60, 80] Treat as unconstrained 𝐷𝐿 𝑃(𝐷𝐿, 𝜄|x𝐺𝑊 ) Inferred from GW strain 𝜄 𝑃(𝐷𝐿, 𝜄|xGW) Inferred from GW strain (RA, DEC) 𝑃(RA, DEC|x𝐺𝑊 ) From GW LAL Inference Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The four free parameters considered in this study and their prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A GW detection providing strain xGW produces a joint posterior distribution for the luminosity distance and inclination angle, 𝑃(𝐷𝐿, 𝜄|xGW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This distribution is treated as the credible region for an merger’s 𝐷𝐿 and 𝜄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Similarly, a GW detection has an associated sky local- ization, 𝑃(RA, DEC|xGW), produced by LAL inference of the GW strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' set, (Θ, S), contains information on the posterior distributions of the input parameters, including 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In this work, we train a neural network to reconstruct the marginal posterior distribution 𝑃(𝐻0) us- ing the data set produced by the forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' However, we found that the noise inherent to our choice of summary statistic hindered training convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We therefore apply intermediate processing in step (v), wherein we recast the data to a basis more amenable to training convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This transformation varies with the global quantities of the dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=', maximum and minimum), so we ap- ply it after the completion of the sampling and forward modelling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Then, in step (vi), a marginal neural ratio estimator (MNRE) trained on these transformed data produces the marginal posterior distribution for 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3 Forward Model 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='1 Observable Generation The forward model generates observable data, x, given a parameter vector, Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Our input parameters consist of 𝐻0, an inclination angle 𝜄 for each merger, and a luminosity distance 𝐷𝐿 for each GW-only merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The observables in this study consist of two parts: GW emission and EM emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The GW strain for a BNS merger provides a joint estimate for its 𝐷𝐿 and 𝜄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Example joint (𝐷𝐿, 𝜄) distributions are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For an EM+GW merger, the measured redshift 𝑧 and sampled 𝐻0 specify a 𝐷𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We then sample an inclination an- gle 𝜄 from the corresponding conditional posterior distribution, 𝑃(𝜄|𝐷𝐿, 𝑥GW), where 𝑥GW represents the GW strain for that merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Meanwhile, a GW-only merger lacks a measured redshift, leaving both 𝐷𝐿 and 𝜄 as free parameters to sample from the joint poste- rior distribution 𝑃(𝐷𝐿, 𝜄|𝑥GW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Following these rules, the forward model fixes the parameters of all mergers in the sample prior to light curve generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For a graphical representation, see the parameter generation panel of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Before generating light curves, we produce GW observables for each merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The forward model uses GWToolbox to produce a GW strain and detection signal-to-noise ratio for each merger in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We count the GW emission of a BNS merger as detected if the merger’s SNR exceeds 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Only GW-detected events contribute to parameter inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Such events supply the GW-inferred joint posterior distribution for 𝐷𝐿 and 𝜄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The forward model then generates realistic EM emission (kilo- novae) for each GW-detected merger using the MOSFiT software package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' MOSFiT uses a number of parameters to specify the fea- tures of a kilonova, for a detailed description of these parameters see Nicholl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' While we sample 𝐷𝐿 and 𝜄 from their prior distributions and pass those samples to the light curve simulator, we fix all other relevant parameters to fiducial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Appendix A1 discusses our assumed values for the following kilonova pa- rameters: the kilonova ejecta component opacities, 𝜅red and 𝜅blue, the neutron star masses, the disk ejection fraction 𝜖disk, the blue ejecta enhancement factor 𝛼, and the geometric parameters 𝜃 and 𝜃𝑐, which respectively describe the half-opening angles of the blue ejecta component and gamma ray burst-shocked region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='2 Summary Statistic The summary statistic S represents the similarity of the simulated observables with the observed data, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Since the simulated observ- ables are generated from the parameter sample Θ, the summary statistic captures the consistency of Θ with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For samples where the simulated and observed data do not meet a minimum consistency threshold, we set the summary statistic to a minimum or ‘null’ value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We refer to such samples as ‘non- informative’ since they do not contribute positively to the inferred posterior density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Our minimum consistency check ensures that only the correct mergers within a sample have detected EM counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' To illustrate this ‘minimum consistency check’ with a basic example, consider a real set of 2 BNS mergers where 1 has a detected EM counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Following our procedure, we sample a parameter sample from prior space and produce mock GW and EM observables from that sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Should zero or both BNS mergers have detected EM counterparts, then we say that the simulated ob- servables are completely inconsistent with the real data, and a null return is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Furthermore, if the wrong merger has a detected EM counterpart in the simulated data, then that is also completely inconsistent, producing a null return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For samples where the minimum consistency threshold is met, we then assess the degree of consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We do this by comparing the sampled 𝐷𝐿 and 𝜄 to the (𝐷𝐿, 𝜄) joint posterior distribution inferred from each merger’s GW signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Appendix A2 discusses the exact form of this comparison, as well as our choice of minimum return for non-informative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3 EM Follow-Up We investigate how imperfect EM follow-up observation attempts influence our ability to correct for EM anisotropy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' To motivate this, consider a scenario where we can image the full sky to some exposure depth in some filter for several days after a BNS merger is detected in GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In this scenario, the only cause for non-detection of the kilonova is the flux from the kilonova does not meet the detection thresholds of the telescope images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Insufficient flux can only be explained by either the distance of the merger or its inclination, both features of the underlying BNS merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Therefore, the EM non- detection places some constraints on these BNS merger parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The existence of these constraints allows for the correction of EM anisotropy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Let us now consider the case where a realistic telescope follow- up is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' When a BNS merger is detected in GWs, LAL inference is used to localize its origin on the sky with a probability density map (Veitch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Telescope follow-up tiling of the localization region is not exhaustive, as they do not cover 100% of the sky, nor are the exposures always deep enough to guarantee kilonova detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This introduces new failure modes for EM de- tection: the possibility that an merger was not within the field of MNRAS 000, 1–14 (2022) 8 Gagnon-Hartman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' view of any images during the follow-up campaign, or the images were not of sufficient depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For any given EM non-detection, it is therefore unclear whether the non-detected was due to factors in- trinsic to the BNS merger (𝐷𝐿, 𝜄) or factors relating to the telescope follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='4 Intermediate Processing We refer to the process of repeatedly simulating observables from parameter samples as a ‘run’ of the forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A run of the forward model produces a table of input parameter vectors, Θ, and their associated summary statistics, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' These tables contain the information necessary to produce a marginal posterior distribution for 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' To correct for EM anisotropy bias, we use information from GW-only mergers to estimate the bias implicit in EM+GW merg- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We do this by inspecting the shift in summary statistics when we include all mergers in the sample versus when the sample only includes EM+GW mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Our approach to EM anisotropy correc- tion requires two full runs of the forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We refer to the process whereby we graft the output of one run onto the other as ‘intermediate processing’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In the first run, the data set includes only EM+GW mergers, and the forward model generates a sample of EM+GW summary statistics called SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In the second run, the data set includes GW- only mergers along with EM+GW mergers, and the forward model generates a second set of summary statistics called SGW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The null statistics in SS indicate Θ for which one or more mergers lack a detected EM counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Meanwhile, the null statistics in SGW may also indicate Θ for which one or more mergers erroneously have a detected EM counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This additional constraint marginalizes over the EM selection effect, debiasing the 𝐻0 posterior prescribed by the summary statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' However, the uncertainties in the GW- only mergers’ redshifts produce greater scatter in SGW than in S𝑆, effectively prescribing a much broader posterior distribution for 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We therefore process both sets of summary statistics into a modified summary statistic set S′, which possesses the unbiasedness of SGW and a precision nearing that of SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We discuss the details of this ‘intermediate processing’ step in Appendix A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='5 Inference Model The inference model learns the marginal posterior distribution of 𝐻0 from the set of input parameter vectors and their associated summary statistics produced by the forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We perform this inference using Marginal Neural Ratio Estimation (MNRE), a simulation-based inference approach (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' MNRE learns the marginal likelihood-to-evidence ratio for a parameter of interest from a set of training data consisting of input parameters and their associated model outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Appendix A4 describes in more detail neural ratio estimation, the operating principle behind MNRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We train the neural network using Adam (Kingma & Ba 2014) to minimize binary cross-entropy loss, as defined in Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Although MNRE tends to produce conservative posterior estimates once it converges, inappropriate selection of training hy- perparameters often leads to convergence issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=', Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Brehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2018, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' To produce reliable posteriors in each training, we adjusted the learning rate schedule and other hy- perparameters, as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We implement MNRE using the swyft software package (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Parameter EM-Only GW-Only Combined 𝑙𝑖 5 · 10−4 5 · 10−5 5 · 10−4 𝑓 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='5 𝑝 5 10 20 𝑒𝑚 25 50 100 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The training hyperparameters used in neural ratio estimation (NRE) for each test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' These are the initial learning rate 𝑙𝑖, the learning rate adjustment factor, 𝑓 , the patience, 𝑝, and the maximum number of epochs 𝑒𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The NRE begins training at the initial learning rate, which is then adjusted by the learning rate scheduler as training progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' At each step in training, a validation loss is calculated and passed to the learning rate scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' If 𝑝 rounds pass without any decrease in validation loss, then the learning rate is multiplied by the factor 𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This continues for 𝑒𝑚 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 5 RESULTS ON MOCK MERGER DATA SETS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='1 GW-only Correction In this test, 100 simulated BNS mergers were generated using GWToolbox assuming the realistic ‘cosmological’ parameter dis- tribution discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This sample of mergers suffers from GW anisotropy bias, as distant mergers are only detected if their inclination angles are nearly face-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This is illustrated in Fig- ure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For this test, the forward model was modified to not evaluate light curves at all, and to always assume that a GW-detected merger is also an EM+GW merger regardless of its luminosity distance or inclination angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This is equivalent to assuming that all mergers lie within the field of view of a telescope with infinite exposure depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The purpose of this modification is to ensure that GW anisotropy bias is corrected in the absence of EM anisotropy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In essence, this test is equivalent to the GW anisotropy bias correction test discussed in Gerardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Two sets of summary statistics are generated using this forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In one set, called Sb, GW anisotropy is not considered, and the full sample of 100 mergers are considered in each summary statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In the other set, called Sc, GW anisotropy is considered, so only a subset of the 100 mergers are considered in each summary statistic, and this subset varies in size and composition from one draw to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The first of these sets is used to produce a biased 𝐻0 posterior, while the second produces the corrected 𝐻0 posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' These posteriors are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The value of 𝐻0 pre- dicted by the biased distribution is 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='54+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='69 km s−1 Mpc−1, which indicates a GW anisotropy bias of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='54+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='69 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The cor- rected distribution yeilds 𝐻0 = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='76 km s−1 Mpc−1, which accounts for 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='33% of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The bias in our distribution is within two standard deviations of the mean bias computed by Ger- ardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021), indicating that our method for merger generation and bias measurement is consistent with their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The degree to which our method corrects for the bias in 𝐻0 is also comparable to their results, which produce corrected distributions with biases consistent with 0 to the 1-𝜎 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='2 EM-only Correction In this test, 10 BNS mergers occur at the same luminosity distance and redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This distance is set to 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='9 Mpc, which has a cor- responding redshift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='031 assuming 𝐻0 = 70 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Each merger is identical except for its inclination angle, 𝜄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Every neutron star is assumed to have a mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='4𝑀⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The inclination MNRAS 000, 1–14 (2022) Debiasing Standard Siren Cosmology 9 Biased Corrected Biased Corrected Biased GW- Corrected Fully- Corrected 66 68 70 72 74 76 78 80 H0 [km s 1Mpc 1] EM Correction Only GW Correction Only Combined Correction True H0 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The 𝐻0 posteriors produced by each of the first three tests in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The posteriors are coloured according to their test, and labelled according to their status as biased, corrected, or partially-corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In the EM-correction only and GW-correction only tests, two posteriors are produced, one biased and one corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In the combined tests, one with and one without sky-sampling considerations, three posteriors are produced, one biased, one with GW anisotropy bias corrected, and one with all biases corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' It is meaningless to correct for EM anisotropy bias without correcting for GW anisotropy bias if the data suffers from GW anisotropy bias, so those posteriors were not produced in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The combined test results pictured here correspond to the C1 test in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Biased GW-Corrected Fully-Corrected Biased GW-Corrected Fully-Corrected 66 68 70 72 74 76 78 80 H0 [km s 1Mpc 1] Realistic Follow-Up Unrealistic Follow-Up True H0 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The 𝐻0 posteriors produced by increasing the EM detection sensitivity on the combined EM+GW correction test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The posteriors are coloured according to their test, and labelled according to their status as biased, corrected, or partially-corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Even when realistic follow-up is considered, EM and GW anisotropy bias can be corrected for using our method, given that the EM detection sensitivity is sufficiently high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' With reference to Table 3, the unrealistic follow-up test corresponds to C3 and the realistic follow-up test corresponds to C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' angles of the mergers are evenly spaced in cos(𝜄), taking values from cos(𝜄) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 to cos(𝜄) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Of these, only the three mergers with the lowest 𝜄 are EM+GW mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' By placing these mergers at the same distance and assuming their GW emission are all detected, we ensure that the only parameters which determine an merger’s status as a standard siren are its inclination angle and 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In this way, we probe EM anisotropy bias in the absence of GW anisotropy bias, and can thus investigate the degree to which our method uniquely corrects for EM anisotropy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In determining the value of the bias, we must compare the mean of the inferred 𝐻0 posterior to some true value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' While in strict terms the true value of 𝐻0 is that assumed in the generation of the BNS mergers, it is inappropriate to use this as the baseline for the bias since it is not necessarily the value which would be inferred if all 10 BNS mergers were EM+GW mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Recall that EM anisotropy bias is the difference between the value of 𝐻0 inferred from a sample of EM+GW mergers and the value inferred a subset of those same mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The biased and corrected posteriors produced in this test are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Both the biased and corrected distributions exhibit a long tail to high 𝐻0, which is due to the low inclinations of the three EM+GW mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Even after our correction scheme is applied, some small probability density is still applied to these high 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The mean of the biased distribution is 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='17+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='31 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='36 km s−1 Mpc−1 while that of the corrected distribution is 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='01+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='75 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The corrected distribution accounts for 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='12% of EM anisotropy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This is greater than the correction level achieved for GW anisotropy bias in both Gerardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021) and the GW MNRAS 000, 1–14 (2022) 10 Gagnon-Hartman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The luminosity distances (𝐷𝐿) and inclination angles of the 100 GW-detected BNS mergers generated using GWToolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Mergers at large distances are unlikely to be detected unless they are nearly face-on (see Figure 3), thus resulting in GW anisotropy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Mergers which would be detected via EM follow-up with a 𝑔-band magnitude cutoff of 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3 are marked in red, and mergers which would be detected with a cutoff of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 are marked in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Test Name Exposure Depth (𝑚AB) Follow-Up Scheme C1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3 Perfect C2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3 GW190425+GOTO C3 25 Perfect C4 25 GW190425+GOTO C5 25 GW190814+CFHT Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The five validation tests performed to evaluate our method’s ability to correct for both EM and GW anisotropy biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' These tests vary the exposure depth of the assumed EM follow-up campaign as well as the coverage of the campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The form of the follow-up schemes listed is (GW localization)+(telescope), except when the follow-up is assumed to be perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A ‘perfect’ follow-up campaign assumes that the targeted merger lies within the imaged region of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The 𝐻0 posteriors inferred in C1 are shown in Figure 8 and those inferred in C3 and C5 are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' anisotropy-only test presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The correction itself manifests as a suppression of high 𝐻0 values, as is expected from our approach to EM anisotropy bias correction (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3 Combined Correction Tests We performed five tests to gauge the ability of our method to si- multaneously correct for EM and GW anisotropy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Each test considers the same sample of mergers used in the GW-only cor- rection test, shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In these tests, we vary the EM follow-up campaign specifications for the same 100 mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In two of these tests, the EM follow-up campaign is assumed to have total sky coverage in the 𝑧 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The tests differ by the exposure depth assumed, which is 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3 𝑚AB in one and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 𝑚AB in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For each exposure depth, we also test how our method can correct for bias if a specific GW localization and follow-up campaign is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We perform this using the GW localization of GW190425 and its GOTO follow-up campaign (Steeghs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Each test and its specifications is laid out in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='1 C1 and C2 In C1 and C2, we test the ability of our method to correct for EM and GW anisotropy bias when the detection threshold for EM follow-up is set to 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3 𝑚AB in the 𝑧-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Three posteriors are produced in C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' First, the EM+GW mergers are used to perform a biased inference on 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Then an intermediate correction is performed by activating the GW anisotropy correction method in the forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The final posterior is fully-corrected, accounting for both GW and EM selection effects in the forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' These posteriors are shown in purple in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We found that for this sample of EM+GW mergers, the bias was negative, with an inferred 𝐻0 of 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='58+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='61 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This is not surprising, and it is fairly common when the EM detection threshold is low compared to the magnitudes of BNS mergers at large distances (≳ 200 Mpc) where GW anisotropy bias becomes important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Gerardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021) demonstrates that negatively-biased EM+GW merger posteriors are common in 100-merger sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Applying GW anisotropy bias correction raises the inferred value of 𝐻0 to 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='89+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='47 km s−1 Mpc−1, correcting for 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='00% of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This demonstrates that our method for GW anisotropy bias correction works whether the bias is positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Including EM anisotropy changes the inferred value of 𝐻0 to 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='93+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='51 km s−1 Mpc−1, increasing the level of bias correction to 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In C2, the GW localization of GW190425 and the GOTO follow-up campaign for that merger is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Due to the poor localization coverage of the follow-up, this resulted in our method being unable to correct for EM anisotropy bias, although the de- gree of GW anisotropy bias correction remained the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This is sensible, since any given GW-only merger is far more likely to lie outside the EM follow-up coverage than to be too dim to be seen by the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='2 C3 and C4 C3 and C4 consider the same sample of 100 mergers as the other combined tests, while assuming that the EM follow-up detection sensitivity reaches 25 𝑚AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' C3 assumes that all mergers lie within the follow-up campaign’s sky coverage, while C4 assumes the sky localization of GW190425 and that merger’s GOTO EM follow- up campaign for all mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The posterior distributions for 𝐻0 produced in C3 is shown in Figure 9, referred to therein as the ‘unrealistic follow-up’ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' When perfect localization coverage is assumed (C3), the biased posterior of 𝐻0 is 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='49+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='51 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Once GW anisotropy correction is applied, this becomes 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='47+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='50 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='48 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' GW anisotropy correction in this sample of mergers acts to broaden the posterior significantly due uncertainty in the level of GW anisotropy bias itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Once EM anisotropy bias is also corrected for, the inferred value of 𝐻0 becomes 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='96+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='16 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='06 km s−1 Mpc−1, constituting a bias correction level of 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='12% at the cost of broadening the posterior by roughly a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Assuming realistic localization coverage (C4), the biased pos- terior of 𝐻0 is 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='61+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='90 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Applying GW anisotropy bias correction changes this to 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='38+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='96, constituting a 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='71% correction in the bias level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Applying the EM anisotropy bias cor- rection in addition to this changes the inferred value of 𝐻0 to 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='81+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='99 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='09, constituting a 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='85% reduction in the bias level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This MNRAS 000, 1–14 (2022) 90 GW-detected 80 EM+GW-detected 70 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3 mAB) Angle [ EM+GW-detected 60 (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 mAB) 50 Inclination 米米 40 30 20 10 米 100 200 300 400 500 600 D, [Mpc]Debiasing Standard Siren Cosmology 11 demonstrates that increasing the depth of follow-up imaging can allow for EM anisotropy bias correction even when the localization coverage is poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='3 C5 C5 is similar to C4, except that the sky localization of GW190814 is used along with that merger’s Canada-France-Hawaii Telescope follow-up campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The purpose of this test is to illustrate the qual- ity with which EM and GW anisotropy biases may be corrected for when exposures are deep and the EM follow-up campaign coverage is extensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In this test, the biased posterior of 𝐻0 is 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='40+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='22 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='14 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Once GW anisotropy correction is applied, this be- comes 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='23+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='96 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Once EM anisotropy bias is also corrected for, the inferred value of 𝐻0 becomes 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='14+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='20 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='34 km s−1 Mpc−1, constituting a bias correction level of 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='67%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The poste- rior distributions for 𝐻0 produced in this test are displayed in Figure 9, labelled therein as the ‘realistic follow-up’ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Due to the in- completeness of the follow-up campaign, all posteriors are broader than those produced in the case where perfect follow-up is assumed (C3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We find that bias correction in a regime where EM follow-up is nearly complete achieves a similar level of bias correction to C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 6 CONCLUSIONS In this study, we have demonstrated how simulation-based inference can be used to produce an unbiased measurement of 𝐻0 using mergers detected with EM+GW in addition to GW-only mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In doing so, we account for both GW and EM selection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Our GW anisotropy bias correction method matches the performance of the SBI method presented by Gerardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021) and further generalizes its inference to account for EM anistropy bias in standard siren measurements of 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The key to EM anisotropy correction is the inclusion of GW- only events – mergers whose GW signal is detected in the absence of a detected EM counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For many such mergers, the fact of EM non-detection places a constraint on the merger’s apparent magnitude, which in turn constrains the range of possible inclination angles and luminosity distances for the merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' These constraints in turn act to temper, and in many cases fully remove, EM anisotropy bias in the inferred value of 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We found that the inclusion of EM anisotropy correction in scenarios where EM anistropy bias is negligible can reduce the bias correction level of our method by a few percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' However, this slight reduction in correction efficacy in low-EM anisotropy bias scenarios is outweighed by the high efficacy of EM anisotropy bias correction in high-EM anisotropy bias scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Without an ab initio method of determining whether a population of BNS mergers is significantly EM anisotropy biased, it is safest to include EM anisotropy correction in the analysis of a merger sample alongside GW anisotropy correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The tests which consider the GW footprint of GW190425 and its GOTO EM follow-up campaign (C3 and C4) illustrate the impor- tance of a comprehensive EM follow-up campaign for each merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In these tests, the probability of a merger not lying within the EM follow-up region is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' When the probability that a GW- only merger only lacks a detected EM counterpart because it was not localized within any telescope pointings is high, only weak constraints can be placed on that merger’s inclination angle and luminosity distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Such was the case for GW190425, where the localization was poor and the GOTO campaign only had ∼25% localization coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This in turn weakens the correction level on the EM anisotropy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Thus, a maximally-informative EM follow- up campaign should seek to maximize its coverage of the BNS merger’s sky localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We make this point by including a test where we consider a more complete EM follow-up scenario, the Canada-France-Hawaii Telescope follow-up of GW190814 (C5), wherein the bias correction level is comparable to that of perfect localization coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Future extensions of this work should focus on more realis- tic inference scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For example, in this work we assumed the neutron star masses of each merger to be precisely measured from the GWs, when in reality a GW measurement provides a joint prob- ability distribution for the merger’s neutron star masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We also assumed the same EM follow-up routine for each merger with the same underlying sky localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A more realistic treatment of EM follow-up should consider the real sky localizations of each merger with merger-specific EM follow-up routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Future work should also address the uncertainty in the kilonova parameters themselves, as incorrectly assuming their values can lead to either insufficient or overzealous bias correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The significance of the biases addressed in this study will only increase as more BNS mergers are detected in the coming years, underscoring the importance of a reliable bias correction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' SBI-based approaches such as ours enable cosmologists to take full advantage of the incoming deluge of BNS merger detections to produce unbiased measurements of the Hubble constant, and possibly resolve the Hubble tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank Sabrina Berger, Michael Matesic, Carter Rhea, Jason Rowe, Nicholas Vieira, and Clovis Vinant-Tang for help- ful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC) through their Canada Graduate Scholarships - Master’s programme, as well as from the Bishop’s University Foundation through their Graduate Entrance Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' acknowledge sup- port from the Canada Research Chairs (CRC) program, the NSERC Discovery Grant program, the FRQNT Nouveaux Chercheurs Grant program, and the Canadian Institute for Advanced Research (CI- FAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' acknowledges support from the Canada Foundation for Innovation (CFI), and the Québec Ministère de l’Économie et de l’Innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Computations were performed on the Cedar and Béluga supercomputing clusters managed by Compute Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article is available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='13662) APPENDIX A: METHODS SUPPLEMENTAL MATERIAL A1 Kilonova Parameters We generate kilonova light curves using the MOSFiT software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We assume that kilonovae have two merger ejecta components, one red and one blue (Metzger 2019), with opacities 𝜅red and 𝜅blue respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We use fiducial values of 𝜅red = 10 cm2 g−1 and 𝜅blue = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='5 cm2 g−1, following the results of Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A kilonova’s peak luminosity scales with the quantity of ejecta produced in the NS merger, and thus to the masses of the NS themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Our light curve simulation requires as inputs the BNS merger chirp mass, M = � (𝑚1𝑚2)3/5 (𝑚1 + 𝑚2)1/5 � , (A1) and NS mass ratio, 𝑞 = 𝑚1/𝑚2, where 𝑚1 < 𝑚2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We compute these quantities from the true NS masses provided by GWToolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The luminosities of red and blue components also depend on their temperatures, which are influenced by the disk ejection frac- tion, 𝜖disk, and the blue component enhancement factor 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 𝜖disk governs the fraction of the BNS remnant accretion disk ejected post-merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We use a fiducial value of 𝜖disk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='15, while Nicholl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021) state that the true value may range anywhere from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='05 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='5, following Metzger (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Meanwhile, surface winds from the merger remnant enhance the temperature of the blue ejecta, en- capsulated by 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Nicholl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021) assign 𝛼 a flat prior from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0, while we set it to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='0 (maximum enhancement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Finally, the geometry of the kilonova influences which com- ponent we actually observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' In MOSFiT, kilonova geometry is sim- plified into two distinct angular regions defined by the half-opening angle of the blue component, 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We follow Nicholl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (2021) by setting 𝜃 = 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' BNS mergers also produce an associated gamma- ray burst (GRB), which shocks ejecta material in its vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For the purposes of this study, we assume that the GRB shock negligibly effects the blue component, thus setting the half-opening angle of the GRB shock to 𝜃𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We do this since the half-opening angle of the shocked cone is uncertain and its physics are poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A2 Summary Statistic We measure the consistency of Θ with x explicitly using the (𝐷𝐿, 𝜄) distributions of each GW-detected merger as measured from its gravitational wave emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The likelihood of guessed parameters 𝐷𝐿 and 𝜄 for any particular merger are taken as L = 𝑃(𝐷𝐿, 𝜄|𝑥GW) MNRAS 000, 1–14 (2022) Debiasing Standard Siren Cosmology 13 where 𝑥GW is the gravitational wave strain for that merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Each merger’s likelihood contributes to the informative summary statistic S, defined as S = 𝑁 ∑︁ 𝑖 log(L(𝐷𝐿,𝑖, 𝜄𝑖|𝑥GW,𝑖)), (A2) or the sum of log-likelihoods of each merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We use this definition of S since it is based on an analytic likelihood, allowing the neural ratio estimator to easily learn the true likelihood-to-evidence ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Furthermore, variance in S originates entirely from uncertainty in the proposed parameters 𝐷𝐿,𝑖 and 𝜄𝑖, meaning that our selected summary statistic does not introduce any new biases that are not already present in the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This follows the principles of summary statistic selection laid out by Raynal & Onnela (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Our choice of the minimum value of S for null returns follows from the form of Equation A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Since the neural ratio estimator is trained to maximize the log-likelihood of the target parameter, a null return must be lower than the lowest informative return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' To this end, we set a minimum allowable likelihood for each merger within a sample, Lmin = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We choose this as the minimum likelihood since it is equivalent to the likelihood of sampling 𝐷𝐿 and 𝜄 from a 2D uniform distribution, given that our likelihood distributions have a resolution of 1000×1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' As such, any sampled (𝐷𝐿, 𝜄) with this likelihood or below is no more informative than a sample drawn from a uniform distribution, and we therefore deem it ‘non-informative’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' If a parameter sample is non-informative for all mergers in a data set (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=', if S < −6 · 𝑁) then the summary statistic is automatically returned as null, or Snull = −6 · 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A careful inspection of Equation A2 reveals that its maximum and minimum vary with the number of mergers in a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Fur- thermore, only a fraction of all BNS mergers within a sample have detected GWs in any given sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' To maintain consistent bounds for the summary statistic, we treat each merger not detected in GWs as contributing their maximum log-likelihood to the summary statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Thus, the summary statistic for each sample is computed as S = 𝐷 ∑︁ 𝑖 𝑙𝑖(Θ|𝐷𝐿,𝑖, 𝜄𝑖) + 𝑁 ∑︁ 𝑖 max(𝑙𝑖(Θ)), (A3) where 𝐷 is the number of GW-detected mergers, 𝑁 is the number of mergers not detected in GWs, 𝑙𝑖(Θ) is the log-likelihood of an merger indexed 𝑖 with the parameter sample Θ, and (𝐷𝐿,𝑖, 𝜄𝑖) are the luminosity distance and inclination angle sampled for the merger 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A3 Intermediate Processing Producing an unbiased 𝐻0 posterior from SGW is not as simple as training another MNRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This is because the nonzero returns in SGW are noise-dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Each summary statistic S is the sum of log-likelihoods for each merger given the sampled parameter vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' EM+GW mergers are governed by only two free variables: 𝐻0 and 𝜄, the former of the two is shared for all mergers within a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This means that at a particular 𝐻0, the variance in log-likelihoods is en- tirely determined by the uncertainty in 𝜄 for each merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The redshift of a EM+GW merger is known from EM observations, fixing the lu- minosity distance when a cosmology is assumed ((𝐻′ 0, 𝑧) −→ 𝐷′ 𝐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Meanwhile, GW-only mergers are governed by an additional pa- rameter, 𝐷𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Without a known redshift, the luminosity distance of a GW-only merger is free to be sampled independently of 𝐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This means that for a particular 𝐻0, the variance in the log-likelihood is driven by both the uncertainty in 𝐷𝐿 and the uncertainty in 𝜄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We find that this drives the variance in nonzero returns to a point where an MNRE is no longer capable of recovering the underlying pattern, rendering direct posterior inference impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' With nonzero returns rendered useless, we are forced to look to the zero returns for answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Zero returns are not non-informative: they show the values of 𝐻0 which are forbidden given a sample of mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Taking the ratio between the number of zero returns and the number of total samples as a function of 𝐻0, the ‘rejection’ rate, or 𝑅(𝐻0), may be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Any set of summary statistics has an associated rejection rate function, and we denote the rejection rate produced from SS as 𝑅S(𝐻0), and that produced from SGW as 𝑅GW(𝐻0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' These rejection rates differ from one another, as including GW-only mergers increases the selectiveness of the forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Crucially, the selectiveness does not change uniformly with 𝐻0, but follows some functional relationship encoded within 𝑅GW(𝐻0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The objective in correcting for EM anisotropy bias is thus to modify the set SS such that it accounts for the information in 𝑅GW(𝐻0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A responsible modification to SS is one that does not introduce any information beyond what is encoded in GW-only mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' The most obvious modification is to apply 𝑅GW(𝐻0) to SS by changing some portion of the informative summary statistics to zero returns according to the target rejection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This does not change the posterior learned from the distribution since the MNRE explicitly ignores null returns in favour of informative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Therefore, the solution must modify the informative samples in SS without setting them to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' We define the modified set of summary statistics as the product of SS and 𝑅GW(𝐻0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' S′ S = SS|SGW = 𝑅GW(𝐻0) · SS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (A4) This modified set of summary statistics, S′ S, includes all in- formation granted by the EM+GW mergers, while also incorporat- ing the rejection rate required by GW-only mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' 𝐻0 posteriors learned from this modified set correct for the bias introduced by EM anisotropy without broadening the distribution by including noisy GW-only mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' A4 Neural Ratio Estimation Neural ratio estimation (NRE) involves training a classifier network 𝑑𝜙 : Θ × 𝑋 −→ [0, 1] to discriminate between pairs (𝜃, 𝑥) sampled from the joint posterior distribution 𝑝(𝜃, 𝑥) and the product of the marginals 𝑝(𝜃)𝑝(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' By the Neyman-Pearson lemma, the proba- bility that a data pair belongs to the joint posterior is proportional to the posterior density (Neyman & Pearson 1933).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' Formally, the neural network optimizes 𝜙∗ = arg min 𝜙 E 𝑝(𝜃,𝑥) 𝑝(𝜃′)[L(𝑑𝜙(𝜃, 𝑥)) + L(1 − 𝑑𝜙(𝜃′, 𝑥))], (A5) where L(𝑝) = − log 𝑝 is the negative log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' For this task, the Bayes optimal classifier uses the decision function 𝑑(𝜃, 𝑥) = 𝑝(𝜃, 𝑥) 𝑝(𝜃, 𝑥) + 𝑝(𝜃)𝑝(𝑥) , (A6) MNRAS 000, 1–14 (2022) 14 Gagnon-Hartman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' which defines the likelihood-to-evidence (LTE) ratio 𝑟(𝜃, 𝑥) = 𝑑(𝜃, 𝑥) 1 − 𝑑(𝜃, 𝑥) = 𝑝(𝜃, 𝑥) 𝑝(𝜃)𝑝(𝑥) = 𝑝(𝑥|𝜃) 𝑝(𝑥) = 𝑝(𝜃|𝑥) 𝑝(𝜃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' (A7) Thus, NRE grants us an estimator log 𝑟𝜙(𝜃, 𝑥) = logit(𝑑𝜙(𝜃, 𝑥)) of the LTE log-ratio and a surrogate ˆ𝑝(𝜃|𝑥) = 𝑟𝜙(𝜃, 𝑥)𝑝(𝜃) for the posterior density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE4T4oBgHgl3EQfvQ3k/content/2301.05241v1.pdf'} diff --git a/q9E2T4oBgHgl3EQf0wi3/content/tmp_files/2301.04145v1.pdf.txt b/q9E2T4oBgHgl3EQf0wi3/content/tmp_files/2301.04145v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb55e6023e5e74abb6f0c9ac239d6183f6d6908b --- /dev/null +++ b/q9E2T4oBgHgl3EQf0wi3/content/tmp_files/2301.04145v1.pdf.txt @@ -0,0 +1,1431 @@ +MNRAS 000, 1–11 (2023) +Preprint 12 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Are the fates of supermassive black holes and galaxies determined by +individual mergers, or by the properties of their host haloes? +Jonathan J. Davies,1★ Andrew Pontzen1† and Robert A. Crain,2 +1Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK +2Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +The fates of massive galaxies are tied to the evolution of their central supermassive black holes (BHs), due to the influence of +AGN feedback. Correlations within simulated galaxy populations suggest that the masses of BHs are governed by properties +of their host dark matter haloes, such as the binding energy and assembly time, at a given halo mass. However, the full picture +must be more complex as galaxy mergers have also been shown to influence the growth of BHs and the impact of AGN. In +this study, we investigate this problem by using the genetic modification technique to adjust the assembly history of a Milky +Way-like galaxy simulated with the EAGLE model. We change the halo assembly time (and hence the binding energy) in the +absence of any disruptive merger events, and find little change in the integrated growth of the BH. We attribute this to the angular +momentum support provided by a galaxy disc, which reduces the inflow of gas towards the BH and effectively decouples the +BH’s growth from the properties of the halo. Introducing major mergers into the assembly history disrupts the disc, causing +the BH to grow ≈ 4× more massive and inject feedback that reduces the halo baryon fraction by a factor of ≈ 2 and quenches +star formation. Merger events appear to be essential to the diversity in BH masses in EAGLE, and we show that they can also +significantly increase the halo binding energy, potentially explaining the correlation between these quantities. +Key words: galaxies: formation – galaxies: evolution – galaxies: haloes – (galaxies:) quasars: supermassive black holes – +methods: numerical +1 INTRODUCTION +Feedback from active galactic nuclei (AGN) is a near-ubiquitous +ingredient of modern galaxy formation models, responsible for reg- +ulating the growth of galaxies at and above the mass of the Milky +Way, quenching star formation in massive galaxies, and maintaining +quiescence in the central galaxies of group and cluster haloes by sup- +pressing cooling flows (e.g. Bower et al. 2006; Croton et al. 2006; +Sijacki et al. 2007; Somerville et al. 2008; Vogelsberger et al. 2014; +Schaye et al. 2015; Dubois et al. 2016; McCarthy et al. 2017; Kaviraj +et al. 2017; Tremmel et al. 2017; Henden et al. 2018; Weinberger +et al. 2018; Davé et al. 2019). +Cosmological simulations predict that AGN feedback has a mini- +mal impact on lower-mass galaxies, as outflows associated with star +formation are able to remove gas from the centre of the galaxy and +prevent the growth of the supermassive black hole (SMBH). How- +ever, above a critical halo mass scale corresponding to that of 𝐿★ +galaxies, the entropy of the shock-heated circumgalactic medium +(CGM) exceeds that of these outflows; this confines gas to the +galaxy centre and allows the SMBH to grow and begin influenc- +ing the galaxy-halo ecosystem (Dubois et al. 2015; Bower et al. +2017; McAlpine et al. 2018; Keller et al. 2020; Habouzit et al. 2021; +Truong et al. 2021). +Above this critical mass scale, the EAGLE, IllustrisTNG and +★ E-mail: astrojdavies@gmail.com +† E-mail: a.pontzen@ucl.ac.uk +SIMBA simulations exhibit diversity in the properties of SMBHs, +galaxies and their CGM. Haloes in which AGN feedback has had +little impact tend to be gas-rich and host star-forming central galax- +ies, whereas haloes hosting overmassive SMBHs that have injected +a lot of AGN feedback energy tend to be gas-poor and host quenched +galaxies (Davies et al. 2019, 2020; Davé et al. 2019; Terrazas et al. +2020; Appleby et al. 2021; Robson & Davé 2021; Sorini et al. 2022). +Understanding why the impact of AGN feedback varies in haloes of +a given mass is therefore key to understanding why diversity exists +in the properties of the ∼ 𝐿★ galaxies in these simulations. +One possible origin of this diversity could lie in differences in the +underlying binding energies (and/or concentrations) of dark matter +haloes (Booth & Schaye 2010, 2011). This idea can be understood +as a consequence of self-regulation; in a more tightly-bound halo, +a SMBH must grow more massive and inject more AGN feedback +energy to expel gas from the halo centre. In the EAGLE and Illus- +trisTNG simulations, haloes with higher binding energies tend to +host more massive SMBHs, providing evidence for this connection +(Davies et al. 2019, 2020). The binding energy of a halo, in turn, +is assumed to be set by its assembly time (e.g. Neto et al. 2007), a +characteristic that is determined by the halo’s initial conditions. +On the other hand, observational evidence is emerging for a con- +nection between AGN feedback and galaxy mergers (e.g. Ellison +et al. 2011, 2019; Satyapal et al. 2014), and simulations have long +predicted that such events can enhance the growth of SMBHs and +the impact of AGN feedback (Di Matteo et al. 2005; Hopkins et al. +© 2023 The Authors +arXiv:2301.04145v1 [astro-ph.GA] 10 Jan 2023 + +2 +J. J. Davies et al. +2006, 2010; Sijacki et al. 2007, 2015; Bellovary et al. 2013; Dubois +et al. 2015; Pontzen et al. 2017; Steinborn et al. 2018; McAlpine +et al. 2020); this merger-induced feedback could explain recent ob- +servations of a quenching excess in post-merger galaxies (Ellison +et al. 2022). Recently, Davies et al. (2022) used a controlled galaxy +formation experiment to show that differences in the stellar mass +ratio (and hence the disruptive influence) of a single merger can have +dramatic effects on the growth of the SMBH at the centre of a galaxy, +transforming the baryon cycle, the properties of the CGM, and the +star formation activity in the central galaxy. +These lines of evidence suggest that the properties of galaxy- +CGM ecosystems depend on both the overall assembly time of the +host halo and on individual disruptive events that occur throughout +the system’s assembly. Identifying the relative importance of these +factors is challenging, as they are likely to be degenerate; haloes that +assemble early tend to reside in more densely clustered environments +(e.g. Sheth & Tormen 2004; Gao et al. 2005; Wechsler & Tinker +2018), and may therefore assemble by undergoing many disruptive +mergers, while later-assembling haloes may have comparatively quiet +histories. +In this study, we perform a controlled galaxy formation experi- +ment using the genetic modification technique (Roth et al. 2016) to +independently assess the role of each of these factors, and unveil +how the assembly history of a dark matter halo is connected to the +evolution of its central galaxy. +2 METHODS +For this study, we have performed a suite of simulations using the +EAGLE version of the gravity and smoothed-particle hydrodynamics +code gadget3 (last described by Springel 2005). Using the ‘zoom’ +technique (e.g. Katz & White 1993; Bertschinger 2001) we simulate +the evolution of an individual galaxy and its local environment at high +resolution, whilst also following the large-scale forces acting on the +system by simulating its wider environment with a low-resolution +periodic volume. In this section, we explain the selection of our +candidate galaxy (Section 2.1), outline how we modify its initial +conditions to adjust its assembly history (Section 2.2), and describe +how we identify and characterise galaxies and haloes within our +simulations (Section 2.3). +We also utilise the flagship EAGLE simulation volume (Ref- +L100N1504) in Section 3.3 to place our genetically-modified galaxies +into the context of the wider population. For detailed descriptions of +this simulation, we refer the reader to the EAGLE reference articles +(Schaye et al. 2015; Crain et al. 2015; McAlpine et al. 2016). +2.1 Construction and evolution of initial conditions +We selected our fiducial galaxy from a periodic simulation volume +evolved with the Reference EAGLE simulation model (Schaye et al. +2015; Crain et al. 2015) from uniform-resolution initial conditions +(ICs) generated by genetIC (Stopyra et al. 2020). We note therefore +that the galaxy was not selected from any of the publicly-available +EAGLE simulations; selecting a galaxy evolved from ICs created +by genetIC simplifies the subsequent genetic modification of these +ICs. This simulation is 50 comoving Mpc on a side, containing 5123 +dark matter particles of mass 3.19 × 107 M⊙ and an initially equal +number of baryonic particles of mass 5.6 × 106 M⊙ (a similar mass +resolution to that of the flagship EAGLE simulations) and adopts the +Planck Collaboration et al. (2016) cosmological parameters. +From this simulation, we selected a present-day star-forming disc +galaxy of stellar mass 𝑀★ = 4.3 × 1010 M⊙, the central galaxy of a +halo of mass 𝑀200 = 3.4×1012 M⊙. We selected this galaxy because +it lies on the star-forming main sequence (sSFR= 10−10.2 yr−1) and +has a CGM mass fraction1 𝑓CGM = 0.31 𝑓 cosmic +b +(where 𝑓 cosmic +b +is +the cosmic baryon fraction, Ωb/Ω0) that is close to the present-day +median 𝑓CGM at this halo mass in the largest EAGLE volume (Davies +et al. 2019). This galaxy is ideal for our purposes, as it resides in a +halo of a mass that far exceeds the critical mass, 𝑀crit +200, above which +BHs are able to grow efficiently in the EAGLE model (Bower et al. +2017; McAlpine et al. 2018), and has a simple merger history after +the halo exceeds this mass, with only one minor merger of stellar +mass ratio2 𝜇 = 0.17 occurring at 𝑧 ≈ 0.74. We will henceforth refer +to this galaxy and its halo as our organic system. +We generate zoomed initial conditions for this system by selecting +all particles within three virial radii3 of the galaxy (at 𝑧 = 0) and +identifying the Lagrangian region defined by these particles in the +ICs (at 𝑧 = 99). We then refine this region with a factor of 27 more +particles, and downsample the simulation volume outside this region +by a factor of 8, yielding particle masses of 𝑚gas = 2.19 × 105 M⊙, +𝑚dm = 1.18×106 M⊙, and 𝑚lr = 3.02×108 M⊙ for gas, dark matter +and low-resolution particles respectively. +We evolve these initial conditions with the EAGLE model, adopt- +ing the recalibrated (Recal) parameter values for the subgrid physics +as defined by Schaye et al. (2015) as these were calibrated for +a near-identical mass resolution to that of our initial conditions +(𝑚gas = 2.26 × 105 M⊙, 𝑚dm = 1.21 × 106 M⊙). The details of +this model and its calibration may be found in the EAGLE reference +articles (Schaye et al. 2015; Crain et al. 2015) and for brevity we do +not repeat them here. However, it is important to note that models +such as EAGLE include stochastic elements; processes such as the +conversion of gas particles to star particles and the injection of feed- +back energy are governed by the drawing of quasi-random numbers +that are compared to probabilities set by the properties of the gas (see +Schaye & Dalla Vecchia 2008; Dalla Vecchia & Schaye 2012). This +stochasticity can cause significant uncertainty in the properties of in- +dividual systems (see e.g. Genel et al. 2019; Keller et al. 2019; Davies +et al. 2021, 2022; Borrow et al. 2022). Since the zoom simulations +in this study are relatively inexpensive to perform, we simulate each +set of initial conditions in our experiment with nine random number +seeds each to quantify this uncertainty. +2.2 Producing genetically-modified galaxies +To adjust the assembly history of the organic galaxy, we use the +genetic modification (GM) technique (Roth et al. 2016; Pontzen et al. +2017) and the genetIC software to generate modified sets of ICs for +the system. From the overdensity field in the original ICs, genetIC +finds the closest possible field that also satisfies certain constraints, +which we design to produce our desired changes to the halo assembly +history. This technique preserves the large-scale environment of the +system, and the modified fields remain consistent with a Λ cold dark +matter (ΛCDM) cosmology. To assess the role of individual merger +events and the overall assembly history of the system independently +1 We define 𝑓CGM ≡ 𝑀CGM/𝑀200, where 𝑀CGM is the total mass of all gas +within the virial radius that is not star-forming. +2 We define the merger stellar mass ratio 𝜇 ≡ 𝑀infall +★ +/𝑀★, where 𝑀infall +★ +and +𝑀★ are the stellar masses of the infalling and primary galaxy respectively. +3 We define the virial radius, 𝑟200, as the radius of a sphere enclosing 200 +times the critical density of the Universe. +MNRAS 000, 1–11 (2023) + +The origin of diversity in SMBH and galaxy growth +3 +of each other, we generate four complementary sets of modified ICs +using this method. +First, to examine the influence of merger events at a fixed assembly +time, we produce a pair of modified ICs, secular and merger, +designed to decrease or increase the stellar mass ratio of the organic +system’s 𝑧 ≈ 0.74 minor merger respectively. This is achieved by +identifying the particles bound to the infalling halo at an earlier time +(𝑧 = 1.73)4, tracing these particles back to their locations in the ICs, +and decreasing or increasing the mean overdensity, ¯𝛿, in the patch +of the field defined by these locations. To preserve the overall mass +accretion history we also apply two further constraints; ¯𝛿 in the patch +defined by particles that comprise the main halo at 𝑧 = 1.73 is kept +fixed, as is ¯𝛿 in the patch corresponding to the 𝑧 = 0 halo to ensure +that the same final halo mass is reached. +We also test the influence of the overall assembly time indepen- +dently of merger events, by comparing the evolution of galaxies that +have differing assembly times and experience no significant mergers +that would be able to drive black hole (BH) growth. To do so, we +compare the secular system with another modified variant of our +galaxy, which assembles earlier and experiences no significant merg- +ers after the 𝑀crit +200 threshold is reached. We produce this behaviour +by assembling more mass into the main progenitor at early times +(𝑧 = 2) whilst keeping the final halo mass fixed, which has the effect +of both accelerating the halo assembly and reducing the significance +of all 𝑧 < 2 mergers. This is achieved in practice by increasing ¯𝛿 in +the patch of the ICs corresponding to the 𝑧 = 2 halo, while keeping +¯𝛿 fixed within the patch corresponding to the 𝑧 = 0 halo. We refer +to the system evolved from these conditions as the early-secular +system. +Finally, so that we can assess how the early-secular system +would evolve if it had a more disruptive evolution after exceeding +𝑀crit +200, we further modify the early-secular ICs to increase ¯𝛿 in +a patch corresponding to an infalling system at 𝑧 = 3 with the aim +of inducing a subsequent major merger of similar mass ratio to that +in the merger system. We refer to the system evolved from these +conditions as the early-merger system. +2.3 Identifying and characterising galaxies and their haloes +Haloes are identified on-the-fly in our simulations by applying the +friends-of-friends (FoF) algorithm to the dark matter distribution, +with a linking length of 0.2 times the mean interparticle separation. +Gas, star and BH particles are then assigned to the FoF halo of their +nearest dark matter particle. In post-processing, we then identify +bound haloes using the subfind algorithm (Springel et al. 2001; +Dolag et al. 2009), and use the analysis packages pynbody (Pontzen +et al. 2013) and tangos (Pontzen & Tremmel 2018) to calculate the +properties of galaxies and their haloes. +We use tangos to construct merger trees that link haloes to their +progenitors and descendants based on the number of particles they +have in common. Starting with our organic system, we identify the +main branch of the tree by calculating the sum of the stellar masses +along each possible branch, and then selecting the branch with the +largest value. We then identify the organic system’s counterparts in +other simulations by matching on the number of particles in com- +mon at 𝑧 = 8, and then tracking this system forwards in time. This +yields a more stable and reliable matching between simulations than +4 We choose this time as it corresponds to the final snapshot output in which +the merging haloes are clearly distinguishable by the SUBFIND algorithm. +attempting to find the organic system’s counterpart at each output +time separately. +We calculate the properties of haloes, such as the halo mass (𝑀200) +and baryon fraction ( 𝑓b ≡ 𝑀b/𝑀200, where 𝑀b is the total mass in +baryons) within one virial radius of the halo centre of mass. We find +the centre of mass with pynbody, using the shrinking-sphere method +(Power et al. 2003). The properties of galaxies, such as the stellar +mass (𝑀★) and specific star formation rate (sSFR) are calculated +within a spherical aperture of radius 30 physical kpc about the centre +of mass. +We calculate the intrinsic inner-halo binding energy (𝐸DMO +2500 , i.e. +that which is set by the halo’s assembly history and initial conditions, +and not by dissipative baryonic processes) by matching each system +to its counterpart in an equivalent dark matter-only simulation and +summing the binding energies of all particles within a radius en- +closing 2500 times the critical density. For our zoom simulations, +we perform this matching using tangos, and for the large-volume +Ref-L100N1504 simulation we use the bijective particle matching +algorithm described by Schaller et al. (2015). +We calculate the total energy injected through stellar feedback, +𝐸★, by summing the energies contributed by all star particles within +a 30 pkpc aperture about the centre of mass; when a gas particle 𝑖 is +converted into a star particle it provides an energy given by +𝐸★,𝑖 = 1.74 × 1049 erg +� 𝑚init +★,𝑖 +1 M⊙ +� +𝑓th,𝑖(𝑛H,𝑖, 𝑍𝑖), +(1) +where 𝑚init +★,𝑖 is the initial stellar mass and 𝑓th,𝑖 is an efficiency that +depends on the density 𝑛H,𝑖 and metallicity 𝑍𝑖 of the gas particle +at the time of conversion (for more information see Schaye et al. +2015, Section 4.5). The total feedback energy injected through AGN +feedback by the galaxy’s central BH is given by +𝐸AGN = +𝜖f𝜖r +1 − 𝜖r +(𝑀BH − 𝑀seed +BH )𝑐2, +(2) +where 𝑀BH is the BH mass, 𝑐 is the speed of light, 𝜖r = 0.1 is the +radiative efficiency of the accretion disc and 𝜖f = 0.15 is the fraction +of the radiated energy that couples to the surrounding gas. 𝑀seed +BH = +105 M⊙/ℎ is the mass of the BH seed, which does not contribute +any feedback energy. We note that these definitions include “ex-situ" +energy injected by stars and BHs in other progenitor galaxies that +subsequently merged with our main system, but since this energy +directly impacts the properties of these progenitors we include it in +our definition. +3 RESULTS +We first examine the impact of a varying halo assembly time in the +absence of merger events in Section 3.1, before exploring how our +systems evolve when mergers do occur in Section 3.2. Finally, in +light of our findings, we discuss the origin of previously-identified +correlations in the EAGLE galaxy population in Section 3.3. +3.1 The impact of halo assembly time in the absence of merger +events +We begin by isolating the effect of a varying assembly time on the +evolution of our system. We do so by comparing the evolution of the +secular and early-secular haloes, which have different assembly +times and experience no significant mergers after their masses exceed +𝑀crit +200, the threshold above which disruptive events may lead to BH +growth. +MNRAS 000, 1–11 (2023) + +4 +J. J. Davies et al. +3.1.1 Halo and galaxy evolution +Figure 1 shows how the organic system and its secular and early- +secular variants evolve in terms of their halo mass (𝑀200, upper +panel), stellar mass (𝑀★, middle panel) and specific star formation +rate (sSFR≡SFR/𝑀★, integrated over the preceding 100 Myr5, lower +panel). In this and other figures of this type, we indicate the first output +after which a significant merger has occurred with vertical lines, +coloured to match the simulation in which the merger occurred. We +define significant mergers to be those in which the stellar mass ratio, +𝜇, is greater than 0.1. We quantify the uncertainty that arises from +EAGLE’s stochastic subgrid physics implementation by simulating +nine realisations of each set of ICs, varying the seed used for the +quasi-random number generator each time. Solid lines indicate the +median value at each output time, and shading shows the interval +between the third and seventh values in the rank-ordered distribution, +a good approximation of the interquartile range (IQR). +The halo mass histories of the organic and secular galaxies are +very similar. We indicate the evolution of 𝑀crit +200 with a dashed line; +both systems exceed this threshold at approximately the same time +(𝑡 ≈ 3.2 Gyr). The early-secular halo assembles more quickly, and +exceeds 𝑀crit +200 earlier, at 𝑡 ≈ 2 Gyr. As intended, all systems converge +to near-identical halo masses of 1012.52 M⊙ (organic, IQR= 0.02 +dex) and 1012.54 M⊙ (secular and early-secular, IQR= 0.02 and +0.01 dex respectively). The secular halo temporarily has a higher +mass than the others at 𝑧 ≈ 0.4 − 0.1; this is due to mass bound to +another halo passing within 𝑟200 of the main halo during this period. +This halo passes closer to the secular system than it does to the +organic and early-secular systems as an unintended side-effect +of our modifications; at its closest approach, its centre of mass lies +approximately at the virial radius of the main halo. +As a result of its earlier assembly time, the early-secular halo +is intrinsically more tightly-bound than the secular halo. When +simulated with purely collisionless dynamics, the binding energy +of the inner halo, 𝐸DMO +2500 , is 30% higher for the early-secular +system; this difference corresponds to ≈ 1𝜎 in the flagship EAGLE +simulation at this halo mass6. +The stellar mass histories of the organic and secular galaxies +also trace each other closely, but diverge slightly at 𝑡 = 7.1 Gyr, as a +result of the organic galaxy’s sole significant merger (after 𝑀crit +200 is +exceeded); a minor merger of stellar mass ratio 𝜇 = 0.17. The ex-situ +stellar mass contributed by this merger elevates the organic galaxy’s +stellar mass above that of its secularly-evolving counterpart, but by +the present day the secular system catches up through predomi- +nantly in-situ star formation, and the organic and secular galaxies +reach stellar masses of 1010.67M⊙ (IQR= 0.07 dex) and 1010.61M⊙ +(IQR= 0.04 dex) respectively. The early-secular galaxy attains a +similar final stellar mass of 1010.66M⊙ (IQR= 0.07 dex), but as- +sembles the majority of that mass earlier as a result of an accelerated +halo assembly history. Neither the secular or early-secular galax- +ies experiences any significant mergers after the halo mass exceeds +𝑀crit +200, and therefore by comparing the properties of these systems we +may examine the influence of the overall halo assembly time in the +absence of mergers. +In the lower panel of Figure 1 we show how these assembly his- +tories influence the specific star formation rates of our galaxies. We +5 Our results are not strongly sensitive to this choice, and we have verified +that using longer timescales of 300 Myr and 1 Gyr yield similar results. +6 We calculated this statistic for haloes in the collisionless (DMONLY) +EAGLE L100N1504 simulation, in a 0.1 dex wide mass window about +𝑀200 = 1012.54 M⊙. +2 +4 +6 +8 +10 +12 +t [Gyr] +11.0 +11.5 +12.0 +12.5 +log10(M200/M⊙) +Mcrit(z) (Bower + 17) +Organic +Secular +Early-secular +2 +4 +6 +8 +10 +12 +t [Gyr] +8.5 +9.0 +9.5 +10.0 +10.5 +log10(M⋆/M⊙) +2 +4 +6 +8 +10 +12 +t [Gyr] +−11 +−10 +−9 +log10(sSFR) [yr−1] +EAGLE green valley +0 +0.2 +0.5 +1 +2 +3 +z +Figure 1. The organic system and our modified secular and early-secular +variants reach approximately the same final halo mass (𝑀200) and stellar +mass (𝑀★), but exhibit differences in their mass histories due to our genetic +modifications. All three systems remain star-forming until 𝑧 = 0, despite +their halo masses being far in excess of 𝑀crit +200. The upper, middle and lower +panels show the evolution of 𝑀200, 𝑀★ and the specific star formation rate +(sSFR) respectively for these three systems. Solid lines show the median +values for nine resimulations of each set of initial conditions with different +random number seeds, and shading indicates the interquartile range. The time +of the first snapshot output following any significant merger (stellar mass ratio +𝜇 > 0.1) is shown with a vertical line. We indicate the critical halo mass, +𝑀crit +200, above which black holes can grow efficiently in EAGLE with a dashed +black line in the top panel. The location of the green valley in the EAGLE +population is shown with green hatching in the lower panel. +also show the location of the green valley as a function of time for +galaxies of a comparable stellar mass in the largest EAGLE simula- +tion volume (Ref-L100N1504). We define this based on the sSFR of +all EAGLE galaxies in a 0.2 dex-wide window about the current stel- +lar mass of the organic galaxy. Following Wright et al. (2019), we +take the locus of the star-forming main sequence to be the mean sSFR +MNRAS 000, 1–11 (2023) + +The origin of diversity in SMBH and galaxy growth +5 +of these galaxies, subject to a floor log10(sSFR/yr−1) > −11 + 0.5𝑧, +and define the green valley to lie within 5% and 50% of this value. +All three systems remain actively star-forming throughout their +evolution. The earlier stellar mass assembly of the early-secular +galaxy is reflected in its sSFR, which is higher at 𝑧 ≈ 3−2, and lower +at 𝑧 ≈ 1.5 − 1, than the later-assembling systems. However, there is +little difference between its final sSFR of 10−10.34 yr−1 (IQR= 0.07 +dex) and the final sSFR of the later-assembling secular system +10−10.2 yr−1, IQR= 0.1 dex). Therefore, while correlations within +the wider EAGLE population indicate that earlier-assembling and +more tightly-bound haloes tend to host central galaxies with lower +sSFR (e.g. Matthee & Schaye 2019; Davies et al. 2019, 2020), we +find that adjusting these quantities alone does not causally affect the +central galaxy’s present-day star formation activity. +3.1.2 Feedback and the baryon fraction +To investigate why changes in the assembly time alone do not change +the present-day star formation rates of our galaxies, we now explore +how the feedback history and the halo baryon content are affected +by our modifications. In the upper panel of Figure 2, we show how +the total AGN feedback energy, 𝐸AGN, injected into our systems as +a function of time, and in the middle panel we show the fraction of +the total feedback energy, 𝐸FB = 𝐸★ + 𝐸AGN, that is contributed +by AGN feedback. In the lower panel, we show how this energy +injection influences the halo baryon fraction ( 𝑓b, normalised to the +cosmic fraction 𝑓 cosmic +b += Ωb/Ω0, lower panel). Details of how each +of these quantities was calculated are given in Section 2.3; since +we calculate 𝐸AGN directly from the central SMBH mass, 𝑀BH, we +show this mass as a second axis in the upper panel. +To place the growth of the BHs in our systems in context, we +briefly review the general behaviour of BHs in the EAGLE model. +McAlpine et al. (2018) proposed that EAGLE BHs grow in three +phases: +(i) In lower-mass haloes, stellar feedback is able to efficiently +expel gas from the central regions of galaxies, keeping the gas density +in the BH’s vicinity, 𝜌BH, low. The BH is therefore unable to grow +and remains close to the mass at which it was seeded. McAlpine et al. +(2018) name this the stellar feedback regulated phase. +(ii) As 𝑀200 reaches 𝑀crit +200 and the halo virial temperature ap- +proaches 105.6 K, stellar feedback-driven outflows have lower en- +tropy than the CGM and are no longer buoyant. These outflows can +no longer remove gas from the galaxy centre, causing 𝜌BH to increase +significantly. EAGLE BHs then accrete gas according to a modified +version of the spherically-symmetric Bondi & Hoyle (1944) formula +(see Rosas-Guevara et al. 2015), growing at a non-linear rate pro- +portional to 𝑀2 +BH for as long as 𝜌BH remains high. McAlpine et al. +(2018) name this the non-linear growth (NLG) phase. Following their +study, we take the NLG phase to be where dlog10(𝑀BH)/d𝑡 > 0.25 +dex Gyr−1 and illustrate it with horizontal bars in Figure 2. +(iii) Once the BH becomes sufficiently massive that its AGN feed- +back can expel gas from the BH’s immediate vicinity, 𝜌BH falls and +the NLG phase ends. Thereafter, the BH can regulate its own growth +(and 𝜌BH) through AGN feedback. McAlpine et al. (2018) name this +the AGN feedback regulated phase. However, as we will demonstrate +in this study, AGN feedback does not necessarily dominate the energy +injected into the galaxy and halo after the NLG phase ends. +During the NLG phase, 𝐸AGN and 𝑀BH rise sharply, and this +occurs markedly earlier for the early-secular system than for +the later-assembling organic and secular systems. However, over +2 +4 +6 +8 +10 +12 +t [Gyr] +58.0 +58.5 +59.0 +59.5 +60.0 +log10(EAGN) [erg] +Organic +Secular +Early-secular +2 +4 +6 +8 +10 +12 +t [Gyr] +−1.2 +−1.0 +−0.8 +−0.6 +−0.4 +−0.2 +log10(EAGN/EFB) +2 +4 +6 +8 +10 +12 +t [Gyr] +0.0 +0.2 +0.4 +0.6 +0.8 +fb/(Ωb/Ω0) +6.0 +6.5 +7.0 +7.5 +log10(MBH/M⊙) +0 +0.2 +0.5 +1 +2 +3 +z +Figure 2. Differences in assembly history make little difference to 𝐸AGN and +its contribution to the overall feedback budget in the absence of disruptive +events, and all systems remain baryon-rich. In the same fashion as Figure 1, +the upper, middle and lower panels show the evolution of the integrated energy +injected by AGN feedback (𝐸AGN), the fraction of the total feedback energy, +𝐸FB, contributed by AGN, and the halo baryon fraction ( 𝑓b, normalised to +the cosmic fraction, Ωb/Ω0) respectively, for the organic, secular and +early-secular systems. We illustrate the phase of non-linear growth (NLG) +undergone by the central black hole in each system with horizontal bars. +the lifetimes of the secular and early-secular systems, the to- +tal energy injected by AGN feedback is not significantly different +(𝐸AGN = 1059.8 erg with IQR= 0.1 dex), and is slightly less than +the energy injected into the organic system (𝐸AGN = 1060.0 erg, +IQR= 0.1 dex). +In the absence of merger events, earlier halo assembly and a 30% +higher binding energy therefore does not necessarily yield a more +massive BH or the injection of more AGN feedback energy. This +is somewhat surprising, since the model of Booth & Schaye (2010, +2011) predicts that BHs will grow until they have injected an energy +set by the halo binding energy, and strong positive correlations exist +between 𝑀BH and 𝐸DMO +2500 at fixed 𝑀200 in the EAGLE population +(Davies et al. 2019). While the final 𝑀BH and 𝐸AGN do not change, +the early-secular BH does grow at a faster rate during the NLG +phase, and we find that it attains a higher 𝜌BH during this period. +This is consistent with the behaviour of the wider EAGLE population, +MNRAS 000, 1–11 (2023) + +6 +J. J. Davies et al. +in which BHs that enter the NLG phase earlier are more luminous +and have higher Eddington ratios during the phase (McAlpine et al. +2018). This may be the result of a higher cosmological infall rate +and/or mean density of the universe at earlier times, or be due to the +higher halo binding energy making it harder for feedback processes +to reduce the central gas density. +AGN feedback does not dominate the energy input into any of these +systems, as shown in the middle panel; it contributes ≈ 30% of the +feedback injected into the secular and early-secular systems and +≈ 40% of the feedback in the organic system. 𝐸AGN/𝐸FB sharply +increases as each system’s BH undergoes its NLG phase, but then +remains fairly constant once this phase is complete; the post-NLG +phase therefore need not be dominated by AGN feedback. Both stellar +and AGN feedback continue to operate in tandem for the remainder +of each system’s evolution, apparently co-existing in equilibrium and +contributing similar rates of energy injection. +The lower panel shows that the feedback injected into these sys- +tems has not led to a strong expulsion of baryons from their haloes. +Prior to each system’s NLG phase, the baryon fraction 𝑓b is approxi- +mately 0.65 𝑓 cosmic +b +, lower than the cosmic fraction due to the effects +of photoionisation and stellar feedback at early times. In all cases, +the AGN feedback injected during the NLG phase causes a small +decrease in 𝑓b, and in the organic halo the additional AGN feed- +back induced by the minor merger causes its 𝑓b to fall below that of +the secular halo. Overall, the haloes remain baryon rich, retaining +approximately 60 − 70% of the cosmic baryon fraction within their +haloes at the present day. +In EAGLE and other galaxy formation models, quenching galaxies +appears to require that a large fraction of the halo’s baryons are +expelled beyond the virial radius by AGN feedback, in order to reduce +the cooling rate of CGM gas onto the interstellar medium (Davies +et al. 2020; Zinger et al. 2020). This does not occur for our secularly- +evolving systems, explaining why they remain star-forming until the +present day. +3.1.3 The role of the galaxy disc +The results discussed so far demonstrate that, in the absence of merg- +ers or other disruptive events, the assembly time and intrinsic binding +energy of a dark matter halo do not strongly influence the present-day +properties of our galaxy and its halo. This result can be understood by +considering the dynamics of the gas in the vicinity of the central BH. +As discussed in Section 3.1.2, EAGLE BHs can only grow rapidly +while the density of gas in their vicinity (𝜌BH) remains high. If gas +is kept away from the BH, spread out in a co-rotating disc, 𝜌BH will +be lower and the BH’s growth will be inhibited. +Using a similar genetic modification experiment, Davies et al. +(2022) showed that suppressing the influence of a merger also sup- +presses the growth of the central BH, because the co-rotational mo- +tion of the gas is preserved in the absence of any disruptive events. +A similar phenomenon is occurring for our secular and early- +secular galaxies, which have persistent, strongly co-rotating discs +throughout their evolution. Following the process outlined in Section +3.1.2, their BHs undergo an NLG phase when 𝑀200 → 𝑀crit +200, before +settling into the AGN feedback-regulated phase. For the remainder +of their evolution, 80-90% of the kinetic energy of the gas within 3 +physical kpc of their BHs is invested in co-rotational motion7, and +so their BHs never grow rapidly again as the rotational support keeps +7 These values are equivalent to the 𝜅co diagnostic (Correa et al. 2017) for +the gas, calculated with the routines of Thob et al. (2019). +𝜌BH low. Figure 2 shows that the halo assembly time and binding +energy influence when the NLG phase occurs, and how rapid the BH +growth is in that phase, but they do not significantly change the AGN +feedback energy required for the BH to reduce 𝜌BH and regulate its +own growth. The final BH mass is therefore similar in each case. +Our findings show that when galaxies are undisturbed by mergers +and retain gaseous discs, they settle into a state where a balance of +stellar feedback and low-level AGN feedback is sufficient to regulate +the growth of the BH and the galaxy, long after the 𝑀crit +200 threshold +has been exceeded. Their BHs inject only the minimum amount of +energy required to end the NLG phase, and afterwards only require +a small amount of AGN feedback energy to regulate the buildup of +gas that migrates to the centre of the disc. This feedback does not +strongly affect the CGM, which can continue cooling onto the galaxy, +where the majority of it settles onto the disc and forms stars, and a +minority fuels slow and steady growth of the BH. +In this state, the growth of the BH and the properties of the host +halo are effectively decoupled. The BH does not need to ‘offset’ the +cooling of gas from the whole halo in order to regulate its growth, as +is the case in more massive group/cluster systems (e.g. McNamara +& Nulsen 2007; Fabian 2012); instead, only a small fraction of the +gas cooling from the halo can fuel the BH. While a higher binding +energy (and deeper potential) may in principle mean that more AGN +feedback is required to expel this gas from the centre of the disc, the +near-identical 𝐸AGN for our secularly-evolving galaxies demonstrates +that this is not a significant effect. As a result, the integrated growth +of the BH, and hence its influence on the CGM and the galaxy, is not +sensitive to changes in the binding energy of the halo. +3.2 The impact of merger events +We now explore how our secular and early-secular galaxies and +their haloes evolve when they do experience a disruptive merger after +exceeding 𝑀crit +200. We therefore turn to the systems evolved from our +merger and early-merger initial conditions. Each of these systems +undergoes a major merger with a stellar mass ratio of 0.46, occurring +at 𝑡 = 6.4 Gyr (merger) and 𝑡 = 4.9 Gyr (early-merger). While +the stellar mass ratios of these mergers are the same, we note that +they are not the same merger, and they differ in their geometry (i.e. +impact parameter and infall angle) and dynamics. +3.2.1 Halo and galaxy evolution +Figure 3 is identical in format to Figure 1, but now focuses on the +evolution of the merger and early-merger halo mass, stellar mass, +and sSFR. To demonstrate the differences induced by mergers relative +to a secularly-evolving case, we also include the secular and early- +secular systems but only show the median evolution, for clarity. +As in previous figures, we highlight the snapshot times after which +significant mergers have occurred for the merger and early-merger +systems with vertical lines. +As shown in the upper panel, the halo mass histories of the secu- +lar and merger haloes are similar to each other, as are those of the +early-secular and early-merger haloes, reaching near-identical +halo masses by the present day. The later-assembling pair of sys- +tems cross the 𝑀crit +200 threshold at approximately the same time, as +do the earlier-assembling pair. The stellar mass histories (middle +panel) are also very similar within the earlier- and later-assembling +pairs of systems. As with the halo mass, the histories differ at the +time of the merger, as the merger and early-merger galaxies gain +a significant ex-situ contribution to their stellar mass, while their +secularly-evolving counterparts form stars more steadily in-situ. +MNRAS 000, 1–11 (2023) + +The origin of diversity in SMBH and galaxy growth +7 +2 +4 +6 +8 +10 +12 +t [Gyr] +11.0 +11.5 +12.0 +12.5 +log10(M200/M⊙) +Mcrit(z) (Bower + 17) +Secular +Early-secular +Merger +Early-merger +2 +4 +6 +8 +10 +12 +t [Gyr] +8.5 +9.0 +9.5 +10.0 +10.5 +log10(M⋆/M⊙) +2 +4 +6 +8 +10 +12 +t [Gyr] +−13 +−12 +−11 +−10 +−9 +log10(sSFR) [yr−1] +EAGLE green valley +0 +0.2 +0.5 +1 +2 +3 +z +Figure 3. Our galaxies modified to undergo major mergers attain similar final +𝑀200 and 𝑀★ to their secularly-evolving counterparts. However, the mergers +cause them to leave the main sequence, enter the green valley, and in many +cases quench. The three panels are equivalent to those in Figure 1, but now +show the evolution of 𝑀200, 𝑀★ and the sSFR for the merger and early- +merger systems. The median evolution for the secular and early-secular +systems is shown for comparison. +The post-merger stellar mass histories of the merger and early- +merger galaxies can be understood by examining the evolution of +their sSFR, shown in the lower panel of Figure 3. Prior to the mergers +taking place, the galaxies are on the main sequence, closely follow- +ing the sSFR of their secularly-evolving counterparts. However, the +sSFR declines following the mergers in both cases, with a stronger +decline seen for the early-merger galaxy. The scatter introduced by +stochasticity is particularly notable here; in the early-merger case, +many realisations of the galaxy rapidly fall through the green val- +ley and quench within the time interval between simulation outputs, +whereas others fall into the green valley and remain there until 𝑧 = 0. +In general, the median early-merger galaxy is quenched due to the +merger it experiences, whereas the median merger galaxy resides in +the green valley. +3.2.2 Feedback and the baryon fraction +Figure 3 shows that individual merger events are able to induce +strong changes in the sSFR of our galaxy, to a degree that adjust- +ing the overall assembly time alone did not. Davies et al. (2022) +showed that mergers induce such changes because they disrupt the +co-rotational motion of gas in the galaxy disc, greatly increasing the +density of gas in the central BH’s vicinity and allowing the BH to +grow more massive (and thus inject more AGN feedback energy) +than in a secularly-evolving system. This phenomenon also occurs +for our merger and early-merger systems; prior to the mergers, +80-90% of the kinetic energy of the gas within 3 physical kpc of +their BHs is invested in co-rotational motion, but in the snapshots +following their merger events, this fraction has fallen to ∼ 30%. This +greatly enhances the growth of the BHs in these galaxies. +We show how this disc disruption affects the BH’s growth in +Figure 4, which has the same format as Figure 2 but now focuses on +the feedback energetics and halo baryon content of the merger and +early-merger systems, again also showing the median evolution +of their secularly-evolving counterparts. The haloes exceed 𝑀crit +200 at +approximately the same time as their secularly-evolving counterparts, +and so their central BHs begin their NLG phases at similar times. The +masses of the BHs then quickly exceed those in the secularly-evolving +galaxies once the merger events begin to influence the amount of +gas available to them. To follow this process in more detail, we +performed additional simulations with 10 Myr output time resolution, +finding that BH growth in the merger system is stimulated at the +infalling halo’s first periapsis over 1 Gyr before the merger, whereas +in the early-merger system it is the final coalescence that triggers +rapid BH growth. The final masses of these BHs are 107.9 M⊙ +(merger, IQR= 0.2 dex) and 108.0 M⊙ (early-merger, IQR= 0.2 +dex), significantly exceeding the masses of the BHs in the secularly- +evolving galaxies (107.4 M⊙, IQR= 0.1 dex in both cases), and +consequently factors of ≈ 3 and ≈ 4 times more AGN feedback +energy is injected in these systems respectively. +The middle panel of Figure 4 shows that mergers significantly +increase the fraction of the total feedback energy that is contributed +by AGN feedback; by the present day AGN feedback accounts for +≈ 55% of the feedback in the merger system and ≈ 59% in the +early-merger system (cf. 30% in the secularly-evolving systems). +This is predominantly the result of a large increase in 𝐸AGN; the +mergers increase the integrated stellar feedback energy by only small +factors of 1.2 and 1.3 for the merger and early-merger galaxies +respectively. This additional AGN feedback expels a large fraction +of the CGM from the merger and early-merger haloes; as shown +in the lower panel they both retain only ≈ 30% (IQR= 20%) of +the cosmic baryon fraction within their haloes at the present day. +This reduction in the baryon content of the CGM causes it to cool +less efficiently, explaining the reduced star formation rates of these +galaxies (see also Davies et al. 2020, 2022). +3.2.3 Two modes self-regulation in galaxies with massive SMBHs +In Section 3.1.2 we outlined the picture described by Bower et al. +(2017) and McAlpine et al. (2018), in which EAGLE’s BHs pre- +dominantly grow in two phases separated by a transitional non-linear +growth phase. McAlpine et al. (2018) name the final, post-NLG phase +the “AGN feedback regulated phase”, implying that the regulation of +MNRAS 000, 1–11 (2023) + +8 +J. J. Davies et al. +BH and galaxy growth is dominated by AGN feedback, but our find- +ings show that this is not necessarily the case. EAGLE galaxies can +remain star-forming and regulated by a mixture of stellar and low- +level AGN feedback long after the NLG phase, so long as no mergers +disrupt the gas disc. We therefore propose that the post-NLG phase +can be further split into two modes: +• Stellar & AGN feedback co-regulation: The presence of a +co-rotating gas disc in the galaxy largely decouples the growth of +the BH from the properties of the host halo. Gas that cools from the +CGM settles into the disc and the BH need only grow enough (and +inject enough AGN feedback) to regulate the gas that migrates to +the centre of the disc. The CGM is minimally affected by this AGN +feedback and the galaxy remains star-forming. The BH may be locally +self-regulating with AGN feedback, but the galaxy and its halo are +regulated by a mixture of stellar and low-level AGN feedback. +• AGN feedback regulation: When the galaxy disc is dis- +rupted/destroyed, the gas cooling from the CGM (in addition to the +cold gas already in the galaxy) can reach the BH’s vicinity. The BH +must therefore regulate inflow from the halo; this involves ejecting a +significant fraction of the CGM in order to reduce the density, and +hence the cooling rate, of the remaining circumgalactic gas. This +regulation mode requires the injection of far more AGN feedback +and the growth of the BH to a higher mass. The amount of energy +required may depend on halo properties such as the binding energy. +The prevention of cooling from the CGM reduces or quenches star +formation in the galaxy, and AGN feedback becomes the dominant +energy injection mechanism. +The BHs in the secular and early-secular galaxies remain in +the first of these modes after they begin to regulate their own growth; +changes to the halo’s formation time and binding energy therefore +have little influence on the energy injected by AGN (and in turn the +halo 𝑓b and galaxy sSFR), as shown in Section 3.1. In the merger +and early-merger systems, however, a merger occurs that induces a +transition to the second of these modes, transforming the subsequent +evolution of the BH, CGM and galaxy, and recoupling the properties +of the central BH to the properties of the halo. The transition between +these two modes is likely essential to the diversity in BH, CGM and +galaxy properties at fixed halo mass in the EAGLE galaxy population. +3.3 Re-interpreting correlations in the EAGLE population +Previous studies of the galaxy populations in the large-volume EA- +GLE (and IllustrisTNG) simulations have concluded that diversity in +the properties of BHs, galaxies and their gaseous haloes at fixed halo +mass stems from differences in halo binding energy, which in turn +is assumed to be the result of differences in assembly/collapse time. +The basis for these conclusions was the clear correlation between the +inner halo binding energy, 𝐸DMO +2500 , and 𝑀BH at fixed 𝑀200. The ex- +planation for this correlation came from self-regulation arguments; +tightly-bound haloes more effectively confine gas at the halo centre, +requiring the injection of more AGN feedback (and hence a higher +𝑀BH) to expel gas and regulate the growth of the BH and galaxy +(e.g. Booth & Schaye 2010, 2011; Davies et al. 2019, 2020, 2021). +However, this explanation neglects the influence of the galaxy disc +and the role of merger events, which we have shown to be crucial to +the evolution of the BH and its host galaxy in this study. We therefore +now revisit this correlation and consider its origin in light of this new +information. +We show the correlation for the galaxy population in the EAGLE +Ref-L100N1504 simulation in Figure 5, plotting 𝑀BH as a function of +𝑀200 for all central galaxies with 𝑀200 > 1011.5 M⊙, and colouring +2 +4 +6 +8 +10 +12 +t [Gyr] +58.0 +58.5 +59.0 +59.5 +60.0 +60.5 +log10(EAGN) [erg] +Secular +Early-secular +Merger +Early-merger +2 +4 +6 +8 +10 +12 +t [Gyr] +−1.2 +−1.0 +−0.8 +−0.6 +−0.4 +−0.2 +log10(EAGN/EFB) +2 +4 +6 +8 +10 +12 +t [Gyr] +0.0 +0.2 +0.4 +0.6 +0.8 +fb/(Ωb/Ω0) +6.0 +6.5 +7.0 +7.5 +8.0 +log10(MBH/M⊙) +0 +0.2 +0.5 +1 +2 +3 +z +Figure 4. Mergers cause the injection of significantly more AGN feedback in +the merger and early-merger systems, leading AGN feedback to constitute +a greater fraction of the feedback energy injected into these systems. The +expulsive effect of this feedback depletes the baryon content of their haloes. +The three panels are equivalent to those in Figure 2, but now show the +evolution of the feedback energetics and halo baryon content for the merger +and early-merger systems, with the median evolution for their secularly- +evolving counterparts shown for comparison. We illustrate the phase of non- +linear growth (NLG) undergone by the central black hole in each system with +horizontal bars. +the data points by the residuals (in log space) of the 𝐸DMO +2500 − 𝑀200 +relation8. The marker colours reveal a very strong positive corre- +lation between 𝐸DMO +2500 +and 𝑀BH at fixed 𝑀200. The locations of +our genetically-modified systems are overlaid with larger symbols, +coloured in the same fashion as the wider population; for each sys- +tem, we calculate 𝐸DMO +2500 and find the residual from the population +median at the system’s halo mass. +The modified systems span the majority of the scatter in 𝑀BH +at 𝑀200 ≈ 1012.5 M⊙, and their 𝐸DMO +2500 values agree well with the +underlying correlation. The systems that experienced merger events +8 The residuals are taken with respect to a running median value obtained +through the locally weighted scatterplot smoothing method (LOWESS, e.g. +Cleveland 1979). +MNRAS 000, 1–11 (2023) + +The origin of diversity in SMBH and galaxy growth +9 +not only host overmassive BHs but also have high inner halo binding +energies, and vice-versa. The merger and secular systems have +very similar assembly times and differ only by the presence, or lack +of, a major merger, yet the merger halo is significantly more tightly +bound for its mass. The inner halo binding energy is therefore not only +set by the assembly and collapse time, but can be strongly increased +by individual mergers. Comparing the secular and early-secular +systems shows that earlier collapse does yield a higher 𝐸DMO +2500 , but +the increase is far smaller than that caused by a major merger, and it +does not cause enhanced BH growth. +This result aligns well with the early predictions of Neto et al. +(2007), who demonstrated that the connection between halo concen- +tration and formation time is clearer when the definition of formation +time includes all of a halo’s major progenitors and not just one, indi- +cating that mergers play a key role in setting the concentration (and +hence the binding energy). More recently, Rey et al. (2019) used ge- +netic modification to increase the variance of the overdensity field in +a halo’s initial conditions, increasing the number of significant merg- +ers in the merger tree; this change also caused the halo concentration +to increase, providing further evidence for this connection. +Our findings suggest that the correlation between the halo binding +energy and the BH mass emerges because mergers help to grow BHs +to high masses and can significantly increase the binding energy of +the host dark matter halo. This picture also provides an explanation +for why earlier-assembling haloes in cosmological simulations tend +to have higher binding energies and host quenched central galaxies +with overmassive central SMBHs (e.g. Matthee & Schaye 2019; +Montero-Dorta et al. 2020); these systems reside in denser, more +clustered environments (Sheth & Tormen 2004; Gao et al. 2005) +and are therefore likely to experience more mergers throughout their +evolution. +4 SUMMARY AND DISCUSSION +In this study, we have used the genetic modification technique (Roth +et al. 2016) to assess how supermassive black hole (BH) growth and +the impact of AGN feedback are influenced by individual galaxy- +galaxy merger events and the overall assembly history of the host +halo. This experiment was motivated by predictions from cosmolog- +ical simulations (see references in Section 1) that mergers can induce +BH growth and AGN feedback, and that the BH mass correlates +strongly and positively with the binding energy of the dark matter +halo, which in turn is correlated with the overall assembly time. +We performed zoom simulations of a star-forming disc galaxy of +stellar mass 𝑀★ = 4.3 × 1010 M⊙ and host halo mass 𝑀200 = 3.4 × +1012 M⊙ with the recalibrated high-resolution version of the EAGLE +galaxy formation model (Schaye et al. 2015; Crain et al. 2015). Using +the initial conditions generator genetIC (Stopyra et al. 2020), we +modified the initial conditions of our fiducial galaxy and its host +halo to generate four new variants of it with systematically adjusted +assembly histories, designed such that we could independently assess +the roles of mergers and the overall assembly time in a controlled +galaxy formation experiment. +First, we produced two modified variants of our galaxy for which +no significant mergers (i.e. where the stellar mass ratio 𝜇 > 0.1) +occur after the host halo reaches the critical mass, 𝑀crit +200, above +which BHs are able to grow via accretion in the EAGLE model (see +Bower et al. 2017; McAlpine et al. 2018). One of these systems +(secular) has a similar assembly time to the unmodified case, while +the other (early-secular) assembles earlier. This earlier assembly +time causes the inner dark matter halo of the early-secular case to +11.6 +11.8 +12.0 +12.2 +12.4 +12.6 +12.8 +13.0 +log10(M200/M⊙) +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +log10(MBH/M⊙) +Secular +Merger +Early-secular +Early-merger +−0.20 +−0.15 +−0.10 +−0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +∆ log10(EDMO +2500 ) +Figure 5. The black hole mass, 𝑀BH, as a function of 𝑀200 for the galaxy pop- +ulation in the Ref-L100N1504 EAGLE simulation. Datapoints are coloured +according to the residuals of the log10(𝐸DMO +2500 ) − log10(𝑀200/M⊙) relation, +where 𝐸DMO +2500 is the binding energy within a sphere enclosing 2500 times the +critical density for each halo’s counterpart in a collisionless dark matter sim- +ulation. At fixed 𝑀200, haloes that are intrinsically more tightly-bound host +more massive black holes. Our genetically-modified systems are overlaid; +our modifications to their assembly histories cause them to span the scatter +in the population. Adjusting the assembly time in the absence of mergers +changes 𝐸DMO +2500 but not 𝑀BH, while mergers can both drive high 𝑀BH and +cause haloes to be more intrinsically tightly-bound. Merger events appear to +be required for establishing both diversity in 𝑀BH and a correlation with the +halo binding energy at fixed 𝑀200. +be intrinsically more tightly-bound; when measured in an equivalent +dark matter-only simulation, its present-day binding energy, 𝐸DMO +2500 , +is 30% higher than the secular case. +Comparing these systems allowed us to isolate how differences in +the halo assembly time influence the growth of the central BH, free +of the influence of significant mergers. We find that these differences +do not drive significant changes to the present-day properties of the +galaxy and its host halo. The secular and early-secular galaxies +both reach the same present-day stellar mass and remain on the star- +forming main sequence (Figure 1), and their central BHs grow to +approximately the same mass and inject the same amount of AGN +feedback energy, both in absolute terms and as a fraction of the total +feedback energy, and their gaseous haloes remain baryon-rich (Figure +2). The BHs grow according to a three-phase process, as is expected +in the EAGLE model (McAlpine et al. 2018); they initially remain +close to the seed mass, undergo non-linear growth (NLG) when the +halo mass exceeds 𝑀crit +200, and then grow more steadily once AGN +feedback is able to expel gas from the BH vicinity. We find that +differences in assembly time influence when the NLG phase occurs +and the typical accretion rate in this phase, but do not change the +total energy injected or the impact of the feedback on the galaxy- +CGM ecosystem. Once the NLG phase ends, the growth of the BH +and galaxy are regulated by a combination of stellar and low-level +AGN feedback, which contribute approximately 70% and 30% of the +feedback energy injected by 𝑧 = 0 respectively. +To test the influence of mergers, we produced two more sets of ini- +MNRAS 000, 1–11 (2023) + +O10 +J. J. Davies et al. +tial conditions, merger and early-merger, designed to yield haloes +with similar assembly times to the secular and early-secular +galaxies respectively, but also experience a major merger (𝜇 = 0.46) +after crossing 𝑀crit +200. At 𝑧 = 0, we find that these systems have approx- +imately the same halo and stellar masses as their secularly-evolving +counterparts, but are either quenched or reside in the green valley +(Figure 3). The mergers cause the BHs in these systems to grow +significantly more massive than the BHs in their secularly-evolving +counterparts, and inject significantly more AGN feedback energy (by +factors of ≈ 3 and ≈ 4 respectively). AGN dominate the feedback +energy injected into these systems, contributing ≈ 55% and ≈ 59% +of the total integrated energy respectively. The expulsive nature of +this feedback depletes the haloes of their baryons, and they retain +only ≈ 30% of the cosmic baryon fraction by the present day (Figure +4). +We attribute these results to the vital importance of the galaxy disc +in determining the conditions in the vicinity of the central BH. The +secular and early-secular galaxies retain strong co-rotating gas +discs throughout their evolution, with 80−90% of the kinetic energy +of the gas invested in co-rotational motion. This rotational support +reduces the inflow rate towards the BH, suppressing its growth and +reducing the feedback energy required to maintain self-regulation. +The BHs in these systems appear to undergo non-linear growth to +a minimum mass at which AGN feedback can expel gas from their +vicinity, and then grow very little thereafter. The majority of the gas +cooling from the halo settles onto the disc and fuels continued star +formation, with a small minority fuelling slow growth of the BH. The +disc therefore decouples the growth of the BH from the properties +of the host halo, and hence changes in assembly time and binding +energy have little influence on our secularly-evolving galaxies. +In our merger and early-merger systems, the disc is disrupted, +allowing gas in the galaxy and halo to fuel BH growth. This increases +the AGN feedback energy required to maintain self-regulation, and +causes a transformation of the CGM and the quenching of star forma- +tion. The disruption of the disc (through mergers or otherwise) there- +fore appears to be key to the establishment of diversity in the masses +of BHs, and to coupling the properties of BHs with those of their host +haloes. We propose that once galaxies host massive central BHs, their +growth can be regulated in one of two modes: (i) co-regulation by +stellar feedback and low-level AGN feedback in secularly-evolving, +star-forming disc galaxies, and (ii) AGN-dominated regulation in +systems without discs, which are likely quenched due to the influ- +ence of integrated AGN feedback on the CGM. The transition of +galaxies between these modes may be essential to the diversity in +galaxy properties seen in the EAGLE simulations. +We concluded by reconsidering the origin of strong positive cor- +relations between the BH mass and the host halo binding energy seen +at fixed halo mass in the EAGLE population, in light of our results. +We placed the properties of our modified systems in the context +of this correlation (Figure 5), and deduced that the correlation likely +emerges because mergers are key to growing BHs to high masses and +they can significantly increase the binding energy of the underlying +dark matter halo. +Our experiment has revealed how mergers and halo properties +influence the fate of a single galaxy. While this behaviour is not +guaranteed to be universal in the galaxy populations of the EAGLE +cosmological simulation volumes, circumstantial evidence for the +essential role of mergers does exist in the EAGLE Ref-L100N1504 +simulation; galaxies residing in ∼ 𝐿★ haloes that have far exceeded +𝑀crit +200 but host undermassive BHs and remain CGM-rich tend to have +rotation-dominated stellar kinematics, while those with overmassive +BHs and gas-poor haloes tend to have dispersion-dominated kine- +matics indicative of disruptive mergers in their past (Davies et al. +2020). +To more firmly establish a connection between mergers and the im- +pact of AGN within the population, one could compare the properties +of two samples of galaxies with similar present-day halo masses, but +where galaxies in one sample have experienced a disruptive merger +when 𝑀200 > 𝑀crit +200 and those in the other have not. Assembling such +samples presents a challenge, however, primarily due to difficulties in +defining what makes a merger ‘disruptive’. Whilst a high stellar mass +ratio is likely a good indicator, other factors such as the morpholo- +gies of the merging galaxies, gas-richness, orbital configuration (i.e. +prograde vs. retrograde) or orbit type (i.e. “spiral-in” or “head-on”) +may be equally important, with even 1:1 mergers often causing little +disruption if they are gas-rich/prograde/spiral-in (e.g. Hopkins et al. +2009; Font et al. 2017; Garrison-Kimmel et al. 2018; Martin et al. +2018; Peschken et al. 2020; Zeng et al. 2021). +In our experiment we were able to systematically adjust the mass +ratio of mergers to make them more or less disruptive, but we did not +control for any of the above extra characteristics of mergers. Some +characteristics are influenced by each other; for example, when we +increase the mass ratio, we find that mergers become more “head- +on” with smaller impact parameters. We expect that in future work, +we will be able to identify the most important characteristics of +mergers by genetically-modifying the angular momentum of a galaxy +and its progenitors, allowing for controlled adjustment of a merger’s +trajectory (Cadiou et al. 2021, 2022). +We have shown in this work that galaxies which retain strong co- +rotating gas discs can remain star-forming and minimally influenced +by AGN feedback at halo masses above the threshold at which AGN +are expected to dominate and transform the system. We have focused +here on the ∼ 𝐿★ mass scale in order to explain diversity in the +properties of Milky Way-like galaxies, however this picture may +also apply at much higher masses and in more extreme systems. +While the likelihood of a galaxy experiencing a disruptive merger +increases with stellar mass (Martin et al. 2018), a rare population +of very massive (𝑀★ > 1011.3 M⊙) blue “super-spiral” galaxies +has been observed at low redshift (Ogle et al. 2019). These galaxies +exhibit a mixture of old and young stellar populations characteristic +of early assembly and a consistently high star formation rate; they +may therefore represent an extreme, high-mass case of the behaviour +seen in our early-secular system. It should be possible to test this +hypothesis using a genetic modification experiment, in which the +assembly of a massive halo (𝑀200 ≈ 1013 M⊙) hosting a quenched +spheroidal galaxy is accelerated and the merger history is made as +secular as possible. If these modifications yield similar changes to +those in our early-secular system, the central galaxy may become +a super-spiral that remains star-forming until 𝑧 = 0, long after one +might expect it to have been quenched by AGN feedback. +ACKNOWLEDGEMENTS +JJD would like to thank Corentin Cadiou and the GMGalaxies team +at UCL for helpful discussions and support. This study was supported +by the European Union’s Horizon 2020 research and innovation pro- +gramme under grant agreement No. 818085 GMGalaxies. AP and +RAC are supported by the Royal Society. This study used computing +equipment funded by the Research Capital Investment Fund (RCIF) +provided by UKRI, and partially funded by the UCL Cosmoparticle +Initiative. It also made use of high performance computing facilities +at Liverpool John Moores University, funded by the Royal Society +and LJMU’s Faculty of Engineering and Technology. We thank the +MNRAS 000, 1–11 (2023) + +The origin of diversity in SMBH and galaxy growth +11 +EAGLE team for making the particle data (The EAGLE team 2017) +and galaxy catalogues (McAlpine et al. 2016) for the Ref-L100N1504 +simulation publicly available. Analysis was performed in Python us- +ing pynbody (Pontzen et al. 2013) and tangos (Pontzen & Tremmel +2018). +DATA AVAILABILITY +The data underlying this article will be shared on reasonable request +to the corresponding author. +REFERENCES +Appleby S., Davé R., Sorini D., Storey-Fisher K., Smith B., 2021, MNRAS, +Bellovary J., Brooks A., Volonteri M., Governato F., Quinn T., Wadsley J., +2013, ApJ, 779, 136 +Bertschinger E., 2001, ApJS, 137, 1 +Bondi H., Hoyle F., 1944, MNRAS, 104, 273 +Booth C. M., Schaye J., 2010, MNRAS, 405, L1 +Booth C. M., Schaye J., 2011, MNRAS, 413, 1158 +Borrow J., Schaller M., Bahe Y. M., Schaye J., Ludlow A. D., Ploeckinger S., +Nobels F. S. 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I., 2019, MNRAS, 487, 3740 +Zeng G., Wang L., Gao L., 2021, MNRAS, 507, 3301 +Zinger E., et al., 2020, MNRAS, 499, 768 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–11 (2023) + diff --git a/q9E2T4oBgHgl3EQf0wi3/content/tmp_files/load_file.txt b/q9E2T4oBgHgl3EQf0wi3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e55a9a60ce1d0b33bd4d9371b98cd2b8f60af24f --- /dev/null +++ b/q9E2T4oBgHgl3EQf0wi3/content/tmp_files/load_file.txt @@ -0,0 +1,1233 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf,len=1232 +page_content='MNRAS 000, 1–11 (2023) Preprint 12 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 Are the fates of supermassive black holes and galaxies determined by individual mergers, or by the properties of their host haloes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Jonathan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Davies,1★ Andrew Pontzen1† and Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Crain,2 1Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK 2Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' in original form ZZZ ABSTRACT The fates of massive galaxies are tied to the evolution of their central supermassive black holes (BHs), due to the influence of AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Correlations within simulated galaxy populations suggest that the masses of BHs are governed by properties of their host dark matter haloes, such as the binding energy and assembly time, at a given halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' However, the full picture must be more complex as galaxy mergers have also been shown to influence the growth of BHs and the impact of AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In this study, we investigate this problem by using the genetic modification technique to adjust the assembly history of a Milky Way-like galaxy simulated with the EAGLE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We change the halo assembly time (and hence the binding energy) in the absence of any disruptive merger events, and find little change in the integrated growth of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We attribute this to the angular momentum support provided by a galaxy disc, which reduces the inflow of gas towards the BH and effectively decouples the BH’s growth from the properties of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Introducing major mergers into the assembly history disrupts the disc, causing the BH to grow ≈ 4× more massive and inject feedback that reduces the halo baryon fraction by a factor of ≈ 2 and quenches star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Merger events appear to be essential to the diversity in BH masses in EAGLE, and we show that they can also significantly increase the halo binding energy, potentially explaining the correlation between these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Key words: galaxies: formation – galaxies: evolution – galaxies: haloes – (galaxies:) quasars: supermassive black holes – methods: numerical 1 INTRODUCTION Feedback from active galactic nuclei (AGN) is a near-ubiquitous ingredient of modern galaxy formation models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' responsible for reg- ulating the growth of galaxies at and above the mass of the Milky Way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' quenching star formation in massive galaxies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' and maintaining quiescence in the central galaxies of group and cluster haloes by sup- pressing cooling flows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Croton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Sijacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Somerville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Vogelsberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Dubois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' McCarthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Kaviraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Tremmel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Henden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Weinberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Cosmological simulations predict that AGN feedback has a mini- mal impact on lower-mass galaxies, as outflows associated with star formation are able to remove gas from the centre of the galaxy and prevent the growth of the supermassive black hole (SMBH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' How- ever, above a critical halo mass scale corresponding to that of 𝐿★ galaxies, the entropy of the shock-heated circumgalactic medium (CGM) exceeds that of these outflows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' this confines gas to the galaxy centre and allows the SMBH to grow and begin influenc- ing the galaxy-halo ecosystem (Dubois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Keller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Habouzit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Truong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Above this critical mass scale, the EAGLE, IllustrisTNG and ★ E-mail: astrojdavies@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='com † E-mail: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='pontzen@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='uk SIMBA simulations exhibit diversity in the properties of SMBHs, galaxies and their CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Haloes in which AGN feedback has had little impact tend to be gas-rich and host star-forming central galax- ies, whereas haloes hosting overmassive SMBHs that have injected a lot of AGN feedback energy tend to be gas-poor and host quenched galaxies (Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Terrazas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Robson & Davé 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Sorini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Understanding why the impact of AGN feedback varies in haloes of a given mass is therefore key to understanding why diversity exists in the properties of the ∼ 𝐿★ galaxies in these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' One possible origin of this diversity could lie in differences in the underlying binding energies (and/or concentrations) of dark matter haloes (Booth & Schaye 2010, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This idea can be understood as a consequence of self-regulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' in a more tightly-bound halo, a SMBH must grow more massive and inject more AGN feedback energy to expel gas from the halo centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In the EAGLE and Illus- trisTNG simulations, haloes with higher binding energies tend to host more massive SMBHs, providing evidence for this connection (Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The binding energy of a halo, in turn, is assumed to be set by its assembly time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Neto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2007), a characteristic that is determined by the halo’s initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' On the other hand, observational evidence is emerging for a con- nection between AGN feedback and galaxy mergers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2011, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Satyapal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2014), and simulations have long predicted that such events can enhance the growth of SMBHs and the impact of AGN feedback (Di Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='04145v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='GA] 10 Jan 2023 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2006, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Sijacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2007, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Bellovary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Dubois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Pontzen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Steinborn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' this merger-induced feedback could explain recent ob- servations of a quenching excess in post-merger galaxies (Ellison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Recently, Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2022) used a controlled galaxy formation experiment to show that differences in the stellar mass ratio (and hence the disruptive influence) of a single merger can have dramatic effects on the growth of the SMBH at the centre of a galaxy, transforming the baryon cycle, the properties of the CGM, and the star formation activity in the central galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' These lines of evidence suggest that the properties of galaxy- CGM ecosystems depend on both the overall assembly time of the host halo and on individual disruptive events that occur throughout the system’s assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Identifying the relative importance of these factors is challenging, as they are likely to be degenerate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' haloes that assemble early tend to reside in more densely clustered environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Sheth & Tormen 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Wechsler & Tinker 2018), and may therefore assemble by undergoing many disruptive mergers, while later-assembling haloes may have comparatively quiet histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In this study, we perform a controlled galaxy formation experi- ment using the genetic modification technique (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2016) to independently assess the role of each of these factors, and unveil how the assembly history of a dark matter halo is connected to the evolution of its central galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2 METHODS For this study, we have performed a suite of simulations using the EAGLE version of the gravity and smoothed-particle hydrodynamics code gadget3 (last described by Springel 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Using the ‘zoom’ technique (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Katz & White 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Bertschinger 2001) we simulate the evolution of an individual galaxy and its local environment at high resolution, whilst also following the large-scale forces acting on the system by simulating its wider environment with a low-resolution periodic volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In this section, we explain the selection of our candidate galaxy (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1), outline how we modify its initial conditions to adjust its assembly history (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2), and describe how we identify and characterise galaxies and haloes within our simulations (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We also utilise the flagship EAGLE simulation volume (Ref- L100N1504) in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='3 to place our genetically-modified galaxies into the context of the wider population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' For detailed descriptions of this simulation, we refer the reader to the EAGLE reference articles (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Crain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1 Construction and evolution of initial conditions We selected our fiducial galaxy from a periodic simulation volume evolved with the Reference EAGLE simulation model (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Crain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015) from uniform-resolution initial conditions (ICs) generated by genetIC (Stopyra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We note therefore that the galaxy was not selected from any of the publicly-available EAGLE simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' selecting a galaxy evolved from ICs created by genetIC simplifies the subsequent genetic modification of these ICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This simulation is 50 comoving Mpc on a side, containing 5123 dark matter particles of mass 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='19 × 107 M⊙ and an initially equal number of baryonic particles of mass 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='6 × 106 M⊙ (a similar mass resolution to that of the flagship EAGLE simulations) and adopts the Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2016) cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' From this simulation, we selected a present-day star-forming disc galaxy of stellar mass 𝑀★ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='3 × 1010 M⊙, the central galaxy of a halo of mass 𝑀200 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='4×1012 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We selected this galaxy because it lies on the star-forming main sequence (sSFR= 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 yr−1) and has a CGM mass fraction1 𝑓CGM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='31 𝑓 cosmic b (where 𝑓 cosmic b is the cosmic baryon fraction, Ωb/Ω0) that is close to the present-day median 𝑓CGM at this halo mass in the largest EAGLE volume (Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This galaxy is ideal for our purposes, as it resides in a halo of a mass that far exceeds the critical mass, 𝑀crit 200, above which BHs are able to grow efficiently in the EAGLE model (Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2018), and has a simple merger history after the halo exceeds this mass, with only one minor merger of stellar mass ratio2 𝜇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='17 occurring at 𝑧 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We will henceforth refer to this galaxy and its halo as our organic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We generate zoomed initial conditions for this system by selecting all particles within three virial radii3 of the galaxy (at 𝑧 = 0) and identifying the Lagrangian region defined by these particles in the ICs (at 𝑧 = 99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We then refine this region with a factor of 27 more particles, and downsample the simulation volume outside this region by a factor of 8, yielding particle masses of 𝑚gas = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='19 × 105 M⊙, 𝑚dm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='18×106 M⊙, and 𝑚lr = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='02×108 M⊙ for gas, dark matter and low-resolution particles respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We evolve these initial conditions with the EAGLE model, adopt- ing the recalibrated (Recal) parameter values for the subgrid physics as defined by Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2015) as these were calibrated for a near-identical mass resolution to that of our initial conditions (𝑚gas = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='26 × 105 M⊙, 𝑚dm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='21 × 106 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The details of this model and its calibration may be found in the EAGLE reference articles (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Crain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015) and for brevity we do not repeat them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' However, it is important to note that models such as EAGLE include stochastic elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' processes such as the conversion of gas particles to star particles and the injection of feed- back energy are governed by the drawing of quasi-random numbers that are compared to probabilities set by the properties of the gas (see Schaye & Dalla Vecchia 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Dalla Vecchia & Schaye 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This stochasticity can cause significant uncertainty in the properties of in- dividual systems (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Genel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Keller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Borrow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Since the zoom simulations in this study are relatively inexpensive to perform, we simulate each set of initial conditions in our experiment with nine random number seeds each to quantify this uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 Producing genetically-modified galaxies To adjust the assembly history of the organic galaxy, we use the genetic modification (GM) technique (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Pontzen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2017) and the genetIC software to generate modified sets of ICs for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' From the overdensity field in the original ICs, genetIC finds the closest possible field that also satisfies certain constraints, which we design to produce our desired changes to the halo assembly history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This technique preserves the large-scale environment of the system, and the modified fields remain consistent with a Λ cold dark matter (ΛCDM) cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' To assess the role of individual merger events and the overall assembly history of the system independently 1 We define 𝑓CGM ≡ 𝑀CGM/𝑀200, where 𝑀CGM is the total mass of all gas within the virial radius that is not star-forming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2 We define the merger stellar mass ratio 𝜇 ≡ 𝑀infall ★ /𝑀★, where 𝑀infall ★ and 𝑀★ are the stellar masses of the infalling and primary galaxy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 3 We define the virial radius, 𝑟200, as the radius of a sphere enclosing 200 times the critical density of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023) The origin of diversity in SMBH and galaxy growth 3 of each other, we generate four complementary sets of modified ICs using this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' First, to examine the influence of merger events at a fixed assembly time, we produce a pair of modified ICs, secular and merger, designed to decrease or increase the stellar mass ratio of the organic system’s 𝑧 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='74 minor merger respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This is achieved by identifying the particles bound to the infalling halo at an earlier time (𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='73)4, tracing these particles back to their locations in the ICs, and decreasing or increasing the mean overdensity, ¯𝛿, in the patch of the field defined by these locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' To preserve the overall mass accretion history we also apply two further constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' ¯𝛿 in the patch defined by particles that comprise the main halo at 𝑧 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='73 is kept fixed, as is ¯𝛿 in the patch corresponding to the 𝑧 = 0 halo to ensure that the same final halo mass is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We also test the influence of the overall assembly time indepen- dently of merger events, by comparing the evolution of galaxies that have differing assembly times and experience no significant mergers that would be able to drive black hole (BH) growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' To do so, we compare the secular system with another modified variant of our galaxy, which assembles earlier and experiences no significant merg- ers after the 𝑀crit 200 threshold is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We produce this behaviour by assembling more mass into the main progenitor at early times (𝑧 = 2) whilst keeping the final halo mass fixed, which has the effect of both accelerating the halo assembly and reducing the significance of all 𝑧 < 2 mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This is achieved in practice by increasing ¯𝛿 in the patch of the ICs corresponding to the 𝑧 = 2 halo, while keeping ¯𝛿 fixed within the patch corresponding to the 𝑧 = 0 halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We refer to the system evolved from these conditions as the early-secular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Finally, so that we can assess how the early-secular system would evolve if it had a more disruptive evolution after exceeding 𝑀crit 200, we further modify the early-secular ICs to increase ¯𝛿 in a patch corresponding to an infalling system at 𝑧 = 3 with the aim of inducing a subsequent major merger of similar mass ratio to that in the merger system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We refer to the system evolved from these conditions as the early-merger system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='3 Identifying and characterising galaxies and their haloes Haloes are identified on-the-fly in our simulations by applying the friends-of-friends (FoF) algorithm to the dark matter distribution, with a linking length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 times the mean interparticle separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Gas, star and BH particles are then assigned to the FoF halo of their nearest dark matter particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In post-processing, we then identify bound haloes using the subfind algorithm (Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Dolag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2009), and use the analysis packages pynbody (Pontzen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2013) and tangos (Pontzen & Tremmel 2018) to calculate the properties of galaxies and their haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We use tangos to construct merger trees that link haloes to their progenitors and descendants based on the number of particles they have in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Starting with our organic system, we identify the main branch of the tree by calculating the sum of the stellar masses along each possible branch, and then selecting the branch with the largest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We then identify the organic system’s counterparts in other simulations by matching on the number of particles in com- mon at 𝑧 = 8, and then tracking this system forwards in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This yields a more stable and reliable matching between simulations than 4 We choose this time as it corresponds to the final snapshot output in which the merging haloes are clearly distinguishable by the SUBFIND algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' attempting to find the organic system’s counterpart at each output time separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We calculate the properties of haloes, such as the halo mass (𝑀200) and baryon fraction ( 𝑓b ≡ 𝑀b/𝑀200, where 𝑀b is the total mass in baryons) within one virial radius of the halo centre of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We find the centre of mass with pynbody, using the shrinking-sphere method (Power et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The properties of galaxies, such as the stellar mass (𝑀★) and specific star formation rate (sSFR) are calculated within a spherical aperture of radius 30 physical kpc about the centre of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We calculate the intrinsic inner-halo binding energy (𝐸DMO 2500 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' that which is set by the halo’s assembly history and initial conditions, and not by dissipative baryonic processes) by matching each system to its counterpart in an equivalent dark matter-only simulation and summing the binding energies of all particles within a radius en- closing 2500 times the critical density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' For our zoom simulations, we perform this matching using tangos, and for the large-volume Ref-L100N1504 simulation we use the bijective particle matching algorithm described by Schaller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We calculate the total energy injected through stellar feedback, 𝐸★, by summing the energies contributed by all star particles within a 30 pkpc aperture about the centre of mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' when a gas particle 𝑖 is converted into a star particle it provides an energy given by 𝐸★,𝑖 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='74 × 1049 erg � 𝑚init ★,𝑖 1 M⊙ � 𝑓th,𝑖(𝑛H,𝑖, 𝑍𝑖), (1) where 𝑚init ★,𝑖 is the initial stellar mass and 𝑓th,𝑖 is an efficiency that depends on the density 𝑛H,𝑖 and metallicity 𝑍𝑖 of the gas particle at the time of conversion (for more information see Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The total feedback energy injected through AGN feedback by the galaxy’s central BH is given by 𝐸AGN = 𝜖f𝜖r 1 − 𝜖r (𝑀BH − 𝑀seed BH )𝑐2, (2) where 𝑀BH is the BH mass, 𝑐 is the speed of light, 𝜖r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1 is the radiative efficiency of the accretion disc and 𝜖f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='15 is the fraction of the radiated energy that couples to the surrounding gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 𝑀seed BH = 105 M⊙/ℎ is the mass of the BH seed, which does not contribute any feedback energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We note that these definitions include “ex-situ" energy injected by stars and BHs in other progenitor galaxies that subsequently merged with our main system, but since this energy directly impacts the properties of these progenitors we include it in our definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 3 RESULTS We first examine the impact of a varying halo assembly time in the absence of merger events in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1, before exploring how our systems evolve when mergers do occur in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Finally, in light of our findings, we discuss the origin of previously-identified correlations in the EAGLE galaxy population in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1 The impact of halo assembly time in the absence of merger events We begin by isolating the effect of a varying assembly time on the evolution of our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We do so by comparing the evolution of the secular and early-secular haloes, which have different assembly times and experience no significant mergers after their masses exceed 𝑀crit 200, the threshold above which disruptive events may lead to BH growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023) 4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1 Halo and galaxy evolution Figure 1 shows how the organic system and its secular and early- secular variants evolve in terms of their halo mass (𝑀200, upper panel), stellar mass (𝑀★, middle panel) and specific star formation rate (sSFR≡SFR/𝑀★, integrated over the preceding 100 Myr5, lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In this and other figures of this type, we indicate the first output after which a significant merger has occurred with vertical lines, coloured to match the simulation in which the merger occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We define significant mergers to be those in which the stellar mass ratio, 𝜇, is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We quantify the uncertainty that arises from EAGLE’s stochastic subgrid physics implementation by simulating nine realisations of each set of ICs, varying the seed used for the quasi-random number generator each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Solid lines indicate the median value at each output time, and shading shows the interval between the third and seventh values in the rank-ordered distribution, a good approximation of the interquartile range (IQR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The halo mass histories of the organic and secular galaxies are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We indicate the evolution of 𝑀crit 200 with a dashed line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' both systems exceed this threshold at approximately the same time (𝑡 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The early-secular halo assembles more quickly, and exceeds 𝑀crit 200 earlier, at 𝑡 ≈ 2 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' As intended, all systems converge to near-identical halo masses of 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='52 M⊙ (organic, IQR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='02 dex) and 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='54 M⊙ (secular and early-secular, IQR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='02 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='01 dex respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The secular halo temporarily has a higher mass than the others at 𝑧 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' this is due to mass bound to another halo passing within 𝑟200 of the main halo during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This halo passes closer to the secular system than it does to the organic and early-secular systems as an unintended side-effect of our modifications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' at its closest approach, its centre of mass lies approximately at the virial radius of the main halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' As a result of its earlier assembly time, the early-secular halo is intrinsically more tightly-bound than the secular halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' When simulated with purely collisionless dynamics, the binding energy of the inner halo, 𝐸DMO 2500 , is 30% higher for the early-secular system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' this difference corresponds to ≈ 1𝜎 in the flagship EAGLE simulation at this halo mass6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The stellar mass histories of the organic and secular galaxies also trace each other closely, but diverge slightly at 𝑡 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1 Gyr, as a result of the organic galaxy’s sole significant merger (after 𝑀crit 200 is exceeded);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' a minor merger of stellar mass ratio 𝜇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The ex-situ stellar mass contributed by this merger elevates the organic galaxy’s stellar mass above that of its secularly-evolving counterpart, but by the present day the secular system catches up through predomi- nantly in-situ star formation, and the organic and secular galaxies reach stellar masses of 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='67M⊙ (IQR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='07 dex) and 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='61M⊙ (IQR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='04 dex) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The early-secular galaxy attains a similar final stellar mass of 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='66M⊙ (IQR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='07 dex), but as- sembles the majority of that mass earlier as a result of an accelerated halo assembly history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Neither the secular or early-secular galax- ies experiences any significant mergers after the halo mass exceeds 𝑀crit 200, and therefore by comparing the properties of these systems we may examine the influence of the overall halo assembly time in the absence of mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In the lower panel of Figure 1 we show how these assembly his- tories influence the specific star formation rates of our galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We 5 Our results are not strongly sensitive to this choice, and we have verified that using longer timescales of 300 Myr and 1 Gyr yield similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 6 We calculated this statistic for haloes in the collisionless (DMONLY) EAGLE L100N1504 simulation, in a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1 dex wide mass window about 𝑀200 = 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='54 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2 4 6 8 10 12 t [Gyr] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 log10(M200/M⊙) Mcrit(z) (Bower + 17) Organic Secular Early-secular 2 4 6 8 10 12 t [Gyr] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 log10(M⋆/M⊙) 2 4 6 8 10 12 t [Gyr] −11 −10 −9 log10(sSFR) [yr−1] EAGLE green valley 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 1 2 3 z Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The organic system and our modified secular and early-secular variants reach approximately the same final halo mass (𝑀200) and stellar mass (𝑀★), but exhibit differences in their mass histories due to our genetic modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' All three systems remain star-forming until 𝑧 = 0, despite their halo masses being far in excess of 𝑀crit 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The upper, middle and lower panels show the evolution of 𝑀200, 𝑀★ and the specific star formation rate (sSFR) respectively for these three systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Solid lines show the median values for nine resimulations of each set of initial conditions with different random number seeds, and shading indicates the interquartile range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The time of the first snapshot output following any significant merger (stellar mass ratio 𝜇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1) is shown with a vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We indicate the critical halo mass, 𝑀crit 200, above which black holes can grow efficiently in EAGLE with a dashed black line in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The location of the green valley in the EAGLE population is shown with green hatching in the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' also show the location of the green valley as a function of time for galaxies of a comparable stellar mass in the largest EAGLE simula- tion volume (Ref-L100N1504).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We define this based on the sSFR of all EAGLE galaxies in a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 dex-wide window about the current stel- lar mass of the organic galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Following Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2019), we take the locus of the star-forming main sequence to be the mean sSFR MNRAS 000, 1–11 (2023) The origin of diversity in SMBH and galaxy growth 5 of these galaxies, subject to a floor log10(sSFR/yr−1) > −11 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5𝑧, and define the green valley to lie within 5% and 50% of this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' All three systems remain actively star-forming throughout their evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The earlier stellar mass assembly of the early-secular galaxy is reflected in its sSFR, which is higher at 𝑧 ≈ 3−2, and lower at 𝑧 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 − 1, than the later-assembling systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' However, there is little difference between its final sSFR of 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='34 yr−1 (IQR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='07 dex) and the final sSFR of the later-assembling secular system 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 yr−1, IQR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Therefore, while correlations within the wider EAGLE population indicate that earlier-assembling and more tightly-bound haloes tend to host central galaxies with lower sSFR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Matthee & Schaye 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2019, 2020), we find that adjusting these quantities alone does not causally affect the central galaxy’s present-day star formation activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 Feedback and the baryon fraction To investigate why changes in the assembly time alone do not change the present-day star formation rates of our galaxies, we now explore how the feedback history and the halo baryon content are affected by our modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In the upper panel of Figure 2, we show how the total AGN feedback energy, 𝐸AGN, injected into our systems as a function of time, and in the middle panel we show the fraction of the total feedback energy, 𝐸FB = 𝐸★ + 𝐸AGN, that is contributed by AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In the lower panel, we show how this energy injection influences the halo baryon fraction ( 𝑓b, normalised to the cosmic fraction 𝑓 cosmic b = Ωb/Ω0, lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Details of how each of these quantities was calculated are given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' since we calculate 𝐸AGN directly from the central SMBH mass, 𝑀BH, we show this mass as a second axis in the upper panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' To place the growth of the BHs in our systems in context, we briefly review the general behaviour of BHs in the EAGLE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2018) proposed that EAGLE BHs grow in three phases: (i) In lower-mass haloes, stellar feedback is able to efficiently expel gas from the central regions of galaxies, keeping the gas density in the BH’s vicinity, 𝜌BH, low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The BH is therefore unable to grow and remains close to the mass at which it was seeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2018) name this the stellar feedback regulated phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (ii) As 𝑀200 reaches 𝑀crit 200 and the halo virial temperature ap- proaches 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='6 K, stellar feedback-driven outflows have lower en- tropy than the CGM and are no longer buoyant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' These outflows can no longer remove gas from the galaxy centre, causing 𝜌BH to increase significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' EAGLE BHs then accrete gas according to a modified version of the spherically-symmetric Bondi & Hoyle (1944) formula (see Rosas-Guevara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015), growing at a non-linear rate pro- portional to 𝑀2 BH for as long as 𝜌BH remains high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2018) name this the non-linear growth (NLG) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Following their study, we take the NLG phase to be where dlog10(𝑀BH)/d𝑡 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='25 dex Gyr−1 and illustrate it with horizontal bars in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (iii) Once the BH becomes sufficiently massive that its AGN feed- back can expel gas from the BH’s immediate vicinity, 𝜌BH falls and the NLG phase ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Thereafter, the BH can regulate its own growth (and 𝜌BH) through AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2018) name this the AGN feedback regulated phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' However, as we will demonstrate in this study, AGN feedback does not necessarily dominate the energy injected into the galaxy and halo after the NLG phase ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' During the NLG phase, 𝐸AGN and 𝑀BH rise sharply, and this occurs markedly earlier for the early-secular system than for the later-assembling organic and secular systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' However, over 2 4 6 8 10 12 t [Gyr] 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 log10(EAGN) [erg] Organic Secular Early-secular 2 4 6 8 10 12 t [Gyr] −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 log10(EAGN/EFB) 2 4 6 8 10 12 t [Gyr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='8 fb/(Ωb/Ω0) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 log10(MBH/M⊙) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 1 2 3 z Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Differences in assembly history make little difference to 𝐸AGN and its contribution to the overall feedback budget in the absence of disruptive events, and all systems remain baryon-rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In the same fashion as Figure 1, the upper, middle and lower panels show the evolution of the integrated energy injected by AGN feedback (𝐸AGN), the fraction of the total feedback energy, 𝐸FB, contributed by AGN, and the halo baryon fraction ( 𝑓b, normalised to the cosmic fraction, Ωb/Ω0) respectively, for the organic, secular and early-secular systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We illustrate the phase of non-linear growth (NLG) undergone by the central black hole in each system with horizontal bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' the lifetimes of the secular and early-secular systems, the to- tal energy injected by AGN feedback is not significantly different (𝐸AGN = 1059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='8 erg with IQR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1 dex), and is slightly less than the energy injected into the organic system (𝐸AGN = 1060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 erg, IQR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1 dex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In the absence of merger events, earlier halo assembly and a 30% higher binding energy therefore does not necessarily yield a more massive BH or the injection of more AGN feedback energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This is somewhat surprising, since the model of Booth & Schaye (2010, 2011) predicts that BHs will grow until they have injected an energy set by the halo binding energy, and strong positive correlations exist between 𝑀BH and 𝐸DMO 2500 at fixed 𝑀200 in the EAGLE population (Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' While the final 𝑀BH and 𝐸AGN do not change, the early-secular BH does grow at a faster rate during the NLG phase, and we find that it attains a higher 𝜌BH during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This is consistent with the behaviour of the wider EAGLE population, MNRAS 000, 1–11 (2023) 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' in which BHs that enter the NLG phase earlier are more luminous and have higher Eddington ratios during the phase (McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This may be the result of a higher cosmological infall rate and/or mean density of the universe at earlier times, or be due to the higher halo binding energy making it harder for feedback processes to reduce the central gas density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' AGN feedback does not dominate the energy input into any of these systems, as shown in the middle panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' it contributes ≈ 30% of the feedback injected into the secular and early-secular systems and ≈ 40% of the feedback in the organic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 𝐸AGN/𝐸FB sharply increases as each system’s BH undergoes its NLG phase, but then remains fairly constant once this phase is complete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' the post-NLG phase therefore need not be dominated by AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Both stellar and AGN feedback continue to operate in tandem for the remainder of each system’s evolution, apparently co-existing in equilibrium and contributing similar rates of energy injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The lower panel shows that the feedback injected into these sys- tems has not led to a strong expulsion of baryons from their haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Prior to each system’s NLG phase, the baryon fraction 𝑓b is approxi- mately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='65 𝑓 cosmic b , lower than the cosmic fraction due to the effects of photoionisation and stellar feedback at early times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In all cases, the AGN feedback injected during the NLG phase causes a small decrease in 𝑓b, and in the organic halo the additional AGN feed- back induced by the minor merger causes its 𝑓b to fall below that of the secular halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Overall, the haloes remain baryon rich, retaining approximately 60 − 70% of the cosmic baryon fraction within their haloes at the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In EAGLE and other galaxy formation models, quenching galaxies appears to require that a large fraction of the halo’s baryons are expelled beyond the virial radius by AGN feedback, in order to reduce the cooling rate of CGM gas onto the interstellar medium (Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Zinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This does not occur for our secularly- evolving systems, explaining why they remain star-forming until the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='3 The role of the galaxy disc The results discussed so far demonstrate that, in the absence of merg- ers or other disruptive events, the assembly time and intrinsic binding energy of a dark matter halo do not strongly influence the present-day properties of our galaxy and its halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This result can be understood by considering the dynamics of the gas in the vicinity of the central BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2, EAGLE BHs can only grow rapidly while the density of gas in their vicinity (𝜌BH) remains high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' If gas is kept away from the BH, spread out in a co-rotating disc, 𝜌BH will be lower and the BH’s growth will be inhibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Using a similar genetic modification experiment, Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2022) showed that suppressing the influence of a merger also sup- presses the growth of the central BH, because the co-rotational mo- tion of the gas is preserved in the absence of any disruptive events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' A similar phenomenon is occurring for our secular and early- secular galaxies, which have persistent, strongly co-rotating discs throughout their evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Following the process outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2, their BHs undergo an NLG phase when 𝑀200 → 𝑀crit 200, before settling into the AGN feedback-regulated phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' For the remainder of their evolution, 80-90% of the kinetic energy of the gas within 3 physical kpc of their BHs is invested in co-rotational motion7, and so their BHs never grow rapidly again as the rotational support keeps 7 These values are equivalent to the 𝜅co diagnostic (Correa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2017) for the gas, calculated with the routines of Thob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 𝜌BH low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Figure 2 shows that the halo assembly time and binding energy influence when the NLG phase occurs, and how rapid the BH growth is in that phase, but they do not significantly change the AGN feedback energy required for the BH to reduce 𝜌BH and regulate its own growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The final BH mass is therefore similar in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Our findings show that when galaxies are undisturbed by mergers and retain gaseous discs, they settle into a state where a balance of stellar feedback and low-level AGN feedback is sufficient to regulate the growth of the BH and the galaxy, long after the 𝑀crit 200 threshold has been exceeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Their BHs inject only the minimum amount of energy required to end the NLG phase, and afterwards only require a small amount of AGN feedback energy to regulate the buildup of gas that migrates to the centre of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This feedback does not strongly affect the CGM, which can continue cooling onto the galaxy, where the majority of it settles onto the disc and forms stars, and a minority fuels slow and steady growth of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In this state, the growth of the BH and the properties of the host halo are effectively decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The BH does not need to ‘offset’ the cooling of gas from the whole halo in order to regulate its growth, as is the case in more massive group/cluster systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' McNamara & Nulsen 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Fabian 2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' instead, only a small fraction of the gas cooling from the halo can fuel the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' While a higher binding energy (and deeper potential) may in principle mean that more AGN feedback is required to expel this gas from the centre of the disc, the near-identical 𝐸AGN for our secularly-evolving galaxies demonstrates that this is not a significant effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' As a result, the integrated growth of the BH, and hence its influence on the CGM and the galaxy, is not sensitive to changes in the binding energy of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 The impact of merger events We now explore how our secular and early-secular galaxies and their haloes evolve when they do experience a disruptive merger after exceeding 𝑀crit 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We therefore turn to the systems evolved from our merger and early-merger initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Each of these systems undergoes a major merger with a stellar mass ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='46, occurring at 𝑡 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='4 Gyr (merger) and 𝑡 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='9 Gyr (early-merger).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' While the stellar mass ratios of these mergers are the same, we note that they are not the same merger, and they differ in their geometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' impact parameter and infall angle) and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1 Halo and galaxy evolution Figure 3 is identical in format to Figure 1, but now focuses on the evolution of the merger and early-merger halo mass, stellar mass, and sSFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' To demonstrate the differences induced by mergers relative to a secularly-evolving case, we also include the secular and early- secular systems but only show the median evolution, for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' As in previous figures, we highlight the snapshot times after which significant mergers have occurred for the merger and early-merger systems with vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' As shown in the upper panel, the halo mass histories of the secu- lar and merger haloes are similar to each other, as are those of the early-secular and early-merger haloes, reaching near-identical halo masses by the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The later-assembling pair of sys- tems cross the 𝑀crit 200 threshold at approximately the same time, as do the earlier-assembling pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The stellar mass histories (middle panel) are also very similar within the earlier- and later-assembling pairs of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' As with the halo mass, the histories differ at the time of the merger, as the merger and early-merger galaxies gain a significant ex-situ contribution to their stellar mass, while their secularly-evolving counterparts form stars more steadily in-situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023) The origin of diversity in SMBH and galaxy growth 7 2 4 6 8 10 12 t [Gyr] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 log10(M200/M⊙) Mcrit(z) (Bower + 17) Secular Early-secular Merger Early-merger 2 4 6 8 10 12 t [Gyr] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 log10(M⋆/M⊙) 2 4 6 8 10 12 t [Gyr] −13 −12 −11 −10 −9 log10(sSFR) [yr−1] EAGLE green valley 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 1 2 3 z Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Our galaxies modified to undergo major mergers attain similar final 𝑀200 and 𝑀★ to their secularly-evolving counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' However, the mergers cause them to leave the main sequence, enter the green valley, and in many cases quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The three panels are equivalent to those in Figure 1, but now show the evolution of 𝑀200, 𝑀★ and the sSFR for the merger and early- merger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The median evolution for the secular and early-secular systems is shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The post-merger stellar mass histories of the merger and early- merger galaxies can be understood by examining the evolution of their sSFR, shown in the lower panel of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Prior to the mergers taking place, the galaxies are on the main sequence, closely follow- ing the sSFR of their secularly-evolving counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' However, the sSFR declines following the mergers in both cases, with a stronger decline seen for the early-merger galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The scatter introduced by stochasticity is particularly notable here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' in the early-merger case, many realisations of the galaxy rapidly fall through the green val- ley and quench within the time interval between simulation outputs, whereas others fall into the green valley and remain there until 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In general, the median early-merger galaxy is quenched due to the merger it experiences, whereas the median merger galaxy resides in the green valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 Feedback and the baryon fraction Figure 3 shows that individual merger events are able to induce strong changes in the sSFR of our galaxy, to a degree that adjust- ing the overall assembly time alone did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2022) showed that mergers induce such changes because they disrupt the co-rotational motion of gas in the galaxy disc, greatly increasing the density of gas in the central BH’s vicinity and allowing the BH to grow more massive (and thus inject more AGN feedback energy) than in a secularly-evolving system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This phenomenon also occurs for our merger and early-merger systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' prior to the mergers, 80-90% of the kinetic energy of the gas within 3 physical kpc of their BHs is invested in co-rotational motion, but in the snapshots following their merger events, this fraction has fallen to ∼ 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This greatly enhances the growth of the BHs in these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We show how this disc disruption affects the BH’s growth in Figure 4, which has the same format as Figure 2 but now focuses on the feedback energetics and halo baryon content of the merger and early-merger systems, again also showing the median evolution of their secularly-evolving counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The haloes exceed 𝑀crit 200 at approximately the same time as their secularly-evolving counterparts, and so their central BHs begin their NLG phases at similar times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The masses of the BHs then quickly exceed those in the secularly-evolving galaxies once the merger events begin to influence the amount of gas available to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' To follow this process in more detail, we performed additional simulations with 10 Myr output time resolution, finding that BH growth in the merger system is stimulated at the infalling halo’s first periapsis over 1 Gyr before the merger, whereas in the early-merger system it is the final coalescence that triggers rapid BH growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The final masses of these BHs are 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='9 M⊙ (merger, IQR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 dex) and 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 M⊙ (early-merger, IQR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 dex), significantly exceeding the masses of the BHs in the secularly- evolving galaxies (107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='4 M⊙, IQR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1 dex in both cases), and consequently factors of ≈ 3 and ≈ 4 times more AGN feedback energy is injected in these systems respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The middle panel of Figure 4 shows that mergers significantly increase the fraction of the total feedback energy that is contributed by AGN feedback;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' by the present day AGN feedback accounts for ≈ 55% of the feedback in the merger system and ≈ 59% in the early-merger system (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 30% in the secularly-evolving systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This is predominantly the result of a large increase in 𝐸AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' the mergers increase the integrated stellar feedback energy by only small factors of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='3 for the merger and early-merger galaxies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This additional AGN feedback expels a large fraction of the CGM from the merger and early-merger haloes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' as shown in the lower panel they both retain only ≈ 30% (IQR= 20%) of the cosmic baryon fraction within their haloes at the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This reduction in the baryon content of the CGM causes it to cool less efficiently, explaining the reduced star formation rates of these galaxies (see also Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='3 Two modes self-regulation in galaxies with massive SMBHs In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 we outlined the picture described by Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2017) and McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2018), in which EAGLE’s BHs pre- dominantly grow in two phases separated by a transitional non-linear growth phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2018) name the final, post-NLG phase the “AGN feedback regulated phase”, implying that the regulation of MNRAS 000, 1–11 (2023) 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' BH and galaxy growth is dominated by AGN feedback, but our find- ings show that this is not necessarily the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' EAGLE galaxies can remain star-forming and regulated by a mixture of stellar and low- level AGN feedback long after the NLG phase, so long as no mergers disrupt the gas disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We therefore propose that the post-NLG phase can be further split into two modes: Stellar & AGN feedback co-regulation: The presence of a co-rotating gas disc in the galaxy largely decouples the growth of the BH from the properties of the host halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Gas that cools from the CGM settles into the disc and the BH need only grow enough (and inject enough AGN feedback) to regulate the gas that migrates to the centre of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The CGM is minimally affected by this AGN feedback and the galaxy remains star-forming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The BH may be locally self-regulating with AGN feedback, but the galaxy and its halo are regulated by a mixture of stellar and low-level AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' AGN feedback regulation: When the galaxy disc is dis- rupted/destroyed, the gas cooling from the CGM (in addition to the cold gas already in the galaxy) can reach the BH’s vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The BH must therefore regulate inflow from the halo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' this involves ejecting a significant fraction of the CGM in order to reduce the density, and hence the cooling rate, of the remaining circumgalactic gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This regulation mode requires the injection of far more AGN feedback and the growth of the BH to a higher mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The amount of energy required may depend on halo properties such as the binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The prevention of cooling from the CGM reduces or quenches star formation in the galaxy, and AGN feedback becomes the dominant energy injection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The BHs in the secular and early-secular galaxies remain in the first of these modes after they begin to regulate their own growth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' changes to the halo’s formation time and binding energy therefore have little influence on the energy injected by AGN (and in turn the halo 𝑓b and galaxy sSFR), as shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In the merger and early-merger systems, however, a merger occurs that induces a transition to the second of these modes, transforming the subsequent evolution of the BH, CGM and galaxy, and recoupling the properties of the central BH to the properties of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The transition between these two modes is likely essential to the diversity in BH, CGM and galaxy properties at fixed halo mass in the EAGLE galaxy population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='3 Re-interpreting correlations in the EAGLE population Previous studies of the galaxy populations in the large-volume EA- GLE (and IllustrisTNG) simulations have concluded that diversity in the properties of BHs, galaxies and their gaseous haloes at fixed halo mass stems from differences in halo binding energy, which in turn is assumed to be the result of differences in assembly/collapse time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The basis for these conclusions was the clear correlation between the inner halo binding energy, 𝐸DMO 2500 , and 𝑀BH at fixed 𝑀200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The ex- planation for this correlation came from self-regulation arguments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' tightly-bound haloes more effectively confine gas at the halo centre, requiring the injection of more AGN feedback (and hence a higher 𝑀BH) to expel gas and regulate the growth of the BH and galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Booth & Schaye 2010, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2019, 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' However, this explanation neglects the influence of the galaxy disc and the role of merger events, which we have shown to be crucial to the evolution of the BH and its host galaxy in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We therefore now revisit this correlation and consider its origin in light of this new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We show the correlation for the galaxy population in the EAGLE Ref-L100N1504 simulation in Figure 5, plotting 𝑀BH as a function of 𝑀200 for all central galaxies with 𝑀200 > 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 M⊙, and colouring 2 4 6 8 10 12 t [Gyr] 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 log10(EAGN) [erg] Secular Early-secular Merger Early-merger 2 4 6 8 10 12 t [Gyr] −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 log10(EAGN/EFB) 2 4 6 8 10 12 t [Gyr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='8 fb/(Ωb/Ω0) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 log10(MBH/M⊙) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 1 2 3 z Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Mergers cause the injection of significantly more AGN feedback in the merger and early-merger systems, leading AGN feedback to constitute a greater fraction of the feedback energy injected into these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The expulsive effect of this feedback depletes the baryon content of their haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The three panels are equivalent to those in Figure 2, but now show the evolution of the feedback energetics and halo baryon content for the merger and early-merger systems, with the median evolution for their secularly- evolving counterparts shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We illustrate the phase of non- linear growth (NLG) undergone by the central black hole in each system with horizontal bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' the data points by the residuals (in log space) of the 𝐸DMO 2500 − 𝑀200 relation8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The marker colours reveal a very strong positive corre- lation between 𝐸DMO 2500 and 𝑀BH at fixed 𝑀200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The locations of our genetically-modified systems are overlaid with larger symbols, coloured in the same fashion as the wider population;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' for each sys- tem, we calculate 𝐸DMO 2500 and find the residual from the population median at the system’s halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The modified systems span the majority of the scatter in 𝑀BH at 𝑀200 ≈ 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 M⊙, and their 𝐸DMO 2500 values agree well with the underlying correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The systems that experienced merger events 8 The residuals are taken with respect to a running median value obtained through the locally weighted scatterplot smoothing method (LOWESS, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Cleveland 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023) The origin of diversity in SMBH and galaxy growth 9 not only host overmassive BHs but also have high inner halo binding energies, and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The merger and secular systems have very similar assembly times and differ only by the presence, or lack of, a major merger, yet the merger halo is significantly more tightly bound for its mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The inner halo binding energy is therefore not only set by the assembly and collapse time, but can be strongly increased by individual mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Comparing the secular and early-secular systems shows that earlier collapse does yield a higher 𝐸DMO 2500 , but the increase is far smaller than that caused by a major merger, and it does not cause enhanced BH growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This result aligns well with the early predictions of Neto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2007), who demonstrated that the connection between halo concen- tration and formation time is clearer when the definition of formation time includes all of a halo’s major progenitors and not just one, indi- cating that mergers play a key role in setting the concentration (and hence the binding energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' More recently, Rey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' (2019) used ge- netic modification to increase the variance of the overdensity field in a halo’s initial conditions, increasing the number of significant merg- ers in the merger tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' this change also caused the halo concentration to increase, providing further evidence for this connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Our findings suggest that the correlation between the halo binding energy and the BH mass emerges because mergers help to grow BHs to high masses and can significantly increase the binding energy of the host dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This picture also provides an explanation for why earlier-assembling haloes in cosmological simulations tend to have higher binding energies and host quenched central galaxies with overmassive central SMBHs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Matthee & Schaye 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Montero-Dorta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' these systems reside in denser, more clustered environments (Sheth & Tormen 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2005) and are therefore likely to experience more mergers throughout their evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 4 SUMMARY AND DISCUSSION In this study, we have used the genetic modification technique (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2016) to assess how supermassive black hole (BH) growth and the impact of AGN feedback are influenced by individual galaxy- galaxy merger events and the overall assembly history of the host halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This experiment was motivated by predictions from cosmolog- ical simulations (see references in Section 1) that mergers can induce BH growth and AGN feedback, and that the BH mass correlates strongly and positively with the binding energy of the dark matter halo, which in turn is correlated with the overall assembly time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We performed zoom simulations of a star-forming disc galaxy of stellar mass 𝑀★ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='3 × 1010 M⊙ and host halo mass 𝑀200 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='4 × 1012 M⊙ with the recalibrated high-resolution version of the EAGLE galaxy formation model (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Crain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Using the initial conditions generator genetIC (Stopyra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2020), we modified the initial conditions of our fiducial galaxy and its host halo to generate four new variants of it with systematically adjusted assembly histories, designed such that we could independently assess the roles of mergers and the overall assembly time in a controlled galaxy formation experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' First, we produced two modified variants of our galaxy for which no significant mergers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' where the stellar mass ratio 𝜇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='1) occur after the host halo reaches the critical mass, 𝑀crit 200, above which BHs are able to grow via accretion in the EAGLE model (see Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' One of these systems (secular) has a similar assembly time to the unmodified case, while the other (early-secular) assembles earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This earlier assembly time causes the inner dark matter halo of the early-secular case to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 log10(M200/M⊙) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='5 log10(MBH/M⊙) Secular Merger Early-secular Early-merger −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='20 ∆ log10(EDMO 2500 ) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The black hole mass, 𝑀BH, as a function of 𝑀200 for the galaxy pop- ulation in the Ref-L100N1504 EAGLE simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Datapoints are coloured according to the residuals of the log10(𝐸DMO 2500 ) − log10(𝑀200/M⊙) relation, where 𝐸DMO 2500 is the binding energy within a sphere enclosing 2500 times the critical density for each halo’s counterpart in a collisionless dark matter sim- ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' At fixed 𝑀200, haloes that are intrinsically more tightly-bound host more massive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Our genetically-modified systems are overlaid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' our modifications to their assembly histories cause them to span the scatter in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Adjusting the assembly time in the absence of mergers changes 𝐸DMO 2500 but not 𝑀BH, while mergers can both drive high 𝑀BH and cause haloes to be more intrinsically tightly-bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Merger events appear to be required for establishing both diversity in 𝑀BH and a correlation with the halo binding energy at fixed 𝑀200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' be intrinsically more tightly-bound;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' when measured in an equivalent dark matter-only simulation, its present-day binding energy, 𝐸DMO 2500 , is 30% higher than the secular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Comparing these systems allowed us to isolate how differences in the halo assembly time influence the growth of the central BH, free of the influence of significant mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We find that these differences do not drive significant changes to the present-day properties of the galaxy and its host halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The secular and early-secular galaxies both reach the same present-day stellar mass and remain on the star- forming main sequence (Figure 1), and their central BHs grow to approximately the same mass and inject the same amount of AGN feedback energy, both in absolute terms and as a fraction of the total feedback energy, and their gaseous haloes remain baryon-rich (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The BHs grow according to a three-phase process, as is expected in the EAGLE model (McAlpine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' they initially remain close to the seed mass, undergo non-linear growth (NLG) when the halo mass exceeds 𝑀crit 200, and then grow more steadily once AGN feedback is able to expel gas from the BH vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We find that differences in assembly time influence when the NLG phase occurs and the typical accretion rate in this phase, but do not change the total energy injected or the impact of the feedback on the galaxy- CGM ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Once the NLG phase ends, the growth of the BH and galaxy are regulated by a combination of stellar and low-level AGN feedback, which contribute approximately 70% and 30% of the feedback energy injected by 𝑧 = 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' To test the influence of mergers, we produced two more sets of ini- MNRAS 000, 1–11 (2023) O10 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' tial conditions, merger and early-merger, designed to yield haloes with similar assembly times to the secular and early-secular galaxies respectively, but also experience a major merger (𝜇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='46) after crossing 𝑀crit 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' At 𝑧 = 0, we find that these systems have approx- imately the same halo and stellar masses as their secularly-evolving counterparts, but are either quenched or reside in the green valley (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The mergers cause the BHs in these systems to grow significantly more massive than the BHs in their secularly-evolving counterparts, and inject significantly more AGN feedback energy (by factors of ≈ 3 and ≈ 4 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' AGN dominate the feedback energy injected into these systems, contributing ≈ 55% and ≈ 59% of the total integrated energy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The expulsive nature of this feedback depletes the haloes of their baryons, and they retain only ≈ 30% of the cosmic baryon fraction by the present day (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We attribute these results to the vital importance of the galaxy disc in determining the conditions in the vicinity of the central BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The secular and early-secular galaxies retain strong co-rotating gas discs throughout their evolution, with 80−90% of the kinetic energy of the gas invested in co-rotational motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This rotational support reduces the inflow rate towards the BH, suppressing its growth and reducing the feedback energy required to maintain self-regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The BHs in these systems appear to undergo non-linear growth to a minimum mass at which AGN feedback can expel gas from their vicinity, and then grow very little thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The majority of the gas cooling from the halo settles onto the disc and fuels continued star formation, with a small minority fuelling slow growth of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The disc therefore decouples the growth of the BH from the properties of the host halo, and hence changes in assembly time and binding energy have little influence on our secularly-evolving galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In our merger and early-merger systems, the disc is disrupted, allowing gas in the galaxy and halo to fuel BH growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This increases the AGN feedback energy required to maintain self-regulation, and causes a transformation of the CGM and the quenching of star forma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The disruption of the disc (through mergers or otherwise) there- fore appears to be key to the establishment of diversity in the masses of BHs, and to coupling the properties of BHs with those of their host haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We propose that once galaxies host massive central BHs, their growth can be regulated in one of two modes: (i) co-regulation by stellar feedback and low-level AGN feedback in secularly-evolving, star-forming disc galaxies, and (ii) AGN-dominated regulation in systems without discs, which are likely quenched due to the influ- ence of integrated AGN feedback on the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' The transition of galaxies between these modes may be essential to the diversity in galaxy properties seen in the EAGLE simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We concluded by reconsidering the origin of strong positive cor- relations between the BH mass and the host halo binding energy seen at fixed halo mass in the EAGLE population, in light of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We placed the properties of our modified systems in the context of this correlation (Figure 5), and deduced that the correlation likely emerges because mergers are key to growing BHs to high masses and they can significantly increase the binding energy of the underlying dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Our experiment has revealed how mergers and halo properties influence the fate of a single galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' While this behaviour is not guaranteed to be universal in the galaxy populations of the EAGLE cosmological simulation volumes, circumstantial evidence for the essential role of mergers does exist in the EAGLE Ref-L100N1504 simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' galaxies residing in ∼ 𝐿★ haloes that have far exceeded 𝑀crit 200 but host undermassive BHs and remain CGM-rich tend to have rotation-dominated stellar kinematics, while those with overmassive BHs and gas-poor haloes tend to have dispersion-dominated kine- matics indicative of disruptive mergers in their past (Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' To more firmly establish a connection between mergers and the im- pact of AGN within the population, one could compare the properties of two samples of galaxies with similar present-day halo masses, but where galaxies in one sample have experienced a disruptive merger when 𝑀200 > 𝑀crit 200 and those in the other have not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Assembling such samples presents a challenge, however, primarily due to difficulties in defining what makes a merger ‘disruptive’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Whilst a high stellar mass ratio is likely a good indicator, other factors such as the morpholo- gies of the merging galaxies, gas-richness, orbital configuration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' prograde vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' retrograde) or orbit type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' “spiral-in” or “head-on”) may be equally important, with even 1:1 mergers often causing little disruption if they are gas-rich/prograde/spiral-in (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Font et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Garrison-Kimmel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Peschken et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' In our experiment we were able to systematically adjust the mass ratio of mergers to make them more or less disruptive, but we did not control for any of the above extra characteristics of mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' Some characteristics are influenced by each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' for example, when we increase the mass ratio, we find that mergers become more “head- on” with smaller impact parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We expect that in future work, we will be able to identify the most important characteristics of mergers by genetically-modifying the angular momentum of a galaxy and its progenitors, allowing for controlled adjustment of a merger’s trajectory (Cadiou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We have shown in this work that galaxies which retain strong co- rotating gas discs can remain star-forming and minimally influenced by AGN feedback at halo masses above the threshold at which AGN are expected to dominate and transform the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We have focused here on the ∼ 𝐿★ mass scale in order to explain diversity in the properties of Milky Way-like galaxies, however this picture may also apply at much higher masses and in more extreme systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' While the likelihood of a galaxy experiencing a disruptive merger increases with stellar mass (Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2018), a rare population of very massive (𝑀★ > 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content='3 M⊙) blue “super-spiral” galaxies has been observed at low redshift (Ogle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' These galaxies exhibit a mixture of old and young stellar populations characteristic of early assembly and a consistently high star formation rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' they may therefore represent an extreme, high-mass case of the behaviour seen in our early-secular system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' It should be possible to test this hypothesis using a genetic modification experiment, in which the assembly of a massive halo (𝑀200 ≈ 1013 M⊙) hosting a quenched spheroidal galaxy is accelerated and the merger history is made as secular as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' If these modifications yield similar changes to those in our early-secular system, the central galaxy may become a super-spiral that remains star-forming until 𝑧 = 0, long after one might expect it to have been quenched by AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' ACKNOWLEDGEMENTS JJD would like to thank Corentin Cadiou and the GMGalaxies team at UCL for helpful discussions and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This study was supported by the European Union’s Horizon 2020 research and innovation pro- gramme under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' 818085 GMGalaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' AP and RAC are supported by the Royal Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' This study used computing equipment funded by the Research Capital Investment Fund (RCIF) provided by UKRI, and partially funded by the UCL Cosmoparticle Initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' It also made use of high performance computing facilities at Liverpool John Moores University, funded by the Royal Society and LJMU’s Faculty of Engineering and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' We thank the MNRAS 000, 1–11 (2023) The origin of diversity in SMBH and galaxy growth 11 EAGLE team for making the particle data (The EAGLE team 2017) and galaxy catalogues (McAlpine et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=', 2019, MNRAS, 487, 3740 Zeng G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=', Wang L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=', Gao L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=', 2021, MNRAS, 507, 3301 Zinger E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=', 2020, MNRAS, 499, 768 This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} +page_content=' MNRAS 000, 1–11 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E2T4oBgHgl3EQf0wi3/content/2301.04145v1.pdf'} diff --git a/sNE1T4oBgHgl3EQfjQRT/content/tmp_files/2301.03260v1.pdf.txt b/sNE1T4oBgHgl3EQfjQRT/content/tmp_files/2301.03260v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..82a9b7b06cf0df91e4a4c94adf883b0701b5841d --- /dev/null +++ b/sNE1T4oBgHgl3EQfjQRT/content/tmp_files/2301.03260v1.pdf.txt @@ -0,0 +1,3210 @@ +arXiv:2301.03260v1 [math.AP] 9 Jan 2023 +ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS OF FRACTIONAL +SEMILINEAR NEUMANN PROBLEM +SOMNATH GANDAL, JAGMOHAN TYAGI +Abstract. We establish the asymptotic behaviour of the least energy solutions of the following nonlocal Neumann +problem: +� +� +� +d(−∆)su + u = |u|p−1 u in Ω, +Nsu = 0 in Rn \ Ω, +u > 0 in Ω, +where Ω ⊂ Rn is a bounded domain of class C1,1, 1 < p < n+s +n−s, n > max {1, 2s} , 0 < s < 1, d > 0 and Nsu is the +nonlocal Neumann derivative. We show that for small d, the least energy solutions ud of the above problem achieves +L∞ bound independent of d. Using this together with suitable Lr-estimates on ud, we show that least energy solution +ud achieve maximum on the boundary of Ω for d sufficiently small. +Contents +1. +Introduction +1 +2. +Auxiliary Results +4 +3. +Regularity and bounds for least energy solution ud +6 +4. +Lr- estimates on ud +11 +5. +Proof of theorem 1.4 +14 +Appendix A +22 +Acknowledgement +22 +Statement +22 +References +22 +1. Introduction +We discuss the asymptotic behaviour of non-constant least energy solutions of the following problem: + + + +d(−∆)su + u = |u|p−1 u in Ω, +Nsu = 0 in CΩ, +u > 0 in Ω, +(1.1) +where Ω ⊂ Rn be a bounded domain of class C1,1, 1 < p < n+s +n−s, n > max {1, 2s} , 0 < s < 1, d > 0, CΩ := Rn \Ω and +Nsu is the nonlocal Neumann derivative, which is defined next. The nonlocal operator (−∆)s is called the fractional +Laplacian which is defined as follows: +(−∆)su(x) = cn,sP.V. +� +Rn +u(x) − u(y) +|x − y|n+2s dy. +(1.2) +Here, by P.V., we mean the Cauchy principal value and cn,s is a normalizing constant, given by +cn,s = +�� +Rn +1 − cosx1 +|x|n+2s dx +�−1 +, +2010 Mathematics Subject Classification. 35J60, 35B09, 35B40, 35J61, 35R11, 35D30. +Key words and phrases. Semilinear Neumann problem; fractional Laplacian; positive solutions; asymptotic behaviour. +Submitted January 10, 2023 +Published—–. +1 + +2 +S. GANDAL, J. TYAGI +see for instance [14] for the details. Recently, Dipierro et al. [16] have introduced a new nonlocal Neumann condition +Ns, which is defined as follows: +Nsu(x) := cn,s +� +Ω +u(x) − u(y) +|x − y|n+2s dy, x ∈ CΩ. +(1.3) +The advantage of this nonlocal Neumann condition is that it has simple probabilistic interpretation and (1.1) has a +variational structure. Further, Nsu approaches to the classical Neumann derivative ∂νu as s goes to 1. +In the last few decades, mathematical analysis of biological phenomena has gained much attention. For example, +the chemotaxis models, which are also known as Keller-Segel models [33], have been widely studied in different +directions in many papers, see [8, 10, 11, 24, 25, 26, 27, 31, 38] and the reference therein. Chemotaxis is the natural +behaviour of an organism in response of surrounding chemical gradients that are frequently separated by the cells +themselves. We refer to [3, 27, 28] for a survey on this subject. The Keller-Segel system with suitable initial data has +blow-up solutions in dimension n ≥ 2 and all solutions are regular in dimension n = 1, see for instance [25, 29, 31, 38] +and the references therein. The analysis on the steady-state for a chemotactic aggregation model with linear or +logarithmic sensitivity function was thoroughly done in many papers, see for instance [32, 34, 39, 40, 43]. Let us +point out that the following semilinear Neumann problem is an example of Keller-Segel model with a logarithmic +chemotactic sensitivity: + + + +−d∆u + u = |u|p−1 u in Ω, +∂u +∂ν = 0 on ∂Ω +u > 0 in Ω, +(1.4) +where d > 0, Ω ⊂ Rn is a bounded domain with smooth boundary and 1 < p ≤ n+2 +n−2 if n ≥ 3 and 1 < p < ∞ if +p = 2, see [34, 43] for the details. Problem (1.4) admits a non-constant solution for d sufficiently small, see [1, 34, 35]. +Lin et al. [34] and C. S. Lin, W. -M. Ni [35] established the solutions of (1.4) in the subcritical case 1 < p < n+2 +n−2. +In the critical case, when p = n+2 +n−2, Adimurthi and G. Mancini [1] obtained a solution of (1.4). There have been +developments on the asymptotic behaviour of solutions to such equations. In the subcritical case, 1 < p < n+2 +n−2, +W. -M. Ni and I. Takagi [40, 41] have studied the shape of least energy solutions of (1.4). They have shown that +the least energy solutions tends to zero as the diffusion constant d goes to zero except at finite number of points. +Moreover, the maximum of a solution ud of (1.4) is attained at a unique point on the boundary of Ω. The critical +case, i.e., p = n+2 +n−2, was examined by Adimurthi et al. [2] using the blow-up analysis. We refer to [23] for the exis- +tence, non-existence and the asymptotic behaviour to critical fractional Choquard equation with a local perturbation. +We mention that Problem (1.1) which we explore in this paper is a nonlocal analogue of the classical problem (1.4). +Recall that the movements of cells of some organisms cannot be described by random jumps. In such situations, +L´evy flights plays an important role. The generalized Keller-Segel model with nonlocal diffusion term d(−∆)s, where +d is a positive constant is used to investigate the chemotaxis. For the fractional Keller-Segel model, we refer to +[18, 30]. In [30], H. Huang and J. Liu studied the existence, stability, uniqueness and regularity for the following +model in dimension n ≥ 2 : + + + + + +ut = d(−∆)su − ∇ · (u∇φ) , +x ∈ Rn, t ≥ 0, +−∆φ = u, +u(x, 0) = u0(x), +(1.5) +where d is a positive constant, u(t, x) is the density of some biological cells and φ(t, x) is the chemical substance +concentration. We mention the work [9], where authors have investigated the asymptotic behaviour of solutions +for nonlinear elliptic problems for fractional Laplacian with Dirichlet boundary condition. We refer to [36] and the +reference therein for in-depth treatment of variational methods to nonlocal fractional problems. +Motivated by the above works and very recent works on nonlocal Neumann problem for fractional Laplacian and +its connections with fractional Keller-Segel models, we have the following natural question to ask: +Question. Can we establish the asymptotic behaviour of least energy solutions of (1.1)? + +ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS +3 +The aim of this paper is to answer the above question. A weak solution of (1.1) can be obtained as a critical point +of the energy functional Jd, which is defined as follows: +Jd(u) := 1 +2 +�dcn,s +2 +� +T (Ω) +|u(x) − u(y)|2 +|x − y|n+2s dxdy + +� +Ω +u2dx +� +− +1 +p + 1 +� +Ω +|u|p+1dx, +u ∈ Hs +Ω. +(1.6) +In the above equation T (Ω) = R2n \ (CΩ)2 and the space Hs +Ω is defined in (2.1). The functional Jd is well-defined +and of class C2 follows from the Theorem 2.1. An application of Mountain-Pass Lemma applying to the functional +Jd yields that +cd := inf +γ∈Γ max +[0,1] Jd(γ(t)) +(1.7) +is a critical value of Jd. In the above equation, by Γ, we mean the following set: +Γ = +� +γ ∈ C([0, 1]; Hs +Ω) | γ(0) = 1, γ(1) = u +� +, +where u ∈ Hs +Ω, u > 0 and satisfying Jd(u) = 0. It turns out that cd is the least positive critical value, see, Lemma 3.3 +next. For the details one may refer, Theorem 6.1 [4] and Theorem 1.1 [6], where authors have obtained a nonnegative +weak solution ud of (1.1) with critical value cd, provided d is sufficiently small. Moreover, ud satisfies +0 < Jd(ud) ≤ Cd +n +2s , +where the constant C is independent of d. Consequently, ud is non-constant. From the proof of Theorem 1.1 [6], it +is immediate to see that the critical points of Jd are not sign changing in Ω. In fact, when ud ≤ 0, we can choose +−ud in order to have a nonnegative solution of (1.1). By the strong maximum principle (see, Theorem 2.6 [12]), one +can see that ud > 0 a.e. in Ω. Further, since ud satisfies the Neumann condition Nsud(x) = 0 in CΩ which implies +that ud > 0 a.e. in Rn. +Definition 1.1. We call a critical point ud of Jd with Jd(ud) = cd, the least energy solution or Mountain-Pass +solution of (1.1). +We show the asymptotic behaviour of least energy solutions of (1.1) following the similar approach as was used +for (1.4) by W. -M. Ni and I. Takagi [40]. They used a positive solution w of non-linear Schr¨odinger equation +−∆u + u = |u|p−1 u in Rn, +1 < p < n + 2 +n − 2 +to study the asymptotic behaviour of the least energy solutions of (1.4). +The fractional non-linear Schr¨odinger +equation +(−∆)su + u = |u|p−1 u in Rn, +(1.8) +where 1 < p < +n+2s +n−2s, n > max {1, 2s} , 0 < s < 1 is thoroughly studied, see for instance [7, 15, 20, 21] and the +references therein. +The main idea of this work is as follows. Let cd be a critical value of Jd, which is defined in (1.7). We use a +positive solution w of (1.8) to observe the asymptotic behaviour of cd as d ↓ 0. More specifically, w is used to build +a suitable function φd to compare cd with maxt≥0 Jd(tφd). In particular, we obtain an inequality +cd < d +n +2s +2 F(w) +for d sufficiently small, where F is the functional associated with (1.8), defined in (2.3). This is closely related to the +location of maximum point of a solution ud of (1.1) on the boundary of Ω. +Now, we summarise the above discussions in terms of the following three main theorems: A priori it is known +that for 1 ≤ p < n+s +n−s, any weak solution u of (1.1) satisfies +∥u∥L∞(Ω) ≤ K, +where K > 0 is some constant depending on Ω, p and d, see Theorem 3.1[37]. In next result we obtain a bound for +least energy solution ud of (1.1) which is independent of d. + +4 +S. GANDAL, J. TYAGI +Theorem 1.2. Let ud be the least energy solution of (1.1). Then +dcn,s +2 +� +T (Ω) +|ud(x) − ud(y)|2 +|x − y|n+2s +dxdy + +� +Ω +u2 +ddx = +� +Ω +up+1 +d +dx ≤ C0d +n +2s , +(1.9) +where C0 > 0 is some constant depending on p. Moreover, there is a constant C1 > 0 depending only on p and Ω +such that +sup +Ω +ud(x) ≤ C1. +(1.10) +In the next theorem, we show that the Lr-norm of the least energy solution ud is bounded by d +n +2s times some +constant independent of d. +Theorem 1.3. Let ud be the least energy solution of (1.1). Then +b(r)d +n +2s ≤ +� +Ω +ur +ddx ≤ B(r)d +n +2s , if 1 ≤ r ≤ ∞. +(1.11) +b(r)d +n +2s ≤ +� +Ω +ur +ddx ≤ B(r)d +nr +2s , if 0 < r < 1, +(1.12) +where b(r) and B(r) are positive constants such that b(r) < B(r) and are independent of d. +We show the asymptotic behaviour in next theorem. +Theorem 1.4. Let Ω ⊂ Rn be a bounded domain of class C1,1. Let ud be the least energy solution of (1.1). If ud +achieves maximum at a point zd ∈ Ω, then for all d sufficiently small, we have the following: +(A) There exists a positive constant K∗ such that ρ(zd, ∂Ω) ≤ K∗d +1 +2s . Here, by ρ we mean the distance between +zd and ∂Ω. +(B) zd ∈ ∂Ω. +The plan of the paper is as follows. In Section 2, we recollect known results which are useful for our analysis. In +Section 3, we study the regularity of least energy solution of (1.1) and complete the proof of Theorem 1.2. In Section +4, we have derived Lr-estimate for the least energy solutions of (1.1). Section 5 is devoted to the proof of Theorem +1.4.The proof of inequality (3.12) (see, next) is a part of Appendix A. +2. Auxiliary Results +Let us recall the important results which are used in this paper. +Theorem 2.1. (Fractional Sobolev Embedding [14]) Let n > 2s and 2∗ +s = +2n +n−2s be the fractional critical exponent. +Then, we have the following inclusions: +(1) for any function u ∈ C0(Rn) and for q ∈ [0, 2∗ +s − 1] : +∥u∥2 +Lq+1(Rn) ≤ B(n, s) +� +Rn +� +Rn +|u(x) − u(y)|2 +|x − y|n+2s dxdy, +for some positive constant B. That means Hs(Rn) is continuously embedded in Lq+1(Rn). +(2) Let Ω ⊂ Rn be a bounded extension domain for Hs(Ω). Then, the space Hs(Ω) is continuously embedded in +Lq+1(Ω) for any q ∈ [0, 2∗ +s − 1], i.e, +∥u∥2 +Lq+1(Ω) ≤ B(n, s, Ω) ∥u∥2 +Hs(Ω) +for some positive constant B. Further, the above embedding is compact for any q ∈ [0, 2∗ +s − 1). +Let T (Ω) := R2n \ (Rn \ Ω)2 be a cross-shaped set on a bounded domain Ω ⊂ Rn. Define +(2.1) +Hs +Ω := +� +u : Rn −→ R measurable : ∥u∥Hs +Ω < ∞ +� +which is equipped with the norm +(2.2) +∥u∥Hs +Ω := +� +∥u∥2 +L2(Ω) + +� +T (Ω) +|u(x) − u(y)|2 +|x − y|n+2s dxdy +� 1 +2 +. + +ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS +5 +Remark 2.2. Hs +Ω is a Hilbert space (see [16], Proposition 3.1). +Let us define the following set: +Ls := +� +u : Rn −→ R measurable : +� +Rn +|u(x)| +1 + |x|n+2s dx < ∞ +� +. +The condition u ∈ Ls is useful to give a sense to pointwise definition of fractional Laplacian 1.2. +Lemma 2.3. (Lemma 2.3 [12]) Let Ω ⊂ Rn be a bounded set. Then Hs +Ω ⊂ Ls. +Next, we recall a few known results about fractional Schr¨odinger equation (1.8). +Definition 2.4. A measurable function u : Rn −→ R is called a weak solution of (1.8) if it satisfies the following +equation +cn,s +2 +� +Rn +� +Rn +(u(x) − u(y))(ψ(x) − ψ(y)) +|x − y|n+2s +dxdy + +� +Rn u(x)ψ(x)dx = +� +Rn |u(x)|p−1 u(x)ψ(x)dx, +for all ψ ∈ C1 +0(Rn). +We define the corresponding energy functional F : Hs(Rn) −→ R as follows: +F(u) := 1 +2 +�cn,s +2 +� +Rn +� +Rn +|u(x) − u(y)|2 +|x − y|n+2s dxdy + +� +Rn u2dx +� +− +1 +p + 1 +� +Rn |u|p+1dx. +(2.3) +The weak solutions of (1.8) corresponds to the critical points of F. +Definition 2.5. A function u ∈ Ls(Rn) ∩ C2s+ǫ(Rn), when 0 < s < 1 +2, 2s + ǫ < 1 or u ∈ C1,2s+ǫ−1(Rn) ∩ Ls(Rn), +when 1 +2 ≤ s < 1, 2s + ǫ − 1 < 1 is said to be a classical solution of (1.8) if it satisfies the equation (1.8) pointwise in +Rn. +Next result gives us a positive, radially symmetric solution of (1.8), which decays at infinity. +Theorem 2.6. (Theorem 3.4 [20]) Let u be a weak solution of (1.8). Then u ∈ Lq(Rn)∩Cα(Rn) for some q ∈ [2, ∞) +and α ∈ (0, 1). Moreover, +lim +|x|→∞ u(x) = 0. +Theorem 2.7. (Theorem 1.3 [20]) Equation (1.8) has a weak solution in Hs(Rn), which satisfies u ≥ 0 a.e. in Rn. +Moreover, u is a classical solution that satisfies u > 0 in Rn. +Following theorem shows that the solutions of (1.8) has a power type of decay at infinity. +Theorem 2.8. (Theorem 1.5 [20]) Let u be a positive classical solution of (1.8) such that +lim +|x|→∞ u(x) = 0. +Then, there exist constants 0 < C1 ≤ C2 such that +C1 +|x|n+2s ≤ u(x) ≤ +C2 +|x|n+2s for all |x| ≥ 1. +(2.4) +One can see that there exist some m > 0 and s0 > 0 such that for f(u) = up − u, we have +f(v) − f(u) +v − u +≤ vp − up +v − u +≤ C(v + u)m for all 0 < u < v < s0, +(2.5) +where C > 0 is some constant. Also, it is simple to see that f : [0, ∞) → R is locally Lipschitz. Consequently, we +have the following result on radial symmetry and monotonicity property of positive solutions of (1.8). +Theorem 2.9. (Theorem 1.2 [21]) Let u be a positive classical solution of (1.8) such that +lim +|x|→∞ u(x) = 0. +Further, assume that there exists +t > max +�2s +m , +n +m + 2 +� + +6 +S. GANDAL, J. TYAGI +such that u satisfies u(x) = O +� +1 +|x|t +� +as |x| → ∞. Then, u is radially symmetric and strictly decreasing about some +point in Rn. +Remark 2.10. Since +C1 +|x|n+2s ≤ u(x) ≤ +C2 +|x|n+2s for all |x| ≥ 1, +we can take t = n + 2s in the above theorem. +Now, Proposition 4.1[44] ascertains that if u ∈ Rn is a weak solution of (1.8) then u satisfies the following Pohozaev +identity: +P(u) := (n − 2s)cn,s +4 +� +Rn +� +Rn +|u(x) − u(y)|2 +|x − y|n+2s dxdy + n +2 +� +Rn u2dx − +n +p + 1 +� +Rn up+1 = 0. +Let us define +G := +� +u ∈ Hs(Rn) \ {0} | P(u) = 0 +� +. +In [7], authors have obtained a weak solution w ∈ Hs(Rn) of (1.8) with least energy among all other solutions. In +particular, they have proved the following result. +Theorem 2.11. (Theorem 1.2 [7]) Equation (1.8) has a weak solution w ∈ Hs(Rn) such that +0 < F(w) = inf +u∈G F(u). +Combining Theorems 2.7, 2.8, 2.9 and 2.11 we have the following result. +Theorem 2.12. Equation (1.8) has a positive classical solution w ∈ Hs(Rn) satisfying +(a) w has a power type of decay at infinity, i.e., there exist constants 0 < C1 ≤ C2 such that +C1 +|x|n+2s ≤ w(x) ≤ +C2 +|x|n+2s for all |x| ≥ 1; +(b) w is radially symmetric, i.e., w(x) = w(r) with r = |x| ; +(c) For any non-negative classical solution u ∈ Hs(Rn) of (1.8), 0 < F(w) ≤ F(u) holds unless u = 0. +Definition 2.13. We call w, given by Theorem 2.12, a ground state solution of (1.8). +3. Regularity and bounds for least energy solution ud +Let s ∈ (0, 1) and Ω ⊂ Rn be a bounded domain of class C1,1. +Definition 3.1. A measurable function u : Rn −→ R is said to be a weak solution of (1.1) if it satisfies the equation +dcn,s +2 +� +T (Ω) +(u(x) − u(y))(ψ(x) − ψ(y)) +|x − y|n+2s +dxdy + +� +Ω +u(x)ψ(x)dx = +� +Ω +|u(x)|p−1 u(x)ψ(x)dx, +(3.1) +for all ψ ∈ Hs +Ω. +We have the following result on the existence of weak solution of (1.1). +Theorem 3.2. (Theorem 6.1 [4], Theorem 1.1 [6]) There exists a nonnegative weak solution ud of (1.1) with critical +value cd, provided d is sufficiently small. Moreover, ud satisfies +0 < Jd(ud) ≤ Cd +n +2s , +where the constant C is independent of d. Consequently, ud is non-constant. +Define +M[v] := sup +t≥0 +Jd(tv), v ∈ Hs +Ω. +In the next lemma, we indicate useful characterization of the critical value cd. We follow the similar lines of proof as +Lemma 3.1 [40]. + +ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS +7 +Lemma 3.3. The critical value cd is independent of the choice of u ∈ Hs +Ω such that u ≥ 0, u ̸≡ 0 and Jd(u) = 0. In +fact, cd is the least positive critical value of Jd, and is given by +cd = inf +� +M[v] | v ∈ Hs +Ω, v ̸≡ 0, v ≥ 0 in Ω +� +. +(3.2) +Proof. For v ∈ Hs +Ω, let +Ω+ = +� +x ∈ Ω | v(x) > 0 +� +. +Now, for all those v satisfying |Ω+| > 0, define +gd(t) := Jd(tv), +for t ≥ 0. +First, we will show that gd(t) has a unique maximum. For this, we have +g′ +d(t) = t +� +dcn,s +2 +� +T (Ω) +|v(x) − v(y)|2 +|x − y|n+2s dxdy + +� +Ω +v2dx +� +− tp +� +Ω +vp+1dx. +Therefore, g′ +d(t0) = 0 for some t0 > 0 if and only if +dcn,s +2 +� +T (Ω) +|v(x) − v(y)|2 +|x − y|n+2s dxdy + +� +Ω +v2dx = tp−1 +0 +� +Ω +vp+1dx. +Note that the right hand side is strictly increasing in t0. And hence there exists unique t0 > 0 such that g′ +d(t0) = 0. +Since gd(t) > 0 for t > 0 small and gd(t) → −∞ as t → +∞, one easily find that gd(t) has a unique maximum. +Let us fix a function u ̸≡ 0, u ≥ 0 in Hs +Ω with Jd(u) = 0. Let ud be a positive solution of (1.1) obtained by applying +Mountain-Pass Lemma and cd the corresponding critical value. We have Jd(ud) = cd and J +′ +d(ud) = 0. Since ud > 0 +and J +′ +d(ud) = 0, we have +M[ud] = cd, +(3.3) +and hence +cd ≥ inf +� +M[v] | v ∈ Hs +Ω, v ̸≡ 0, v ≥ 0 in Ω +� +. +(3.4) +On the contrary, assume that the strict inequality occurs in (3.4). Then, we have +M[v0] < cd, +for some v0 ≥ 0, v0 ̸≡ 0 in Hs +Ω. Therefore, there exists some t1 > 0 such that t1v0 = u0 satisfies Jd(u0) = 0. Denote +by U the subspace of Hs +Ω spanned by u and u0. Consider the subset of U defined as follows: +U + := +� +αu + βu0 | α, β ≥ 0 +� +. +Let S be a circle on U of radius R so large that R > max +� +∥u∥ , ∥u0∥ +� +and Jd ≤ 0 on S ∩U +. Let γ be the path made +up of the line segment with endpoints 0 and +Ru0 +∥u0∥, the circular arc S ∩ U + and the line segment with endpoints +Ru +∥u∥ +and u. One can easily notice that, along γ, Jd is positive only on the line segment joining 0 and u0. Hence, we have +max +v∈γ Jd(v) = M[v0] < cd, +a contradiction to (1.7). Thus, we have the equality in (3.4), i.e., +cd = inf +� +M[v] | v ∈ Hs +Ω, v ̸≡ 0, v ≥ 0 in Ω +� +. +(3.5) +Note that Jd(v) = Jd(−v) for any v ∈ Hs +Ω. Since any nontrivial critical point of Jd is either positive or negative +almost everywhere in Ω, from the above discussion one can see that cd is the least positive critical value of Jd. This +completes the proof. +□ +The following lemma gives us the regularity estimate. The similar result is already proved in Lemma 3.6 [12], +Remark 4.9 [13]. +Lemma 3.4. Let u ∈ Hs +Ω be a weak solution of (1.1). If u ∈ L∞(Ω) then u ∈ L∞(Rn). Moreover, +(1) For 0 < s < 1 +2, u ∈ C2(Ω) if p > 3 − 2s and u ∈ C1,p−2+2s(Ω) if 2 < p ≤ 3 − 2s. +(2) For 1 +2 ≤ s < 1, u ∈ C2(Ω). + +8 +S. GANDAL, J. TYAGI +Now, we prove that the least energy solution ud is bounded by some constant independent of d. +Proof of Theorem 1.2. The proof of the first inequality of Theorem 1.2 is fairly standard and simple, which can +be seen in the literature, for instance, see Theorem 1.1 [7]. Since it is short, for the sake of completeness, we include +it here. For this, we have +Jd(ud) := 1 +2 +� +cn,sd +2 +� +T (Ω) +|ud(x) − ud(y)|2 +|x − y|n+2s +dxdy + +� +Ω +u2dx +� +− +1 +p + 1 +� +Ω +up+1 +d +dx. +(3.6) +Since ud is a critical point of Jd, we have +Jd +′(ud) = 0 on Hs +Ω. +(3.7) +This implies that +dcn,s +2 +� +T (Ω) +|ud(x) − ud(y)|2 +|x − y|n+2s +dxdy + +� +Ω +u2 +ddx = +� +Ω +up+1 +d +dx. +(3.8) +Hence from above equations, we get +Jd(ud) = +�1 +2 − +1 +p + 1 +� � +Ω +up+1 +d +dx +(3.9) += (p − 1) +2(p + 1) +� +Ω +up+1 +d +dx. +(3.10) +Now, by Theorem 3.2, we have Jd(ud) ≤ Cd +n +2s , where the constant C depends only on p. Using this inequality in the +above equation, we get +� +Ω +up+1 +d +dx ≤ 2(p + 1) +p − 1 Cd +n +2s . +Taking C0 = 2(p+1) +p−1 C, proves the first inequality of Theorem 1.2. +The proof of second inequality of Theorem 1.2 is little constructive. We claim that +sup +Ω +ud(x) ≤ C1 +for some constant C1 > 0 depending on p and Ω only. Multiplying (1.1) by u2t−1 +d +and integrating over Ω, we get +cn,sd +2 +� +T (Ω) +(ud(x) − ud(y))(u2t−1 +d +(x) − u2t−1 +d +(y)) +|x − y|n+2s +dxdy + +� +Ω +u2t +d dx = +� +Ω +up+2t−1 +d +dx. +(3.11) +Now, we use the following inequality. We have given the proof of this inequality in appendix. Let x, y ≥ 0 are real +numbers and k ≥ 1, then we have +1 +k (xk − yk)2 ≤ (x − y)(x2k−1 − y2k−1). +(3.12) +Consequently, we have +1 +t +� +T (Ω) +(ut +d(x) − ut +d(y))2 +|x − y|n+2s +dxdy ≤ +� +T (Ω) +(ud(x) − ud(y))(u2t−1 +d +(x) − u2t−1 +d +(y)) +|x − y|n+2s +dxdy. +(3.13) +From (3.11) and (3.13), we get +dcn,s +2t +� +T (Ω) +(ut +d(x) − ut +d(y))2 +|x − y|n+2s +dxdy + +� +Ω +u2t +d dx ≤ +� +Ω +up+2t−1 +d +dx. +(3.14) +Now, by the fractional Sobolev embedding Theorem 2.1, +�� +Ω +|v|2∗ +s +�2/2∗ +s ≤ A +d +� +dcn,s +2 +� +Ω +� +Ω +|v(x) − v(y)|2 +|x − y|n+2s dxdy + +� +Ω +|v|2 dx +� +, +(3.15) +where d ∈ (0, d0) for some d0 > 0, A > 0 some constant, v ∈ Hs(Ω), and 2∗ +s = +2n +n−2s. The embedding constant A +depends only on n, s, d0, and Ω. To see this, let us define +Ωd := +� +y : +y +d1/2s ∈ Ω +� +and w(y) := v +� +y +d1/2s +� +, where y ∈ Ωd. + +ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS +9 +Now, we have +d +� +Ω +� +Ω +|v(x) − v(y)|2 +|x − y|n+2s dxdy + +� +Ω +v2dx = 1 +d +n +2s +�� +Ωd +� +Ωd +���v( +� +x′ +d +1 +2s +� +− v( +� +y′ +d +1 +2s +���� +2 +|x′ − y′|n+2s +dx′dy′ + +� +Ωd +v +� x′ +d +1 +2s +�2 +dx′ +� +(3.16) += 1 +d +n +2s +�� +Ωd +� +Ωd +|w(x′) − w(y′)|2 +|x′ − y′|n+2s +dx′dy′ + +� +Ωd +w(x′)2dx′ +� +(3.17) +≥ A +d +n +2s +�� +Ωd +|w|2∗ +sdx′� 2 +2∗s +(3.18) += Ad +� +2 +2∗s −1 +� +n +2s �� +Ω +|v|2∗ +sdx +� 2 +2∗s . +(3.19) +Therefore, we observe that A is uniform for d ∈ (0, d0). +Note that Ω × Ω ⊂ T (Ω). Then by virtue of (3.14) and (3.15), we have +�� +Ω +|ud|t2∗ +s +� 2 +2∗s ≤ tA +d +� +Ω +up+2t−1 +d +dx. +(3.20) +Now, we define two sequences +� +Lj +� +and +� +Mj +� +by the following recurrence relations: +p − 1 + 2L0 = 2∗ +s, +p − 1 + 2Lj+1 = 2∗ +sLj, +j = 0, 1, 2, ... +(3.21) +M0 = (AC0) +2∗s +2 , +Mj+1 = (ALjMj) +2∗s +2 , +j = 0, 1, 2, ... +(3.22) +We note that Lj is explicitly given by +Lj = +1 +(2∗s − 2) +��2∗ +s +2 +�j+1 +(2∗ +s − p − 1) + p − 1 +� +. +(3.23) +Since 1 < p < 2∗ +s − 1, it follows that Lj ≥ 1 for all j ≥ 0 and Lj → ∞ as j → ∞. We shall show that +� +Ω +up−1+2Lj +d +dx ≤ Mjd +n +2s +for all j ≥ 0, +(3.24) +and +Mj ≤ emLj−1 +(3.25) +for some constant m > 0. Then, we have +sup +Ω +ud(x) ≤ C1, +where C1 > 0 depending only on C0 and Ω. In fact (3.23) and (3.24) entail us +∥u∥L2∗sLj−1 (Ω) ≤ +� +emLj−1d +n +2s +� +1 +(2∗s Lj−1) += e +m +2∗s d +(n−2s) +4Lj−1 +(3.26) +and hence letting j → ∞, we obtain +∥u∥L∞(Ω) ≤ e +m +2∗s . + +10 +S. GANDAL, J. TYAGI +First, we verify (3.24). By virtue of (1.9) and (3.15), we have +�� +Ω +|ud|2∗ +s +� 2 +2∗s ≤ A +d +�cn,sd +2 +� +T (Ω) +|ud(x) − ud(y)|2 +|x − y|n+2s +dxdy + +� +Ω +|ud|2 dx +� +≤ A +d C0d +n +2s += AC0d +n +s2∗s . +(3.27) +Hence, (3.24) holds for j = 0. Suppose that we have proved (3.24) for j ≥ 0. Then by (3.20), we have +� +Ω +|ud|p−1+2Lj+1dx ≤ +�LjA +d +� +Ω +up+2Lj−1 +d +dx +� 2∗s +2 +≤ +� +ALjd−1Mjd +n +2s +� 2∗s +2 += +� +ALjMj +� 2∗ +s +2 d +n +2s . +(3.28) +This implies that (3.24) is also true for j + 1. Therefore it remains to show (3.25). Put +λj = 2∗ +s +2 · log(ALj) and ηj = log(Mj). +(3.29) +Hence +ηj+1 = 2∗ +s +2 · ηj + λj. +(3.30) +The explicit value of Lj is given by +Lj = (2∗ +s − 2)−1� +(2−12∗ +s)j+1(2∗ +s − p − 1) + p − 1 +� +. +(3.31) +Now, we have +λj = 2∗ +s +2 log +� +A +(2∗s − 2) +� +(2−12∗ +s)j+1(2∗ +s − p − 1) + p − 1 +�� +(3.32) += 2∗ +s +2 +� +log(A(2∗ +s − 2)) + log +� +(2−12∗ +s)j+1(2∗ +s − p − 1) + p − 1 +�� +. +(3.33) +Therefore, we can find some C∗ such that +λj ≤ C∗(j + 1). +(3.34) +We now define a sequence +� +γj +� +by +γ0 = η0 and γj+1 = 2∗ +s +2 γj + C∗(j + 1) +(3.35) +for j ≥ 1. Clearly, ηj ≤ γj for all j ≥ 0. Moreover, since +γj = +�2∗ +s +2 +�j� +η0 + 2C∗2∗ +s(2∗ +s − 2)−2� +−2C∗(2∗ +s − 2)−1� +j + 2∗ +s(2∗ +s − 2) +� +, +in view of (3.31), there exists m > 0 such that γj ≤ mLj−1. Hence log(Mj) ≤ mLj−1 and we obtain (3.25). Note +that m depends only on η0, 2∗ +s and C∗; whereas C∗ depends only on 2∗ +s, p and A. This completes the proof. +□ +Remark 3.5. It is known that if u ∈ Ls(Rn) ∩ C2s+ǫ(Ω), when 0 < s < 1 +2, 2s + ǫ < 1 or u ∈ Ls(Rn) ∩ C1,2s+ǫ−1(Ω), +when 1 +2 ≤ s < 1, 2s + ǫ − 1 < 1, one can compute (−∆)su(x) pointwise for all x in Ω. In fact, one can write +(−∆)su(x) = cn,sP.V. +� +Rn +u(x) − u(y) +|x − y|n+2s dy +Definition 3.6. We say that u : Rn −→ R is a classical solution of (1.1) if it satisfies the following: +(1) u ∈ Ls(Rn)∩C2s+ǫ(Ω), when 0 < s < 1 +2, 2s+ǫ < 1 or u ∈ Ls(Rn)∩C1,2s+ǫ−1(Ω), when 1 +2 ≤ s < 1, 2s+ǫ−1 < +1. + +ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS +11 +(2) Nsu(x) = 0, x ∈ Rn \ Ω; +(3) d(−∆)su(x) + u(x) = |u(x)|p−1 u(x) pointwise for all x ∈ Ω. +We make similar remarks as in [5], which offers a relation between the weak and classical solutions of (1.1). +Remark 3.7. Let ud be a least energy solution of (1.1) in Hs +Ω. Then by Lemma 2.3, Theorem 1.2 and Lemma 3.4, we +have +(1) for 0 < s < 1 +2, ud ∈ Ls(Rn) ∩ C2(Ω) if p > 3 − 2s and ud ∈ Ls(Rn) ∩ C1,p−2+2s(Ω) if 2 < p ≤ 3 − 2s; +(2) for 1 +2 ≤ s < 1, ud ∈ Ls(Rn) ∩ C2(Ω). +Now, using nonlocal integration by parts formulae given in [16], one can easily check that +d(−∆)sud(x) + ud(x) = |ud(x)|p−1 ud(x) +holds pointwise in Ω. This implies that ud is a classical solution of (1.1). Conversely, if ud is a classical solution of +(1.1) satisfying ud ∈ Hs +Ω, then ud is a weak solution of (1.1). +The following lemma shows that the maximum of least energy solution is always greater than unity. +Lemma 3.8. Let ud be the least energy solution of (1.1). Let +Md = sup +x∈Ω +ud(x). +(3.36) +Then Md > 1. +Proof. Since ud is a weak solution of (1.1), we get +dcn,s +2 +� +T (Ω) +(ud(x) − ud(y))(w(x) − w(y)) +|x − y|n+2s +dxdy + +� +Ω +udwdx = +� +Ω +up +dwdx holds, ∀ w ∈ Hs +Ω. +(3.37) +Taking w = 1 in the above equation, we get +� +Ω +ud(x)dx = +� +Ω +up +d(x)dx. +This implies that +� +Ω +ud(x)(1 − up−1 +d +(x))dx = 0. +Now, if ud(x) ≤ 1, for all x ∈ Ω, then +1 − ud(x) ≥ 0, ∀x ∈ Ω. +Thus from the above equation, we get that ud(x) = 1 a.e. in Ω. Now, by Lemma 3.4, we can assume that ud is +continuous and hence ud ≡ 1 in Ω, a contradiction to our assumption that ud is a non-constant solution. Therefore, +there exists x0 in Ω such that ud(x0) > 1. Thus Md > 1. +□ +4. Lr- estimates on ud +Here, we derive Lr-estimate for ud. Following results are generalization to the nonlocal case of Proposition 2.2 +and Lemma 2.3 [34]. +Proposition 4.1. For d0 > 0 fixed, there is a constant K0 such that +dcn,s +2 +� +T (Ω) +(ud(x) − ud(y))2 +|x − y|n+2s +dxdy + +� +Ω +u2 +ddx ≥ K0d +n +2s , +(4.1) +where ud is the least energy solution of (1.1) with 0 < d < d0. +Proof. On contrary, suppose that there is a sequence +� +dk +� +contained in the interval (0, d0) and a sequence of positive +solutions +� +uk +� +to (1.1) with d = dk such that +ζk := +1 +d +n +2s +� +dcn,s +2 +� +T (Ω) +(uk(x) − uk(y))2 +|x − y|n+2s +dxdy + +� +Ω +u2 +kdx +� +→ 0 as k → ∞. +(4.2) + +12 +S. GANDAL, J. TYAGI +We are going to follow the same arguments as used in the proof of Lemma 1.2 to prove this proposition. Once again +define the sequences +� +Lk +� +and +� +Mj +� +as defined earlier in (3.21) and (3.22), respectively. Instead of C0, we write ζk +in the definition of +� +Mj +� +: +p − 1 + 2L0 = 2∗ +s, +p − 1 + 2Lj+1 = 2∗ +sLj, +j = 0, 1, 2, . . . +(4.3) +and +M0 = (Aζk) +2∗ +s +2 , +Mj+1 = (ALjMj) +2∗ +s +2 , +j = 0, 1, 2, . . . +(4.4) +Further, define the sequences +� +λj +� +, +� +ηj +� +, and +� +γj +� +as defined earlier in (3.29) and (3.35). From (3.24), we have +�� +Ω +u2∗ +sLj−1 +k +dx +�(2∗ +sLj−1) +≤ +� +Mjdn/2s +k +�1/(2∗ +sLj−1) +. +(4.5) +Since +log(Mj) = ηj ≤ γj, +we have +log (Mj) +2∗sLj−1 +≤ +ηj +2∗sLj−1 +. +(4.6) +Now, +lim +j→∞ +ηj +2∗sLj−1 += lim +j→∞ +� +2∗ +s +2 +�j� +η0 + 2C∗2∗ +s(2∗ +s − 2)−2� +− 2C∗(2∗ +s − 2)−1� +j + 2∗ +s(2∗ +s − 2) +� +2∗ +s +(2∗ +s−2) +�� +2∗ +s +2 +�j +(2∗s − p − 1) + p − 1 +� += (2∗ +s − 2)(η0 + 2C∗2∗ +s(2∗ +s − 2)−2) +2∗s(2∗s − p − 1) +. +Letting j → ∞ in (4.5), we get +∥uk∥L∞(Ω) ≤ ea1(η0+a2), +(4.7) +with a1 and a2 depending only on 2∗ +s, p and C∗. Since +η0 = log(M0) = 2∗ +s +2 log(Aζk). +Therefore, as k → ∞, η0 → −∞. Thus, in view of (4.7), we get +∥uk∥L∞(Ω) → 0, +which leads to a contradiction to Lemma 3.8. +□ +Proof of the Theorem 1.3: First, we will show the second part of Inequality (1.11). +Case-I. r ≥ 2∗ +s = +2n +n−2s. +Let +� +Lj +� +be the sequence defined in (3.21). If r ∈ +� +2∗ +sLj +� +, then the second inequality of (1.11) follows from (3.24). +So assume that 2∗ +sLj < r < 2∗ +sLj+1 for some j ≥ 0. We have +r = t2∗ +sLj + (1 − t)2∗ +sLj+1, for some t ∈ (0, 1). + +ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS +13 +Using H¨olders inequality and (3.24), we get +� +Ω +ur +ddx = +� +Ω +ut2∗ +sLj+(1−t)2∗ +sLj+1 +d +dx, +≤ +�� +Ω +u2∗ +sLj +d +dx +�t �� +Ω +u2∗ +sLj+1 +d +dx +�1−t +≤ +� +Mj−1dn/2s�t +(Mjdn/2s)1−t += M t +j−1M 1−t +j +d +n +2s . +Case-II. 2 ≤ r ≤ 2∗ +s. +We write +r = 2t + (1 − t)2∗ +s, +for some t ∈ [0, 1]. Then, using H¨older’s inequality, from Equations (1.9) and (3.24) with j = 0, we get +� +Ω +ur +ddx ≤ +�� +Ω +u2 +ddx +�t �� +Ω +u2∗ +s +d dx +�1−t +≤ Ct +0M (1−t) +0 +d +n +2s , +where the constant C0 is independent of d. +Case-III. 1 ≤ r < p + 1. +Integrating both sides of (1.1) and using the condition Nsu(x) = 0, for x ∈ CΩ, we get +� +Ω +uddx = +� +Ω +up +ddx. +(4.8) +It is easy to see that +p = t + (1 − t)(p + 1) with t = 1 +p ∈ (0, 1). +Notice that p + 1 ∈ (2, 2∗ +s). Therefore, using the H¨older’s inequality and (4.8), we get +� +Ω +up +ddx ≤ +�� +Ω +uddx +�t �� +Ω +up+1 +d +dx +�(1−t) +, +� +Ω +up +ddx ≤ +� +Ω +up+1 +d +dx ≤ C0d +n +2s +(by (1.9)) , +where the constant C0 depends only upon p + 1. +Also, in the view of (4.8) and (1.9), we observe that the second inequality of (1.11) holds for r = 1. Now, repeating +the interpolation between 1 and p + 1, we see that the second inequality of (1.11) holds for all r ≥ 1. +Case-IV. Let 0 < r ≤ 1. Taking F = ur +d, G = 1, p = 1 +r, q = +1 +1−r and using the H¨olders inequality, we get +� +Ω +ur +ddx ≤ ∥F∥p ∥G∥q = |Ω|1−r +�� +Ω +uddx +�r +≤ |Ω|1−r B(1)rd +nr +2s . +This proves the second inequality of (1.12). +Now, let us prove the first inequality of (1.11) and (1.12). In view of equations (3.8) and (4.1), we see that +� +Ω +up+1 +d +≥ K0d +n +2s . +(4.9) +Since +sup +Ω +ud(x) ≤ C1, for some constant C1 > 0, +we have +K0d +n +2s ≤ +� +Ω +up+1 +d += +� +Ω +� +up+1−r +d +� +(ur +d) dx +≤ Cp+1−r +1 +� +Ω +ur +ddx. + +14 +S. GANDAL, J. TYAGI +This implies that +� +Ω +ur +ddx ≥ K0Cr−p−1 +1 +d +n +2s , r < p + 1. +For r > p + 1, we write p + 1 = 1 + (1 − t)r. Therefore, we get +K0d +n +2s ≤ +� +Ω +up+1 +d +dx += +� +Ω +u1+(1−t)r +d +dx +≤ +� +uddx +�t� +ur +ddx +�1−t +≤ +� +B(1)d +n +2s +�t� +ur +ddx +�1−t +. +This yields that +� +Ω +ur +ddx ≥ (K0B(1)−t) +1 +1−t d +n +2s . +□ +5. Proof of theorem 1.4 +We prove Theorem 1.4 in this section. Its proof is more involved and requires some scaling and compactness +arguments. We prove the statements of theorem one by one. Let zd ∈ Ω be a point of maximum of ud. The basic +idea for its proof is simple. We approximate ud around zd by a scaled positive radial solution of (1.8). It gives us an +upper bound on cd, which is closely related to the location of point zd. +Proof of (A). If the inequality in (A) is not true, then there is a decreasing sequence dj ↓ 0 such that +ρj := ρ(zj, ∂Ω) +d +1 +2s +j +→ +∞ as j → ∞, +(5.1) +where zj := zdj is a point of maximum of udj on Ω. Define +φj(y) := udj(yd +1 +2s +j ++ zj) for y ∈ Rn. +Since ud is a classical solution of (1.1), we have +(−∆)sφj + φj = φp +j in Bρj, +(5.2) +and +(1) φj ∈ C0,2s+ǫ(Bρj), when 0 < s < 1 +2, 2s + ǫ < 1 +(2) φj ∈ C1,2s+ǫ−1(Bρj), when 1 +2 ≤ s < 1, 2s + ǫ − 1 < 1. +First, we claim that the sequence +� +φj +� +contains a convergent subsequence. Let +� +Rk +� +be a monotone increasing +sequence of positive numbers with Rk → +∞ as k → ∞. Therefore, we have for each k, there is a number jk such +that 4Rk < ρj whenever j ≥ jk. Since ud ∈ L∞(Rn) ∩ Ls(Rn), we have φj ∈ L∞(Rn) ∩ Ls(Rn) for each j ≥ 1. Now, +we can use Theorem 1.4 [19] to get the following estimates: +For 0 < s < 1 +2, 2s + ǫ < 1 +i) 4s + ǫ < 1, then +∥φj∥C0,4s+ǫ(B2Rk) ≤ C +� +∥φj∥L∞(Rn) + +��φp +j − φj +�� +C0,2s+ǫ(B4Rk ) +� +ii) 1 < 4s + ǫ < 2, then +∥φj∥C1,4s+ǫ−1(B2Rk) ≤ C +� +∥φj∥L∞(Rn) + +��φp +j − φj +�� +C0,2s+ǫ(B4Rk ) +� +, +and for 1 +2 ≤ s < 1, 2s + ǫ − 1 < 1 +iii) 4s + ǫ − 1 < 1, then +∥φj∥C1,4s+ǫ−1(B2Rk ≤ C +� +∥φj∥L∞(Rn) + +��φp +j − φj +�� +C1,2s+ǫ−1(B4Rk ) +� + +ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS +15 +iv) 1 < 4s + ǫ − 1 < 2, then +∥φj∥C2,4s+ǫ−1(B2Rk ) ≤ C +� +∥φj∥L∞(Rn) + +��φp +j − φj +�� +C1,2s+ǫ−1(B4Rk) +� +, +where the constant C > 0 is independent of j. +Let us recall the inequality (1.9) here: +dcn,s +2 +� +T (Ω) +|ud(x) − ud(y)|2 +|x − y|n+2s +dxdy + +� +Ω +u2dx = +� +Ω +up+1 +d +≤ C0d +n +2s , +where C0 is independent of d. This yields +� +Bρj +φp+1 +j +≤ C0, +(5.3) +and +∥φj∥Hs(Bρj ) ≤ C0, for all j ≥ 1. +(5.4) +Also, by Theorem 1.3 we have +� +Ω +ur +d ≤ B(r)d +n +2s +for all r ≥ 1 +which implies that +� +Bρj +φr +j ≤ B(r), +for all j ≥ 1 and r ≥ 1. +(5.5) +By Lemma 3.4 and Theorem 1.2, we have +∥ud∥L∞(Rn) ≤ C1, +(5.6) +where the constant C1 is independent of the diffusion constant d. So the equations (5.5), (5.6) and Theorem 1.3 [19] +imply that +∥φj∥Xs(BRk ) < C2 +for all j ≥ jk, +where the constant C2 > 0 is independent of j and the space Xs(BRk) is identified with one of the spaces +C0,4s+ǫ(BRk), C1,4s+ǫ−1(BRk) or C2,4s+ǫ−1(BRk) with same assumptions on s and ǫ as above. Therefore +� +φj +� +is a relatively compact set in Xs(BRk), hence by the standard diagonal process, one can extract a convergent +subsequence of +� +φj +� +, we continue to denote such a subsequence by +� +φj +� +itself such that +φj → v in C0,2s+ǫ +loc +(Rn) when 0 < s < 1 +2, 2s + ǫ < 1 +or +φj → v in C1,2s+ǫ−1 +loc +(Rn) when 1 +2 < s < 1, 2s + ǫ − 1 < 1 +for some v. The limit v ∈ C0,2s+ǫ(Rn) ∩ Hs(Rn) when 0 < s < 1 +2, 2s + ǫ < 1 or v ∈ C1,2s+ǫ−1(Rn) ∩ Hs(Rn) when +1 +2 < s < 1, 2s + ǫ − 1 < 1 follows from (5.4). Consequently, we have +lim +|x|→∞ v(x) = 0. +Using Theorem 1.1 [17], we have (−∆)sφj(x) converges to (−∆)sv(x) point-wise in Rn. Consequently, we see that +the limit v satisfies the equation +(−∆)sv + v = vp in Rn. +(5.7) +Clearly, v ≥ 0 because each φj ≥ 0. Since by Lemma 3.8, we have φj(0) = udj(zj) > 1 for each j ≥ 1, one can see +that v ̸≡ 0. +Using Theorem 2.9, one can see that v is radially symmetric and decreasing about some point in Rn. Since +∇v(0) = lim +j→∞ ∇φj(0) = 0 + +16 +S. GANDAL, J. TYAGI +so it implies that v is radially symmetric about the origin. And by Theorem 2.8, v has a power type of decay at +infinity, i.e., +v(r) ≤ +C2 +rn+2s , r ≥ 1. +Now, we derive a lower bound on the critical value cdj. Let us define +δR := +C2 +Rn+2s , +(5.8) +where R > 0 arbitrarily large real number. Then, there exists a positive integer jR such that if j ≥ jR then ρj ≥ 2R +and +∥φj − v∥C2(B2R) ≤ δR. +(5.9) +By Lemma 3.3, we have +cdj = M[udj] = Jdj(udj). +Using this fact and (3.9), we obtain +cdj = +�1 +2 − +1 +p + 1 +� � +Ω +up+1 +dj dx +(5.10) +≥ +�1 +2 − +1 +p + 1 +� � +|x−zj| 0 is some constant. This implies that +|Fj| ≤ +�1 +2 − +1 +p + 1 +� +C |BR| δR = C3RnδR, +where +C3 = +�1 +2 − +1 +p + 1 +� wn +n C +and wn denotes the surface area of the unit sphere in Rn. Consequently, Inequality (5.11) becomes +cdj ≥ d +n +2s +j +��1 +2 − +1 +p + 1 +� � +BR +vp+1dy − C3RnδR +� +. +(5.13) +Now, it is easy to see that +�1 +2 − +1 +p + 1 +� � +BR +vp+1dy = F(v) − +�1 +2 − +1 +p + 1 +� � +|y|>R +vp+1dy, +(5.14) +where F(v) is defined earlier in (2.3). Simplifying the second term on right hand side, we get +�1 +2 − +1 +p + 1 +� � +|y|>R +vp+1dy = +�1 +2 − +1 +p + 1 +� � ∞ +R +rn−1wn +r(n+2s)(p+1) dr += +�1 +2 − +1 +p + 1 +� +wn +(n + 2s)p + 2s +1 +R(n+2s)p+2s = +C4 +R(n+2s)p+2s . +Therefore, one can write + +ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS +17 +�1 +2 − +1 +p + 1 +� � +BR +vp+1dy = F(v) − +C4 +R(n+2s)p+2s . +(5.15) +On combining Equations (5.8), (5.13) and (5.15), we get for j ≥ jR, +cdj ≥ d +n +2s +j +� +F(v) − +C4 +R(n+2s)p+2s − C2C3 +R2s +� +≥ d +n +2s +j +� +F(v) − C5 +R2s +� +, +(5.16) +where C5 is independent of j and R. +Now, we derive an upper bound on the critical value cdj. Without loss of generality, we may assume that the domain +Ω is a subset of Rn ++ and 0 ∈ ∂Ω. Given Definition 2.13, let w be a ground state solution of (1.8). Define +Ωd := +� x +d +1 +2s | x ∈ Ω +� +, +wd(x) := w +� x +d +1 +2s +� +, for x ∈ Rn. +Since w ≥ 0 so this implies that wd ≥ 0. Define +gd(t) := Jd(twd), +t ≥ 0. +Then by Lemma 3.3, there exists unique t0 = t0(d) > 0 at which gd attains maximum. Note that t0(d) → 1 as d ↓ 0. +Hence, we have +M[wd] = Jd (t0wd) += t2 +0 +2 +� +dcn,s +2 +� +T (Ω) +|wd(x) − wd(y)|2 +|x − y|n+2s +dxdy + +� +Ω +w2 +ddx +� +− tp+1 +0 +p + 1 +� +Ω +wp+1 +d +dx += t2 +0 +2 + +dcn,s +2 +� +T (Ω) +���w +� +x +d +1 +2s +� +− w +� +y +d +1 +2s +���� +2 +|x − y|n+2s +dxdy + +� +Ω +w2 +� x +d +1 +2s +� +dx + + − tp+1 +0 +p + 1 +� +Ω +wp+1 +� x +d +1 +2s +� +dx. +The change of variables +x +d +1 +2s = a, +y +d +1 +2s = b, +gives us +M[wd] = d +n +2s +� +t2 +0 +2 +� +cn,s +2 +� +T (Ωd) +|w(a) − w(b)|2 +|a − b|n+2s +dadb + +� +Ωd +w2da +� +− tp+1 +0 +p + 1 +� +Ωd +wp+1da +� += d +n +2s +� +t2 +0 +2 +� +cn,s +2 +� +Ωd +� +Ωd +|w(a) − w(b)|2 +|a − b|n+2s +dadb + 2cn,s +� +CΩd +� +Ωd +|w(a) − w(b)|2 +|a − b|n+2s +dadb + +� +Ωd +w2da +� +− tp+1 +0 +p + 1 +� +Ωd +wp+1da +� +. +Let us denote by Id : +Id = t2 +0 +2 +� +cn,s +2 +� +Ωd +� +Ωd +|w(a) − w(b)|2 +|a − b|n+2s +dadb + 2cn,s +� +CΩd +� +Ωd +|w(a) − w(b)|2 +|a − b|n+2s +dadb + +� +Ωd +w2da +� +− tp+1 +0 +p + 1 +� +Ωd +wp+1da. +Since +t0(d) → 1 as d ↓ 0, +we get +Id = 1 +2 +� +cn,s +2 +� +Rn ++ +� +Rn ++ +|w(a) − w(b)|2 +|a − b|n+2s +dadb + 2cn,s +� +CRn ++ +� +Rn ++ +|w(a) − w(b)|2 +|a − b|n+2s +dadb + +� +Rn ++ +w2da +� +− +1 +p + 1 +� +Rn ++ +wp+1da + o(1) +as d ↓ 0. Further, w is a nonnegative and radially symmetric implies that +� +Rn ++ +w2da = 1 +2 +� +Rn w2da, +� +Rn ++ +wp+1da = 1 +2 +� +Rn wp+1da, + +18 +S. GANDAL, J. TYAGI +� +Rn ++ +� +Rn ++ +|w(a) − w(b)|2 +|a − b|n+2s +dadb = 1 +4 +� +Rn +� +Rn +|w(a) − w(b)|2 +|a − b|n+2s +dadb, +� +CRn ++ +� +Rn ++ +|w(a) − w(b)|2 +|a − b|n+2s +dadb = 1 +4 +� +Rn +� +Rn +|w(a) − w(b)|2 +|a − b|n+2s +dadb. +Using these estimates, we get +Id < 1 +2 +� +1 +2 +� +cn,s +2 +� +Rn +� +Rn +|w(a) − w(b)|2 +|a − b|n+2s +dadb + +� +Rn w2da +� +− +1 +p + 1 +� +Rn wp+1da +� ++ o(1) = 1 +2F(w) + o(1), +as d ↓ 0. Thus, we have +M[wd] = d +n +2s Id < d +n +2s +2 F(w) + o(1), +(5.17) +as d ↓ 0. Using (c) of Theorem 2.12, we have 0 < F(w) ≤ F(v) for any nonnegative nonzero classical solution v of +(1.8) and by Lemma 3.3, we have +cdj ≤ M[wdj] < +d +n +2s +j +2 F(v) +for dj sufficiently small. By taking R sufficiently large in (5.16), we thus obtain a contradiction. This proves (A). +Remark 5.1. In the classical case [40], authors have defined diffeomorphisms which straightens a boundary portion +near Q ∈ ∂Ω. Further, using scaling and translations of least energy solutions ud of (1.4), the classical problem (1.4) +gets transferred into new elliptic equation. Due to the nonlocal nature of the fractional Laplacian and boundary +condition in our problem, it is almost impossible to introduce such scaling and translation arguments. +Proof of (B). Now, we claim that zd ∈ ∂Ω. Suppose that there is a decreasing sequence dj ↓ 0 such that zdj := zj ∈ Ω. +We have from Lemma 1.4 that the sequence {zj} converges to some z ∈ ∂Ω. Without loss of generality, let us assume +that z = 0. Define +�uj(x) := +� +udj(x) +in Rn ++, +udj(x′, −xn) +in Rn +−, +where +x′ = (x1, x2, . . . , xn−1), Rn ++ = +� +(x′, xn) | xn ≥ 0 +� +, Rn +− = +� +(x′, xn) | xn ≤ 0 +� +. +Also, define a scaled function +ψj(y) := �uj +� +yd +1 +2s +j ++ zj +� +for y ∈ Rn. +(5.18) +Note that for zj = (z′ +j, zjn), we can write zjn = αjd +1 +2s +j +for some αj > 0. The sequence {αj} is bounded, which follows +from Lemma 1.4. Let +ρj := ρ(zj, ∂Ω) +d +1 +2s +j +, +(5.19) +where ρ(zj, ∂Ω) denotes the distance between zj and ∂Ω. Note that the function ψj satisfies the equation +(−∆)sψj(y) + ψj(y) = ψj(y)p + djh(y) in Bρj, +(5.20) + +ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS +19 +for some function h of y. To see this, let y ∈ Bρj, we have +(−∆)sψj(y) = cn,sP.V. +� +Rn +ψj(y) − ψj(x) +|y − x|n+2s dx = cn,s lim +ǫ→0 +� +CBǫ(y) +ψj(y) − ψj(x) +|y − x|n+2s dx += cn,s lim +ǫ→0 + + +� +CBǫ(y) +�uj(yd +1 +2s +j ++ zj) − �uj(xd +1 +2s +j ++ zj) +|y − x|n+2s +dx + + += cn,s lim +ǫ→0 +�� +� +xn≥−αj +� � CBǫ(y) +�uj(yd +1 +2s +j ++ zj) − �uj(xd +1 +2s +j ++ zj) +|y − x|n+2s +dx ++ +� +� +xn≤−αj +� � CBǫ(y) +�uj(yd +1 +2s +j ++ zj) − �uj(xd +1 +2s +j ++ zj) +|y − x|n+2s +dx +� +. +(5.21) +For yn ≥ −αj, we have +(−∆)sψj(y) = cn,s lim +ǫ→0 +�� +� +xn≥−αj +� � CBǫ(y) +uj(yd +1 +2s +j ++ zj) − uj(xd +1 +2s +j ++ zj) +|y − x|n+2s +dx ++ +� +� +xn≤−αj +� � CBǫ(y) +uj(yd +1 +2s +j ++ zj) − uj(x′d +1 +2s +j ++ z′ +j, −(xn + αjd +1 +2s +j )) +|y − x|n+2s +dx +� += cn,s lim +ǫ→0 +�� +� +xn≥−αj +� � CBǫ(y) +uj(yd +1 +2s +j ++ zj) − uj(xd +1 +2s +j ++ zj) +|y − x|n+2s +dx ++ +� +� +xn≤−αj +� � CBǫ(y) +uj(yd +1 +2s +j ++ zj) − uj(xd +1 +2s +j ++ zj) + uj(xd +1 +2s +j ++ zj) − uj(x′d +1 +2s +j ++ z′ +j, −(xn + αjd +1 +2s +j )) +|y − x|n+2s +dx +� += cn,s lim +ǫ→0 +�� +CBǫ(y) +uj(yd +1 +2s +j ++ zj) − uj(xd +1 +2s +j ++ zj) +|y − x|n+2s +dx ++ +� +� +xn≤−αj +� � CBǫ(y) +uj(xd +1 +2s +j ++ zj) − uj(x′d +1 +2s +j ++ z′ +j, −(xn + αjd +1 +2s +j )) +|y − x|n+2s +dx +� += cn,s lim +ǫ→0 + + +� +CBǫ(y) +uj(yd +1 +2s +j ++ zj) − uj(xd +1 +2s +j ++ zj) +|y − x|n+2s +dx + +� +� +xn≤−αj +� � CBǫ(y) +uj(xd +1 +2s +j ++ zj) − �uj(xd +1 +2s +j ++ zj) +|y − x|n+2s +dx + + . +(5.22) +Making change of variables +yd +1 +2s +j ++ zj = a and xd +1 +2s +j ++ zj = b, +we get +(−∆)sψj(y) = dj(−∆)suj(a) + djcn,s lim +η→0 +� +� +bn≤0 +� � CBη(a) +uj(b) − �uj(b) +|a − b|n+2s db += dj(−∆)suj(a) + djh(a), +(5.23) +where +η = ǫd +1 +2s +j +and +h(a) = cn,s lim +η→0 +� +� +bn≤0 +� � CBη(a) +uj(b) − �uj(b) +|a − b|n+2s db. + +20 +S. GANDAL, J. TYAGI +Note that a ∈ Ω. +Now, consider the case yn ≤ −αj. Equation (5.21) becomes +(−∆)sψj(y) = cn,s lim +ǫ→0 +�� +� +xn≥−αj +� � CBǫ(y) +uj(y′d +1 +2s +j ++ z′ +j, −(ynd +1 +2s +j ++ αjd +1 +2s +j )) − uj(xd +1 +2s +j ++ zj) +|y − x|n+2s +dx ++ +� +� +xn≤−αj +� � CBǫ(y) +uj(y′d +1 +2s +j ++ z′ +j, −(ynd +1 +2s +j ++ αjd +1 +2s +j )) − uj(x′d +1 +2s +j ++ z′ +j, −(xnd +1 +2s +j ++ αjd +1 +2s +j )) +|y − x|n+2s +dx +� += I1 + I2, +(5.24) +where I1 and I2 denote the first and second integral on the right hand side, respectively. Let us introduce some of +notations. We write �x = (x′, −xn), �x = (�x′, �xn) and �xn = −xn for x = (x′, xn) ∈ Rn, n > 1. Using these, let us +compute +I2 = cn,s lim +ǫ→0 +�� +� +�xn≥αj +� � CBǫ(�y) +uj(�y′d +1 +2s +j ++ ˆz′ +j, �ynd +1 +2s +j ++ �αjd +1 +2s +j ) − uj(�x′d +1 +2s +j ++ �z′ +j, �xnd +1 +2s +j ++ �αjd +1 +2s +j ) +|�y − �x|n+2s +d�x +� += cn,s lim +ǫ→0 +�� +� +�xn≥αj +� � CBǫ(�y) +uj(�yd +1 +2s +j ++ �zj) − uj(�xd +1 +2s +j ++ �zj) +|�y − �x|n+2s +d�x +� += cn,s lim +ǫ→0 +�� +� +�xn≥αj +� � CBǫ(�y) +uj(�yd +1 +2s +j ++ �zj) − uj(�xd +1 +2s +j ++ �zj) +|�y − �x|n+2s +d�x +� +. +(5.25) +Now, we simplify I1 : +I1 = cn,s lim +ǫ→0 +�� +� +xn≥−αj +� � CBǫ(y) +uj(y′d +1 +2s +j ++ z′ +j, −(ynd +1 +2s +j ++ αjd +1 +2s +j )) − uj(x′d +1 +2s +j ++ z′ +j, −(xnd +1 +2s +j ++ αjd +1 +2s +j )) +|y − x|n+2s ++ +� +{xn≥−αj}∩CBǫ(y) +uj(w′d +1 +2s +j ++ z′ +j, −(xnd +1 +2s +j ++ αjd +1 +2s +j )) − uj(xd +1 +2s +j ++ zj) +|y − x|n+2s +dx +� += cn,s lim +ǫ→0 +�� +� +�xn≤αj +� � CBǫ(�y) +uj(�yd +1 +2s +j ++ �zj) − uj(�xd +1 +2s +j ++ �zj) +|�y − �x|n+2s +d�x + +� +� +xn≥−αj +� +∩CBǫ(�y) +uj(�xd +1 +2s +j ++ �zj) − �uj(�xd +1 +2s +j ++ �zj) +|�y − �x|n+2s +d�x +� +. +(5.26) +Using these estimates for I1 and I2 in equation (5.24), we get +(−∆)sψj(y) = cn,sP.V. +� +Rn +uj(�yd +1 +2s +j ++ �zj) − uj(�xd +1 +2s +j ++ �zj) +|�y − �x|n+2s +d�x + cn,s lim +ǫ→0 +� +� +xn≥−αj +� � CBǫ(�y) +uj(�xd +1 +2s +j ++ �zj) − �uj(�xd +1 +2s +j ++ �zj) +|�y − �x|n+2s +d�x. +(5.27) +By the change of variables +�yd +1 +2s +j ++ �zj = e and �wd +1 +2s +j ++ �zj = f, +we get +(−∆)sψj(y) = dj(−∆)suj(e) + djcn,s lim +η→0 +� +� +fn≤0 +� � CBη(e) +uj(f) − �uj(f) +|e − f|n+2s df, wherefn is the n-th coordinate of f += dj(−∆)suj(e) + djh(e). +(5.28) +Note that e ∈ Ω. Further, for y ∈ Bρj, we have +ψj(y) = �uj(yd +1 +2s +j ++ zj) = +� +udj(yd +1 +2s +j ++ zj) +if yn ≥ −αj, +udj(y′d +1 +2s +j ++ z′ +j, −(ynd +1 +2s +j ++ αjd +1 +2s +j )) +if yn ≤ −αj. +We can write +(y′d +1 +2s +j ++ z′ +j, −(ynd +1 +2s +j ++ αjd +1 +2s +j )) = �yd +1 +2s +j ++ �zj. + +ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS +21 +Again renaming the variables yd +1 +2s +j ++ zj and �yd +1 +2s +j ++ �zj by a and e, respectively, we get +ψj(y) = +� +udj(a) +if yn ≥ −αj, +udj(e) +if yn ≤ −αj. +We know that uj satisfies the equation (1.1) in the point-wise sense as well. Therefore, combining above equation +with the equations (5.23), (5.28), we have for y ∈ Bρj, +(−∆)sψj(y) + ψj(y) = ψj(y)p + djh(y). +(5.29) +Now, arguing as in the proof of (A) with minor modifications, one can obtain a convergent subsequence of +� +ψj +� +, +which we denote again by +� +ψj +� +such that ψj → v in C2 +loc(Rn). Therefore as dj ↓ 0, one obtain +(−∆)sv + v = vp in Rn. +(5.30) +Since v ∈ Hs(Rn) and v is radially decreasing so that v is spherically symmetric to y = 0. Moreover, v has the power +type decay at infinity, which follows from Theorem 2.8, i.e., +v(r) ≤ +C2 +rn+2s , r ≥ 1, +(5.31) +for some constant C2 > 0. Let us define δR as in (5.8), i.e., +δR := +C2 +Rn+2s , +for R sufficiently large to be defined later. Then, there exists an integer jR such that for j ≥ jR, +∥ψj − v∥C2(B4R) ≤ δR. +(5.32) +We choose R sufficiently large that R > αj for all j, where αj’s are same as defined earlier right after the equation +(5.18). We can choose such a R because {αj} is a bounded sequence. The following lemma is very useful to prove +our claim that zd ∈ ∂Ω. +Lemma 5.2. (see Lemma 4.2 [40]) Let f ∈ C2(Bt) be a radial function. Assume that f satisfies f ′(0) = 0 and +f ′′ < 0 for 0 ≤ r ≤ t. Then there exists a η > 0 such that if g ∈ C2(Bt) satisfies +(1) ∇g(0) = 0 +(2) ∥f − g∥C2(Bt) < η, +then ∇g ̸= 0 for x ̸= 0. +Now, we use this lemma to show that ψj has only one local maximum point in BR. For this, we choose two +numbers k, l (0 < k < l) such that v′′(r) < 0 for 0 ≤ r ≤ k. Note that v′′(0) < 0 and v(k) < 1. Let us define +θ = min +� +|v′(r)| | k ≤ r ≤ l +� +. +It is easy to observe that θ > 0 because v′ < 0 for r > 0. Then for δR < θ, we have by (5.32) that +0 < θ − δR ≤ |∇v(y)| − |∇ψj(y) − ∇v(y)| ≤ |∇ψj(y)| for k ≤ |y| ≤ l. +Applying Lemma 5.2 in the ball Bk, we conclude that y = 0 is the only local maximum point of ψj in Bl. If yj +is a maximum point of ψj in BR, then by Lemma 3.8, we have ψj ≥ 1. Choose R > 0 sufficiently large so that +δR < 1 − v(l). Therefore +ψj(y) ≤ v(y) + δR ≤ v(l) + δR < 1. +Hence yj ∈ Bl, and therefore yj = 0. +If αj > 0, then by the definition of �uj, z∗ +R = (z′ +j, −αjd +1 +2s +j ) is also a maximum point of �uj. This implies that (0, −αj) +is another maximum point of ψj in BR, a contradiction. This proves our claim. +□ + +22 +S. GANDAL, J. TYAGI +Appendix A +Proof of the Inequality (3.12): We show that for real numbers x, y ≥ 0 and k ≥ 1, +1 +k (xk − yk)2 ≤ (x − y)(x2k−1 − y2k−1). +(5.33) +Clearly, the inequality holds when either x or y or both are zero. Thus, without loss of generality we may assume +that x > y > 0. Now our claim is reduced to show that +1 +k +� +1 − +�y +x +�k�2 +≤ +� +1 − y +x +� � +1 − +�y +x +�2k−1� +i.e., to show that +(1 − ak)2 ≤ k(1 − a)(1 − a2k−1), +where 0 < a := y +x < 1. Consider +f(a) := k(1 − a)(1 − a2k−1) − (1 − ak)2 +≥ (1 − ak) +� +k(1 − a) − (1 − ak) +� +≥ (1 − ak)(1 − a) +� +k − (1 + a + a2 + · · · + ak−1) +� +≥ (1 − ak)(1 − a)(k − k) = 0. +This proves the inequality +□. +Acknowledgement +The first author thanks CSIR for the financial support under the scheme 09/1031(0009)/2019-EMR-I. 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Soc. 367 (2015), no. 1, +67–102. +Somnath Gandal +Indian Institute of Technology Gandhinagar +Palaj, Gnadhinagar Gujarat India-382355. +Email address: gandal somnath@iitgn.ac.in +JagmohanTyagi +Indian Institute of Technology Gandhinagar +Palaj, Gandhinagar Gujarat, India-382355. +Email address: jtyagi@iitgn.ac.in, jtyagi1@gmail.com + diff --git a/sNE1T4oBgHgl3EQfjQRT/content/tmp_files/load_file.txt b/sNE1T4oBgHgl3EQfjQRT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c213e310b5404b132b4f27f81db5d10189e6400f --- /dev/null +++ b/sNE1T4oBgHgl3EQfjQRT/content/tmp_files/load_file.txt @@ -0,0 +1,1202 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf,len=1201 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='03260v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='AP] 9 Jan 2023 ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS OF FRACTIONAL SEMILINEAR NEUMANN PROBLEM SOMNATH GANDAL, JAGMOHAN TYAGI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We establish the asymptotic behaviour of the least energy solutions of the following nonlocal Neumann problem: � � � d(−∆)su + u = |u|p−1 u in Ω, Nsu = 0 in Rn \\ Ω, u > 0 in Ω, where Ω ⊂ Rn is a bounded domain of class C1,1, 1 < p < n+s n−s, n > max {1, 2s} , 0 < s < 1, d > 0 and Nsu is the nonlocal Neumann derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We show that for small d, the least energy solutions ud of the above problem achieves L∞ bound independent of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Using this together with suitable Lr-estimates on ud, we show that least energy solution ud achieve maximum on the boundary of Ω for d sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Auxiliary Results 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Regularity and bounds for least energy solution ud 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Lr- estimates on ud 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Proof of theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4 14 Appendix A 22 Acknowledgement 22 Statement 22 References 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Introduction We discuss the asymptotic behaviour of non-constant least energy solutions of the following problem: \uf8f1 \uf8f2 \uf8f3 d(−∆)su + u = |u|p−1 u in Ω, Nsu = 0 in CΩ, u > 0 in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) where Ω ⊂ Rn be a bounded domain of class C1,1, 1 < p < n+s n−s, n > max {1, 2s} , 0 < s < 1, d > 0, CΩ := Rn \\Ω and Nsu is the nonlocal Neumann derivative, which is defined next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The nonlocal operator (−∆)s is called the fractional Laplacian which is defined as follows: (−∆)su(x) = cn,sP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' � Rn u(x) − u(y) |x − y|n+2s dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2) Here, by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=', we mean the Cauchy principal value and cn,s is a normalizing constant, given by cn,s = �� Rn 1 − cosx1 |x|n+2s dx �−1 , 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' 35J60, 35B09, 35B40, 35J61, 35R11, 35D30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Semilinear Neumann problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' fractional Laplacian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' positive solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' asymptotic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Submitted January 10, 2023 Published—–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' 1 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' GANDAL, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' TYAGI see for instance [14] for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Recently, Dipierro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' [16] have introduced a new nonlocal Neumann condition Ns, which is defined as follows: Nsu(x) := cn,s � Ω u(x) − u(y) |x − y|n+2s dy, x ∈ CΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3) The advantage of this nonlocal Neumann condition is that it has simple probabilistic interpretation and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) has a variational structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Further, Nsu approaches to the classical Neumann derivative ∂νu as s goes to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In the last few decades, mathematical analysis of biological phenomena has gained much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' For example, the chemotaxis models, which are also known as Keller-Segel models [33], have been widely studied in different directions in many papers, see [8, 10, 11, 24, 25, 26, 27, 31, 38] and the reference therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Chemotaxis is the natural behaviour of an organism in response of surrounding chemical gradients that are frequently separated by the cells themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We refer to [3, 27, 28] for a survey on this subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The Keller-Segel system with suitable initial data has blow-up solutions in dimension n ≥ 2 and all solutions are regular in dimension n = 1, see for instance [25, 29, 31, 38] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The analysis on the steady-state for a chemotactic aggregation model with linear or logarithmic sensitivity function was thoroughly done in many papers, see for instance [32, 34, 39, 40, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let us point out that the following semilinear Neumann problem is an example of Keller-Segel model with a logarithmic chemotactic sensitivity: \uf8f1 \uf8f2 \uf8f3 −d∆u + u = |u|p−1 u in Ω, ∂u ∂ν = 0 on ∂Ω u > 0 in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4) where d > 0, Ω ⊂ Rn is a bounded domain with smooth boundary and 1 < p ≤ n+2 n−2 if n ≥ 3 and 1 < p < ∞ if p = 2, see [34, 43] for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4) admits a non-constant solution for d sufficiently small, see [1, 34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' [34] and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Lin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' -M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Ni [35] established the solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4) in the subcritical case 1 < p < n+2 n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In the critical case, when p = n+2 n−2, Adimurthi and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Mancini [1] obtained a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' There have been developments on the asymptotic behaviour of solutions to such equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In the subcritical case, 1 < p < n+2 n−2, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' -M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Ni and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Takagi [40, 41] have studied the shape of least energy solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' They have shown that the least energy solutions tends to zero as the diffusion constant d goes to zero except at finite number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Moreover, the maximum of a solution ud of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4) is attained at a unique point on the boundary of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The critical case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=', p = n+2 n−2, was examined by Adimurthi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' [2] using the blow-up analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We refer to [23] for the exis- tence, non-existence and the asymptotic behaviour to critical fractional Choquard equation with a local perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We mention that Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) which we explore in this paper is a nonlocal analogue of the classical problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Recall that the movements of cells of some organisms cannot be described by random jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In such situations, L´evy flights plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The generalized Keller-Segel model with nonlocal diffusion term d(−∆)s, where d is a positive constant is used to investigate the chemotaxis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' For the fractional Keller-Segel model, we refer to [18, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In [30], H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Huang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Liu studied the existence, stability, uniqueness and regularity for the following model in dimension n ≥ 2 : \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ut = d(−∆)su − ∇ · (u∇φ) , x ∈ Rn, t ≥ 0, −∆φ = u, u(x, 0) = u0(x), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='5) where d is a positive constant, u(t, x) is the density of some biological cells and φ(t, x) is the chemical substance concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We mention the work [9], where authors have investigated the asymptotic behaviour of solutions for nonlinear elliptic problems for fractional Laplacian with Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We refer to [36] and the reference therein for in-depth treatment of variational methods to nonlocal fractional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Motivated by the above works and very recent works on nonlocal Neumann problem for fractional Laplacian and its connections with fractional Keller-Segel models, we have the following natural question to ask: Question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Can we establish the asymptotic behaviour of least energy solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS 3 The aim of this paper is to answer the above question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' A weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) can be obtained as a critical point of the energy functional Jd, which is defined as follows: Jd(u) := 1 2 �dcn,s 2 � T (Ω) |u(x) − u(y)|2 |x − y|n+2s dxdy + � Ω u2dx � − 1 p + 1 � Ω |u|p+1dx, u ∈ Hs Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='6) In the above equation T (Ω) = R2n \\ (CΩ)2 and the space Hs Ω is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The functional Jd is well-defined and of class C2 follows from the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' An application of Mountain-Pass Lemma applying to the functional Jd yields that cd := inf γ∈Γ max [0,1] Jd(γ(t)) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='7) is a critical value of Jd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In the above equation, by Γ, we mean the following set: Γ = � γ ∈ C([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Hs Ω) | γ(0) = 1, γ(1) = u � , where u ∈ Hs Ω, u > 0 and satisfying Jd(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' It turns out that cd is the least positive critical value, see, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3 next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' For the details one may refer, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1 [4] and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1 [6], where authors have obtained a nonnegative weak solution ud of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) with critical value cd, provided d is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Moreover, ud satisfies 0 < Jd(ud) ≤ Cd n 2s , where the constant C is independent of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Consequently, ud is non-constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' From the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1 [6], it is immediate to see that the critical points of Jd are not sign changing in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In fact, when ud ≤ 0, we can choose −ud in order to have a nonnegative solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' By the strong maximum principle (see, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='6 [12]), one can see that ud > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Further, since ud satisfies the Neumann condition Nsud(x) = 0 in CΩ which implies that ud > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We call a critical point ud of Jd with Jd(ud) = cd, the least energy solution or Mountain-Pass solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We show the asymptotic behaviour of least energy solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) following the similar approach as was used for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4) by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' -M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Ni and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Takagi [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' They used a positive solution w of non-linear Schr¨odinger equation −∆u + u = |u|p−1 u in Rn, 1 < p < n + 2 n − 2 to study the asymptotic behaviour of the least energy solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The fractional non-linear Schr¨odinger equation (−∆)su + u = |u|p−1 u in Rn, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) where 1 < p < n+2s n−2s, n > max {1, 2s} , 0 < s < 1 is thoroughly studied, see for instance [7, 15, 20, 21] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The main idea of this work is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let cd be a critical value of Jd, which is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We use a positive solution w of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) to observe the asymptotic behaviour of cd as d ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' More specifically, w is used to build a suitable function φd to compare cd with maxt≥0 Jd(tφd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In particular, we obtain an inequality cd < d n 2s 2 F(w) for d sufficiently small, where F is the functional associated with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8), defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' This is closely related to the location of maximum point of a solution ud of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) on the boundary of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Now, we summarise the above discussions in terms of the following three main theorems: A priori it is known that for 1 ≤ p < n+s n−s, any weak solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) satisfies ∥u∥L∞(Ω) ≤ K, where K > 0 is some constant depending on Ω, p and d, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In next result we obtain a bound for least energy solution ud of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) which is independent of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' GANDAL, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' TYAGI Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let ud be the least energy solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then dcn,s 2 � T (Ω) |ud(x) − ud(y)|2 |x − y|n+2s dxdy + � Ω u2 ddx = � Ω up+1 d dx ≤ C0d n 2s , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9) where C0 > 0 is some constant depending on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Moreover, there is a constant C1 > 0 depending only on p and Ω such that sup Ω ud(x) ≤ C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='10) In the next theorem, we show that the Lr-norm of the least energy solution ud is bounded by d n 2s times some constant independent of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let ud be the least energy solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then b(r)d n 2s ≤ � Ω ur ddx ≤ B(r)d n 2s , if 1 ≤ r ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='11) b(r)d n 2s ≤ � Ω ur ddx ≤ B(r)d nr 2s , if 0 < r < 1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='12) where b(r) and B(r) are positive constants such that b(r) < B(r) and are independent of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We show the asymptotic behaviour in next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let Ω ⊂ Rn be a bounded domain of class C1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let ud be the least energy solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' If ud achieves maximum at a point zd ∈ Ω, then for all d sufficiently small, we have the following: (A) There exists a positive constant K∗ such that ρ(zd, ∂Ω) ≤ K∗d 1 2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Here, by ρ we mean the distance between zd and ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (B) zd ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The plan of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In Section 2, we recollect known results which are useful for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In Section 3, we study the regularity of least energy solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) and complete the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In Section 4, we have derived Lr-estimate for the least energy solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Section 5 is devoted to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='The proof of inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='12) (see, next) is a part of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Auxiliary Results Let us recall the important results which are used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (Fractional Sobolev Embedding [14]) Let n > 2s and 2∗ s = 2n n−2s be the fractional critical exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then, we have the following inclusions: (1) for any function u ∈ C0(Rn) and for q ∈ [0, 2∗ s − 1] : ∥u∥2 Lq+1(Rn) ≤ B(n, s) � Rn � Rn |u(x) − u(y)|2 |x − y|n+2s dxdy, for some positive constant B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' That means Hs(Rn) is continuously embedded in Lq+1(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (2) Let Ω ⊂ Rn be a bounded extension domain for Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then, the space Hs(Ω) is continuously embedded in Lq+1(Ω) for any q ∈ [0, 2∗ s − 1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='e, ∥u∥2 Lq+1(Ω) ≤ B(n, s, Ω) ∥u∥2 Hs(Ω) for some positive constant B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Further, the above embedding is compact for any q ∈ [0, 2∗ s − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let T (Ω) := R2n \\ (Rn \\ Ω)2 be a cross-shaped set on a bounded domain Ω ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) Hs Ω := � u : Rn −→ R measurable : ∥u∥Hs Ω < ∞ � which is equipped with the norm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2) ∥u∥Hs Ω := � ∥u∥2 L2(Ω) + � T (Ω) |u(x) − u(y)|2 |x − y|n+2s dxdy � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS 5 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Hs Ω is a Hilbert space (see [16], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let us define the following set: Ls := � u : Rn −→ R measurable : � Rn |u(x)| 1 + |x|n+2s dx < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The condition u ∈ Ls is useful to give a sense to pointwise definition of fractional Laplacian 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3 [12]) Let Ω ⊂ Rn be a bounded set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then Hs Ω ⊂ Ls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Next, we recall a few known results about fractional Schr¨odinger equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' A measurable function u : Rn −→ R is called a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) if it satisfies the following equation cn,s 2 � Rn � Rn (u(x) − u(y))(ψ(x) − ψ(y)) |x − y|n+2s dxdy + � Rn u(x)ψ(x)dx = � Rn |u(x)|p−1 u(x)ψ(x)dx, for all ψ ∈ C1 0(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We define the corresponding energy functional F : Hs(Rn) −→ R as follows: F(u) := 1 2 �cn,s 2 � Rn � Rn |u(x) − u(y)|2 |x − y|n+2s dxdy + � Rn u2dx � − 1 p + 1 � Rn |u|p+1dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3) The weak solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) corresponds to the critical points of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' A function u ∈ Ls(Rn) ∩ C2s+ǫ(Rn), when 0 < s < 1 2, 2s + ǫ < 1 or u ∈ C1,2s+ǫ−1(Rn) ∩ Ls(Rn), when 1 2 ≤ s < 1, 2s + ǫ − 1 < 1 is said to be a classical solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) if it satisfies the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) pointwise in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Next result gives us a positive, radially symmetric solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8), which decays at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4 [20]) Let u be a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then u ∈ Lq(Rn)∩Cα(Rn) for some q ∈ [2, ∞) and α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Moreover, lim |x|→∞ u(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3 [20]) Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) has a weak solution in Hs(Rn), which satisfies u ≥ 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Moreover, u is a classical solution that satisfies u > 0 in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Following theorem shows that the solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) has a power type of decay at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='5 [20]) Let u be a positive classical solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) such that lim |x|→∞ u(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then, there exist constants 0 < C1 ≤ C2 such that C1 |x|n+2s ≤ u(x) ≤ C2 |x|n+2s for all |x| ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4) One can see that there exist some m > 0 and s0 > 0 such that for f(u) = up − u, we have f(v) − f(u) v − u ≤ vp − up v − u ≤ C(v + u)m for all 0 < u < v < s0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='5) where C > 0 is some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Also, it is simple to see that f : [0, ∞) → R is locally Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Consequently, we have the following result on radial symmetry and monotonicity property of positive solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2 [21]) Let u be a positive classical solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) such that lim |x|→∞ u(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Further, assume that there exists t > max �2s m , n m + 2 � 6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' GANDAL, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' TYAGI such that u satisfies u(x) = O � 1 |x|t � as |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then, u is radially symmetric and strictly decreasing about some point in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Since C1 |x|n+2s ≤ u(x) ≤ C2 |x|n+2s for all |x| ≥ 1, we can take t = n + 2s in the above theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Now, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1[44] ascertains that if u ∈ Rn is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) then u satisfies the following Pohozaev identity: P(u) := (n − 2s)cn,s 4 � Rn � Rn |u(x) − u(y)|2 |x − y|n+2s dxdy + n 2 � Rn u2dx − n p + 1 � Rn up+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let us define G := � u ∈ Hs(Rn) \\ {0} | P(u) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In [7], authors have obtained a weak solution w ∈ Hs(Rn) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) with least energy among all other solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In particular, they have proved the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2 [7]) Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) has a weak solution w ∈ Hs(Rn) such that 0 < F(w) = inf u∈G F(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Combining Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='7, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='11 we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) has a positive classical solution w ∈ Hs(Rn) satisfying (a) w has a power type of decay at infinity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=', there exist constants 0 < C1 ≤ C2 such that C1 |x|n+2s ≤ w(x) ≤ C2 |x|n+2s for all |x| ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (b) w is radially symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=', w(x) = w(r) with r = |x| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (c) For any non-negative classical solution u ∈ Hs(Rn) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8), 0 < F(w) ≤ F(u) holds unless u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We call w, given by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='12, a ground state solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Regularity and bounds for least energy solution ud Let s ∈ (0, 1) and Ω ⊂ Rn be a bounded domain of class C1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' A measurable function u : Rn −→ R is said to be a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) if it satisfies the equation dcn,s 2 � T (Ω) (u(x) − u(y))(ψ(x) − ψ(y)) |x − y|n+2s dxdy + � Ω u(x)ψ(x)dx = � Ω |u(x)|p−1 u(x)ψ(x)dx, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) for all ψ ∈ Hs Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We have the following result on the existence of weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1 [4], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1 [6]) There exists a nonnegative weak solution ud of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) with critical value cd, provided d is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Moreover, ud satisfies 0 < Jd(ud) ≤ Cd n 2s , where the constant C is independent of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Consequently, ud is non-constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Define M[v] := sup t≥0 Jd(tv), v ∈ Hs Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In the next lemma, we indicate useful characterization of the critical value cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We follow the similar lines of proof as Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS 7 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The critical value cd is independent of the choice of u ∈ Hs Ω such that u ≥ 0, u ̸≡ 0 and Jd(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In fact, cd is the least positive critical value of Jd, and is given by cd = inf � M[v] | v ∈ Hs Ω, v ̸≡ 0, v ≥ 0 in Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' For v ∈ Hs Ω, let Ω+ = � x ∈ Ω | v(x) > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Now, for all those v satisfying |Ω+| > 0, define gd(t) := Jd(tv), for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' First, we will show that gd(t) has a unique maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' For this, we have g′ d(t) = t � dcn,s 2 � T (Ω) |v(x) − v(y)|2 |x − y|n+2s dxdy + � Ω v2dx � − tp � Ω vp+1dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Therefore, g′ d(t0) = 0 for some t0 > 0 if and only if dcn,s 2 � T (Ω) |v(x) − v(y)|2 |x − y|n+2s dxdy + � Ω v2dx = tp−1 0 � Ω vp+1dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Note that the right hand side is strictly increasing in t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' And hence there exists unique t0 > 0 such that g′ d(t0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Since gd(t) > 0 for t > 0 small and gd(t) → −∞ as t → +∞, one easily find that gd(t) has a unique maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let us fix a function u ̸≡ 0, u ≥ 0 in Hs Ω with Jd(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let ud be a positive solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) obtained by applying Mountain-Pass Lemma and cd the corresponding critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We have Jd(ud) = cd and J ′ d(ud) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Since ud > 0 and J ′ d(ud) = 0, we have M[ud] = cd, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3) and hence cd ≥ inf � M[v] | v ∈ Hs Ω, v ̸≡ 0, v ≥ 0 in Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4) On the contrary, assume that the strict inequality occurs in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then, we have M[v0] < cd, for some v0 ≥ 0, v0 ̸≡ 0 in Hs Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Therefore, there exists some t1 > 0 such that t1v0 = u0 satisfies Jd(u0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Denote by U the subspace of Hs Ω spanned by u and u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Consider the subset of U defined as follows: U + := � αu + βu0 | α, β ≥ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let S be a circle on U of radius R so large that R > max � ∥u∥ , ∥u0∥ � and Jd ≤ 0 on S ∩U +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let γ be the path made up of the line segment with endpoints 0 and Ru0 ∥u0∥, the circular arc S ∩ U + and the line segment with endpoints Ru ∥u∥ and u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' One can easily notice that, along γ, Jd is positive only on the line segment joining 0 and u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Hence, we have max v∈γ Jd(v) = M[v0] < cd, a contradiction to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Thus, we have the equality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=', cd = inf � M[v] | v ∈ Hs Ω, v ̸≡ 0, v ≥ 0 in Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='5) Note that Jd(v) = Jd(−v) for any v ∈ Hs Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Since any nontrivial critical point of Jd is either positive or negative almost everywhere in Ω, from the above discussion one can see that cd is the least positive critical value of Jd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' □ The following lemma gives us the regularity estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The similar result is already proved in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='6 [12], Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9 [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let u ∈ Hs Ω be a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' If u ∈ L∞(Ω) then u ∈ L∞(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Moreover, (1) For 0 < s < 1 2, u ∈ C2(Ω) if p > 3 − 2s and u ∈ C1,p−2+2s(Ω) if 2 < p ≤ 3 − 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (2) For 1 2 ≤ s < 1, u ∈ C2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' 8 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' GANDAL, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' TYAGI Now, we prove that the least energy solution ud is bounded by some constant independent of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The proof of the first inequality of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2 is fairly standard and simple, which can be seen in the literature, for instance, see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Since it is short, for the sake of completeness, we include it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' For this, we have Jd(ud) := 1 2 � cn,sd 2 � T (Ω) |ud(x) − ud(y)|2 |x − y|n+2s dxdy + � Ω u2dx � − 1 p + 1 � Ω up+1 d dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='6) Since ud is a critical point of Jd, we have Jd ′(ud) = 0 on Hs Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='7) This implies that dcn,s 2 � T (Ω) |ud(x) − ud(y)|2 |x − y|n+2s dxdy + � Ω u2 ddx = � Ω up+1 d dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) Hence from above equations, we get Jd(ud) = �1 2 − 1 p + 1 � � Ω up+1 d dx (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9) = (p − 1) 2(p + 1) � Ω up+1 d dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='10) Now, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2, we have Jd(ud) ≤ Cd n 2s , where the constant C depends only on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Using this inequality in the above equation, we get � Ω up+1 d dx ≤ 2(p + 1) p − 1 Cd n 2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Taking C0 = 2(p+1) p−1 C, proves the first inequality of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The proof of second inequality of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2 is little constructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We claim that sup Ω ud(x) ≤ C1 for some constant C1 > 0 depending on p and Ω only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Multiplying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) by u2t−1 d and integrating over Ω, we get cn,sd 2 � T (Ω) (ud(x) − ud(y))(u2t−1 d (x) − u2t−1 d (y)) |x − y|n+2s dxdy + � Ω u2t d dx = � Ω up+2t−1 d dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='11) Now, we use the following inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We have given the proof of this inequality in appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let x, y ≥ 0 are real numbers and k ≥ 1, then we have 1 k (xk − yk)2 ≤ (x − y)(x2k−1 − y2k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='12) Consequently, we have 1 t � T (Ω) (ut d(x) − ut d(y))2 |x − y|n+2s dxdy ≤ � T (Ω) (ud(x) − ud(y))(u2t−1 d (x) − u2t−1 d (y)) |x − y|n+2s dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='13) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='13), we get dcn,s 2t � T (Ω) (ut d(x) − ut d(y))2 |x − y|n+2s dxdy + � Ω u2t d dx ≤ � Ω up+2t−1 d dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='14) Now, by the fractional Sobolev embedding Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1, �� Ω |v|2∗ s �2/2∗ s ≤ A d � dcn,s 2 � Ω � Ω |v(x) − v(y)|2 |x − y|n+2s dxdy + � Ω |v|2 dx � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='15) where d ∈ (0, d0) for some d0 > 0, A > 0 some constant, v ∈ Hs(Ω), and 2∗ s = 2n n−2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The embedding constant A depends only on n, s, d0, and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' To see this, let us define Ωd := � y : y d1/2s ∈ Ω � and w(y) := v � y d1/2s � , where y ∈ Ωd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS 9 Now, we have d � Ω � Ω |v(x) − v(y)|2 |x − y|n+2s dxdy + � Ω v2dx = 1 d n 2s �� Ωd � Ωd ���v( � x′ d 1 2s � − v( � y′ d 1 2s ���� 2 |x′ − y′|n+2s dx′dy′ + � Ωd v � x′ d 1 2s �2 dx′ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='16) = 1 d n 2s �� Ωd � Ωd |w(x′) − w(y′)|2 |x′ − y′|n+2s dx′dy′ + � Ωd w(x′)2dx′ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='17) ≥ A d n 2s �� Ωd |w|2∗ sdx′� 2 2∗s (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='18) = Ad � 2 2∗s −1 � n 2s �� Ω |v|2∗ sdx � 2 2∗s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='19) Therefore, we observe that A is uniform for d ∈ (0, d0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Note that Ω × Ω ⊂ T (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then by virtue of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='15), we have �� Ω |ud|t2∗ s � 2 2∗s ≤ tA d � Ω up+2t−1 d dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='20) Now, we define two sequences � Lj � and � Mj � by the following recurrence relations: p − 1 + 2L0 = 2∗ s, p − 1 + 2Lj+1 = 2∗ sLj, j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='21) M0 = (AC0) 2∗s 2 , Mj+1 = (ALjMj) 2∗s 2 , j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='22) We note that Lj is explicitly given by Lj = 1 (2∗s − 2) ��2∗ s 2 �j+1 (2∗ s − p − 1) + p − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='23) Since 1 < p < 2∗ s − 1, it follows that Lj ≥ 1 for all j ≥ 0 and Lj → ∞ as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We shall show that � Ω up−1+2Lj d dx ≤ Mjd n 2s for all j ≥ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='24) and Mj ≤ emLj−1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='25) for some constant m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then, we have sup Ω ud(x) ≤ C1, where C1 > 0 depending only on C0 and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In fact (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='23) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='24) entail us ∥u∥L2∗sLj−1 (Ω) ≤ � emLj−1d n 2s � 1 (2∗s Lj−1) = e m 2∗s d (n−2s) 4Lj−1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='26) and hence letting j → ∞, we obtain ∥u∥L∞(Ω) ≤ e m 2∗s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' GANDAL, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' TYAGI First, we verify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' By virtue of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='15), we have �� Ω |ud|2∗ s � 2 2∗s ≤ A d �cn,sd 2 � T (Ω) |ud(x) − ud(y)|2 |x − y|n+2s dxdy + � Ω |ud|2 dx � ≤ A d C0d n 2s = AC0d n s2∗s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='27) Hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='24) holds for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Suppose that we have proved (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='24) for j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='20), we have � Ω |ud|p−1+2Lj+1dx ≤ �LjA d � Ω up+2Lj−1 d dx � 2∗s 2 ≤ � ALjd−1Mjd n 2s � 2∗s 2 = � ALjMj � 2∗ s 2 d n 2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='28) This implies that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='24) is also true for j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Therefore it remains to show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Put λj = 2∗ s 2 · log(ALj) and ηj = log(Mj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='29) Hence ηj+1 = 2∗ s 2 · ηj + λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='30) The explicit value of Lj is given by Lj = (2∗ s − 2)−1� (2−12∗ s)j+1(2∗ s − p − 1) + p − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='31) Now, we have λj = 2∗ s 2 log � A (2∗s − 2) � (2−12∗ s)j+1(2∗ s − p − 1) + p − 1 �� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='32) = 2∗ s 2 � log(A(2∗ s − 2)) + log � (2−12∗ s)j+1(2∗ s − p − 1) + p − 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='33) Therefore, we can find some C∗ such that λj ≤ C∗(j + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='34) We now define a sequence � γj � by γ0 = η0 and γj+1 = 2∗ s 2 γj + C∗(j + 1) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='35) for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Clearly, ηj ≤ γj for all j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Moreover, since γj = �2∗ s 2 �j� η0 + 2C∗2∗ s(2∗ s − 2)−2� −2C∗(2∗ s − 2)−1� j + 2∗ s(2∗ s − 2) � , in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='31), there exists m > 0 such that γj ≤ mLj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Hence log(Mj) ≤ mLj−1 and we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Note that m depends only on η0, 2∗ s and C∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' whereas C∗ depends only on 2∗ s, p and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' It is known that if u ∈ Ls(Rn) ∩ C2s+ǫ(Ω), when 0 < s < 1 2, 2s + ǫ < 1 or u ∈ Ls(Rn) ∩ C1,2s+ǫ−1(Ω), when 1 2 ≤ s < 1, 2s + ǫ − 1 < 1, one can compute (−∆)su(x) pointwise for all x in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In fact, one can write (−∆)su(x) = cn,sP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' � Rn u(x) − u(y) |x − y|n+2s dy Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We say that u : Rn −→ R is a classical solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) if it satisfies the following: (1) u ∈ Ls(Rn)∩C2s+ǫ(Ω), when 0 < s < 1 2, 2s+ǫ < 1 or u ∈ Ls(Rn)∩C1,2s+ǫ−1(Ω), when 1 2 ≤ s < 1, 2s+ǫ−1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS 11 (2) Nsu(x) = 0, x ∈ Rn \\ Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3) d(−∆)su(x) + u(x) = |u(x)|p−1 u(x) pointwise for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We make similar remarks as in [5], which offers a relation between the weak and classical solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let ud be a least energy solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) in Hs Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4, we have (1) for 0 < s < 1 2, ud ∈ Ls(Rn) ∩ C2(Ω) if p > 3 − 2s and ud ∈ Ls(Rn) ∩ C1,p−2+2s(Ω) if 2 < p ≤ 3 − 2s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (2) for 1 2 ≤ s < 1, ud ∈ Ls(Rn) ∩ C2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Now, using nonlocal integration by parts formulae given in [16], one can easily check that d(−∆)sud(x) + ud(x) = |ud(x)|p−1 ud(x) holds pointwise in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' This implies that ud is a classical solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Conversely, if ud is a classical solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) satisfying ud ∈ Hs Ω, then ud is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The following lemma shows that the maximum of least energy solution is always greater than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let ud be the least energy solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let Md = sup x∈Ω ud(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='36) Then Md > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Since ud is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1), we get dcn,s 2 � T (Ω) (ud(x) − ud(y))(w(x) − w(y)) |x − y|n+2s dxdy + � Ω udwdx = � Ω up dwdx holds, ∀ w ∈ Hs Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='37) Taking w = 1 in the above equation, we get � Ω ud(x)dx = � Ω up d(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' This implies that � Ω ud(x)(1 − up−1 d (x))dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Now, if ud(x) ≤ 1, for all x ∈ Ω, then 1 − ud(x) ≥ 0, ∀x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Thus from the above equation, we get that ud(x) = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Now, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4, we can assume that ud is continuous and hence ud ≡ 1 in Ω, a contradiction to our assumption that ud is a non-constant solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Therefore, there exists x0 in Ω such that ud(x0) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Thus Md > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Lr- estimates on ud Here, we derive Lr-estimate for ud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Following results are generalization to the nonlocal case of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' For d0 > 0 fixed, there is a constant K0 such that dcn,s 2 � T (Ω) (ud(x) − ud(y))2 |x − y|n+2s dxdy + � Ω u2 ddx ≥ K0d n 2s , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) where ud is the least energy solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) with 0 < d < d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' On contrary, suppose that there is a sequence � dk � contained in the interval (0, d0) and a sequence of positive solutions � uk � to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) with d = dk such that ζk := 1 d n 2s � dcn,s 2 � T (Ω) (uk(x) − uk(y))2 |x − y|n+2s dxdy + � Ω u2 kdx � → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2) 12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' GANDAL, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' TYAGI We are going to follow the same arguments as used in the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2 to prove this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Once again define the sequences � Lk � and � Mj � as defined earlier in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='21) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='22), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Instead of C0, we write ζk in the definition of � Mj � : p − 1 + 2L0 = 2∗ s, p − 1 + 2Lj+1 = 2∗ sLj, j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3) and M0 = (Aζk) 2∗ s 2 , Mj+1 = (ALjMj) 2∗ s 2 , j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4) Further, define the sequences � λj � , � ηj � , and � γj � as defined earlier in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='29) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='24), we have �� Ω u2∗ sLj−1 k dx �(2∗ sLj−1) ≤ � Mjdn/2s k �1/(2∗ sLj−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='5) Since log(Mj) = ηj ≤ γj, we have log (Mj) 2∗sLj−1 ≤ ηj 2∗sLj−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='6) Now, lim j→∞ ηj 2∗sLj−1 = lim j→∞ � 2∗ s 2 �j� η0 + 2C∗2∗ s(2∗ s − 2)−2� − 2C∗(2∗ s − 2)−1� j + 2∗ s(2∗ s − 2) � 2∗ s (2∗ s−2) �� 2∗ s 2 �j (2∗s − p − 1) + p − 1 � = (2∗ s − 2)(η0 + 2C∗2∗ s(2∗ s − 2)−2) 2∗s(2∗s − p − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Letting j → ∞ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='5), we get ∥uk∥L∞(Ω) ≤ ea1(η0+a2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='7) with a1 and a2 depending only on 2∗ s, p and C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Since η0 = log(M0) = 2∗ s 2 log(Aζk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Therefore, as k → ∞, η0 → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Thus, in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='7), we get ∥uk∥L∞(Ω) → 0, which leads to a contradiction to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' □ Proof of the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3: First, we will show the second part of Inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Case-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' r ≥ 2∗ s = 2n n−2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let � Lj � be the sequence defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' If r ∈ � 2∗ sLj � , then the second inequality of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='11) follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' So assume that 2∗ sLj < r < 2∗ sLj+1 for some j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We have r = t2∗ sLj + (1 − t)2∗ sLj+1, for some t ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS 13 Using H¨olders inequality and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='24), we get � Ω ur ddx = � Ω ut2∗ sLj+(1−t)2∗ sLj+1 d dx, ≤ �� Ω u2∗ sLj d dx �t �� Ω u2∗ sLj+1 d dx �1−t ≤ � Mj−1dn/2s�t (Mjdn/2s)1−t = M t j−1M 1−t j d n 2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Case-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' 2 ≤ r ≤ 2∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We write r = 2t + (1 − t)2∗ s, for some t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then, using H¨older’s inequality, from Equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='24) with j = 0, we get � Ω ur ddx ≤ �� Ω u2 ddx �t �� Ω u2∗ s d dx �1−t ≤ Ct 0M (1−t) 0 d n 2s , where the constant C0 is independent of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Case-III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' 1 ≤ r < p + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Integrating both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) and using the condition Nsu(x) = 0, for x ∈ CΩ, we get � Ω uddx = � Ω up ddx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) It is easy to see that p = t + (1 − t)(p + 1) with t = 1 p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Notice that p + 1 ∈ (2, 2∗ s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Therefore, using the H¨older’s inequality and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8), we get � Ω up ddx ≤ �� Ω uddx �t �� Ω up+1 d dx �(1−t) , � Ω up ddx ≤ � Ω up+1 d dx ≤ C0d n 2s (by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9)) , where the constant C0 depends only upon p + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Also, in the view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9), we observe that the second inequality of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='11) holds for r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Now, repeating the interpolation between 1 and p + 1, we see that the second inequality of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='11) holds for all r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Case-IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let 0 < r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Taking F = ur d, G = 1, p = 1 r, q = 1 1−r and using the H¨olders inequality, we get � Ω ur ddx ≤ ∥F∥p ∥G∥q = |Ω|1−r �� Ω uddx �r ≤ |Ω|1−r B(1)rd nr 2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' This proves the second inequality of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Now, let us prove the first inequality of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='11) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' In view of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1), we see that � Ω up+1 d ≥ K0d n 2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9) Since sup Ω ud(x) ≤ C1, for some constant C1 > 0, we have K0d n 2s ≤ � Ω up+1 d = � Ω � up+1−r d � (ur d) dx ≤ Cp+1−r 1 � Ω ur ddx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' 14 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' GANDAL, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' TYAGI This implies that � Ω ur ddx ≥ K0Cr−p−1 1 d n 2s , r < p + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' For r > p + 1, we write p + 1 = 1 + (1 − t)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Therefore, we get K0d n 2s ≤ � Ω up+1 d dx = � Ω u1+(1−t)r d dx ≤ � uddx �t� ur ddx �1−t ≤ � B(1)d n 2s �t� ur ddx �1−t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' This yields that � Ω ur ddx ≥ (K0B(1)−t) 1 1−t d n 2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Proof of theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4 We prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4 in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Its proof is more involved and requires some scaling and compactness arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We prove the statements of theorem one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let zd ∈ Ω be a point of maximum of ud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The basic idea for its proof is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' We approximate ud around zd by a scaled positive radial solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' It gives us an upper bound on cd, which is closely related to the location of point zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Proof of (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' If the inequality in (A) is not true, then there is a decreasing sequence dj ↓ 0 such that ρj := ρ(zj, ∂Ω) d 1 2s j → +∞ as j → ∞, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1) where zj := zdj is a point of maximum of udj on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Define φj(y) := udj(yd 1 2s j + zj) for y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Since ud is a classical solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1), we have (−∆)sφj + φj = φp j in Bρj, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2) and (1) φj ∈ C0,2s+ǫ(Bρj), when 0 < s < 1 2, 2s + ǫ < 1 (2) φj ∈ C1,2s+ǫ−1(Bρj), when 1 2 ≤ s < 1, 2s + ǫ − 1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' First, we claim that the sequence � φj � contains a convergent subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let � Rk � be a monotone increasing sequence of positive numbers with Rk → +∞ as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Therefore, we have for each k, there is a number jk such that 4Rk < ρj whenever j ≥ jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Since ud ∈ L∞(Rn) ∩ Ls(Rn), we have φj ∈ L∞(Rn) ∩ Ls(Rn) for each j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Now, we can use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4 [19] to get the following estimates: For 0 < s < 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' 2s + ǫ < 1 i) 4s + ǫ < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' then ∥φj∥C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4s+ǫ(B2Rk) ≤ C � ∥φj∥L∞(Rn) + ��φp j − φj �� C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2s+ǫ(B4Rk ) � ii) 1 < 4s + ǫ < 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' then ∥φj∥C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4s+ǫ−1(B2Rk) ≤ C � ∥φj∥L∞(Rn) + ��φp j − φj �� C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2s+ǫ(B4Rk ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' and for 1 2 ≤ s < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' 2s + ǫ − 1 < 1 iii) 4s + ǫ − 1 < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' then ∥φj∥C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4s+ǫ−1(B2Rk ≤ C � ∥φj∥L∞(Rn) + ��φp j − φj �� C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2s+ǫ−1(B4Rk ) � ASYMPTOTIC BEHAVIOUR OF THE LEAST ENERGY SOLUTIONS 15 iv) 1 < 4s + ǫ − 1 < 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' then ∥φj∥C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4s+ǫ−1(B2Rk ) ≤ C � ∥φj∥L∞(Rn) + ��φp j − φj �� C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2s+ǫ−1(B4Rk) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' where the constant C > 0 is independent of j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let us recall the inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9) here: dcn,s 2 � T (Ω) |ud(x) − ud(y)|2 |x − y|n+2s dxdy + � Ω u2dx = � Ω up+1 d ≤ C0d n 2s , where C0 is independent of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' This yields � Bρj φp+1 j ≤ C0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3) and ∥φj∥Hs(Bρj ) ≤ C0, for all j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4) Also, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3 we have � Ω ur d ≤ B(r)d n 2s for all r ≥ 1 which implies that � Bρj φr j ≤ B(r), for all j ≥ 1 and r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='5) By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='2, we have ∥ud∥L∞(Rn) ≤ C1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='6) where the constant C1 is independent of the diffusion constant d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' So the equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='5), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='6) and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3 [19] imply that ∥φj∥Xs(BRk ) < C2 for all j ≥ jk, where the constant C2 > 0 is independent of j and the space Xs(BRk) is identified with one of the spaces C0,4s+ǫ(BRk), C1,4s+ǫ−1(BRk) or C2,4s+ǫ−1(BRk) with same assumptions on s and ǫ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Therefore � φj � is a relatively compact set in Xs(BRk), hence by the standard diagonal process, one can extract a convergent subsequence of � φj � , we continue to denote such a subsequence by � φj � itself such that φj → v in C0,2s+ǫ loc (Rn) when 0 < s < 1 2, 2s + ǫ < 1 or φj → v in C1,2s+ǫ−1 loc (Rn) when 1 2 < s < 1, 2s + ǫ − 1 < 1 for some v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' The limit v ∈ C0,2s+ǫ(Rn) ∩ Hs(Rn) when 0 < s < 1 2, 2s + ǫ < 1 or v ∈ C1,2s+ǫ−1(Rn) ∩ Hs(Rn) when 1 2 < s < 1, 2s + ǫ − 1 < 1 follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Consequently, we have lim |x|→∞ v(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='1 [17], we have (−∆)sφj(x) converges to (−∆)sv(x) point-wise in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Consequently, we see that the limit v satisfies the equation (−∆)sv + v = vp in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='7) Clearly, v ≥ 0 because each φj ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Since by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8, we have φj(0) = udj(zj) > 1 for each j ≥ 1, one can see that v ̸≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9, one can see that v is radially symmetric and decreasing about some point in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Since ∇v(0) = lim j→∞ ∇φj(0) = 0 16 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' GANDAL, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' TYAGI so it implies that v is radially symmetric about the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' And by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8, v has a power type of decay at infinity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=', v(r) ≤ C2 rn+2s , r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Now, we derive a lower bound on the critical value cdj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Let us define δR := C2 Rn+2s , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='8) where R > 0 arbitrarily large real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Then, there exists a positive integer jR such that if j ≥ jR then ρj ≥ 2R and ∥φj − v∥C2(B2R) ≤ δR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9) By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='3, we have cdj = M[udj] = Jdj(udj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content=' Using this fact and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='9), we obtain cdj = �1 2 − 1 p + 1 � � Ω up+1 dj dx (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE1T4oBgHgl3EQfjQRT/content/2301.03260v1.pdf'} +page_content='10) ≥ �1 2 − 1 p + 1 � � |x−zj|0 = 0 and the masses of the polar and nematic fields are +Dr and 4Dr − γ, respectively. To compute the first-order cor- +rection to mean field, we evaluate +γ +2N ⟨ ˆf2 ˆfk−2 − ˆf−2 ˆfk+2⟩ + +vO +roDr +0.2 +0.1 +po +0.0 +0.0 +0.5 +1.0 +1.530 +roDr +20 +10 +po +0 +0.0 +0.5 +1.0 +1.521.0 +0.5 +0.0P(e1.0 +0.5 +0.03 +to leading order in N −1. +Rewriting the Fourier modes as +ˆfk = f 0 +k + δfk and using that f 0 +k = 0 in the high-temperature +phase, one can expand the correlators to get +∂tfk = −k2Drfk+ kγ +2N (⟨δf2δfk−2⟩ − ⟨δf−2δfk+2⟩) . (5) +Then, the linearized dynamics of the fluctuations in Fourier +space read ∂tδfk = − +� +k2Dr − γδk,±2 +� +δfk + ξk. Using It¯o +calculus, one then finds the steady-state correlators +⟨δfkδfq⟩ = +Λkq +(k2 + q2)Dr − γ(δ|k|,2 + δ|q|,2) , +(6) +with Λkq = ⟨ˆΛkq⟩. This yields a renormalized dynamics for +the modes given by ∂tfk = −mkfk + O(fk/N 2). For the +polar and nematic fields, we get the renormalized masses +m1 = Dr − +4γDr(γ − Dr) +N(5Dr − γ)(13Dr − γ) + O +� 1 +N 2 +� +, +(7a) +m2 = 4Dr − γ + +16Drγ +N(20Dr − γ) + O +� 1 +N 2 +� +. +(7b) +As expected, fluctuations increase the mass of the nematic +field, shifting the ordering transition to temperatures lower +than the mean-field prediction Dr = γ/4. Surprisingly, how- +ever, the mass of the polar field is reduced by fluctuations +when Dr < γ < 4Dr. Fluctuations thus suppress the ne- +matic order while they favor the polar one. While our results +are perturbative, microscopic simulations reported in supple- +mentary Fig. S2 show these effects to hold non-pertubatively. +Note that our results rely on the exact expression of the noise +statistics, Eq. (3), which we derived in Eq. (2) from the micro- +scopic dynamics, Eq. (1). As shown in [45], complementing a +mean-field Landau theory by phenomenological noises would +(wrongly) lead to the opposite prediction of an increase of m1 +due to fluctuations. +The interplay between fluctuations and nematic torques +thus leads to a reduction of the polar field mass. This general +result, which also holds in equilibrium, implies that nematic +torques enhance the persistence of active particles. Together +with the phase diagrams shown in Fig. 1, this qualitatively ex- +plains how phase separation is either favored or suppressed by +nematic torques. We now turn to check the validity of our pre- +dictions as well as their scope. Before considering the com- +plex many-body dynamics of Eq. (1), we consider a simpler +problem in which persistence plays a key role. +Boundary accumulation. +A typical trait of active par- +ticles is their tendency to accumulate at confining bound- +aries [2, 18, 47–50] where they spend a typical time of or- +der τ before escaping back to the bulk of the system. Di- +mensional analysis predicts that the ratio between surface and +bulk densities—which scale as inverse surface and volume, +respectively—should behave as ρs/ρb ∝ v0τ = ℓp, where +ℓp is the persistence length. In Fig. 2a, we show the results +of simulations of Eq. (1) without interparticle forces, in the +presence of a confining potential. As γ is increased up to +b +a +FIG. 2. (a) Ratio between boundary density ρs and bulk density ρb, +normalized by the bare persistence ℓ0 +p, as a function of γ. Red trian- +gles are measured in numerical simulations of Eq. (1) in the absence +of pairwise forces. (b) Boundary accumulation vs effective rotational +diffusivities Dc +r(γ) (red triangles), extracted from the autocorrelation +function. Non-interacting particles with bare rotational diffusivities +Dr = Dc +r(γ) lead to similar boundary accumulation (blue circle). +See [45] for simulation details. +γ ≃ 5Dr, the fraction of particles at the walls increases by +a factor of 2 while the system remains disordered. Our ana- +lytical computations suggest a simple qualitative explanation: +aligning interactions lead to a reduced effective rotational dif- +fusivity Dc +r. In turn, this yields an enhanced persistence length +ℓc +p = v0/Dc +r and thus an increased boundary accumulation. +To test this hypothesis, we measured the auto-correlation +function of the global orientation in the presence of align- +ing torques: CM(t) ≡ ⟨ ˆM(t) · ˆM(0)⟩, where ˆM(t) = +� +dr ˆm(r, t). The autocorrelation function is well fitted by an +exponential decay CM(t) = M2 +0 exp(−Dc +rt) from which we +extracted Dc +r(γ). As predicted, Dc +r(γ) is a decreasing func- +tion of γ. We then compared the boundary accumulation with +that observed in simulations of non-interacting particles with +rotational diffusivity Dr = Dc +r(γ). The excess densities are +identical (Fig. 2b). Remarkably, the renormalization of the +mass of the orientation field, which is a hydrodynamic ef- +fect, quantitatively accounts for the accumulation of particles +at confining boundaries, despite the microscopic nature of this +phenomenon. +Nematic torques and MIPS. Let us now show that the renor- +malization of the polar-field mass also quantitatively accounts +for the emergence of MIPS at finite γ. To do so, we first de- +rive the relaxation dynamics of the density field. For k = 0, +Eq. (2) reads ˙ρ = −∇ · J, where J(r) = v0m(r) + µI0(r), +m = ⟨ ˆm⟩, and I0(r) = ⟨ +� +dr′ ˆρ(r)F(r − r′)ˆρ(r′)⟩. The +dynamics of m then stems from that of f1 as +∂tm = −∇· +� +v0 +� +Q+ρI +2 +� ++µI1 +� +−Drm+ +�γ +ˆρ +ˆQ· ˆm−2γ +ˆρ ˆχ· ˆQ +� +(8) +where Iαβ = δαβ, I1,αβ(r) = ⟨ +� +dr′ ˆmβ(r)Fα(r − r′)ˆρ(r′)⟩, +ˆQαβ(r) = � +i(ui,αui,β − δαβ +2 )δ(r − ri) is the nematic order +field, Q = ⟨ ˆQ⟩, and ˆχαβγ is a third-order tensor [45]. In +general, closing Eq. (8) for the field m is a difficult task. In +light of our results, we predict that, in the high-temperature +disordered phase, the non-linear aligning terms can simply be +accounted for by a renormalization of the bare mass Dr of the +polar field. The non-conserving terms in Eq. (8) thus reduce + +Ps/(pbeo +1.5 - +1.0 +/Dr +2 +6Ps/(pbeo +1.5 +1.0- +D/D +1.0 +1.5 +2.0 +2.54 +b +a +FIG. 3. Onset of MIPS. (a) Measurement of the active (red line) +and passive (green line) components, Ka and KIK respectively, of +the generalized bulk modulus K (blue line) as the density ρ0 is +varied, in the absence of nematic torques. The density is normal- +ized by ρ∗, the density at which the effective self-propulsion speed +v(ρ) ≡ ⟨˙ri · u(θi)⟩ vanishes. From the measurement of m1(ρ0, γ), +Eq. (11) predicts the evolution of Kth(ρ0, γ) when γ is increased. +Five representative curves areshown in inset, with γ ranging from 0 +to 0.6. The solid line corresponds to γc = 0.52. (b): Phase dia- +gram corresponding to simulations of Eq. (1) as ρ0 and γ are varied. +Blue squares correspond to homogenous disorder systems. Red tri- +angles correspond to the MIPS region. Green circles correspond to +the emergence of nematic order. The solid green line corresponds to +the theoretical prediction γc = 0.52. See [45] for numerical details. +to −m1(ρ)m. A fast variable treatment on m then allows us +to rewrite the dynamics of ρ as +˙ρ = ∇ · +� +µDr +m1 +∇ · σa + µ∇ · σIK� +, +(9) +where we have introduced σa = v2 +0(Q+ ρI +2 )/(µDr)+I1/Dr +and followed Irving and Kirkwood to rewrite the contribution +of pairwise forces as I0 = −∇ · σIK [51, 52]. To assess +the stability of an isotropic, homogeneous phase at density +ρ0, we compute the linearized dynamics in Fourier space of a +fluctuation along, say, the ˆx axis, which reads +∂tδρq = −µq2�Dr +m1 +σa +xx +′(ρ0) + σIK +xx +′(ρ0) +� +δρq , +(10) +where the prime denotes derivative with respect to ρ0. +When γ = 0, σa is the contribution of the active forces +to the stress tensor [49, 53, 54], m1 = Dr, and Eq. (9) re- +duces to ˙ρ = ∇ · (µ∇ · σ) [54]. The mechanical pressure +exerted by active particles then satisfies an equation of state +given by P = −Trσ/2 and an isotropic homogeneous pro- +file at density ρ0 is linearly unstable whenever P ′(ρ0) = +−σa +xx +′(ρ0) − σIK +xx +′(ρ0) < 0. +This is the standard theory +for MIPS in systems of self-propelled particles interacting +via pairwise forces [25, 54–56]. The system is thus unsta- +ble when its bulk modulus is negative: K = ρ0P ′(ρ0) = +Ka(ρ0) + KIK(ρ0) < 0, where K has been split into ac- +tive and passive components Ka(ρ0) ≡ −ρ0σa +xx +′(ρ0) and +KIK(ρ0) ≡ −ρ0σIK +xx +′(ρ0). +In the presence of aligning torques, despite the lack of +equation of state [57], Ka and KIK still control the stabil- +ity of homogeneous profiles through Eq. (10). This suggests +defining a ‘generalized bulk modulus’—without connection to +mechanics—as K≡ Dr +m1 Ka + KIK. Like in equilibrium, nega- +tive values of K then lead to a spinodal decomposition. In the +disordered phase, we expect that γ barely alters the values of +Ka and KIK (see Fig. S3a-b). Their measurements at γ = 0, +shows that Ka favors instability whereas KIK stabilizes ho- +mogeneous phases (see Fig 3a). Their sum is positive and the +system is stable. As γ increases, we estimate the generalized +bulk modulus as +Kth(ρ0, γ) ≡ KIK(ρ0) + +Dr +m1(ρ0, γ)Ka(ρ0) . +(11) +The renormalization of m1 thus enhances the contribution of +Ka by a factor of Dr +m1 . The inset of Fig 3a shows Kth(ρ0, γ) +for several values of γ. For γ > γc = 0.52, the active bulk +modulus dominates and we predict the occurence of MIPS. +This is successfully compared with simulations of Eq. (1) in +the (γ, ρ0) plane in Fig. 3b. All in all, the renormalization of +m1 due to the nematic torques thus induces MIPS by increas- +ing the active contribution to the bulk modulus. +Mixture of active and passive particles. To show that our re- +sults apply more broadly, we consider mixtures of active and +passive particles interacting via purely repulsive forces, which +have attracted a lot of attention recently [58–64]. Active par- +ticles are characterized by positions ra +i and orientations u(θi) +while the positions of passive particles are denoted by rp +i . The +spatial dynamics read +˙ra +i = v0ui − µ +� +|ra +i −ra +j| r0. In fact, rvir is attributed +to a radius that includes a volume in which the average +halo density reaches 200 to 500 times the critical density +of the Universe. Under such assumptions, the total mass +enclosed by a volume of radius r is determined as follows +M(r) = 4πρ0r0 +� r +0 +rdr +(1 + r/r0)2 = 4πρ0r3 +0g(r/r0), +(6) +where g(x) = log(1+x)−x/(1+x). It needs to be men- +tioned that the contribution of dark-matter spike and +central SMBH are negligible in comparison to the total +mass of dark matter halo. Moreover, the concentration +parameter determines the central density of dark matter +halos, which is defined as C ≡ rvir/r0. Hence, the virial +mass takes the following form +Mvir = 4πρ0r3 +0g(C). +(7) +Also, the circular velocity of dark matter particles +reaches the maximum value at a distance rm = Cmr0 = +2.16 r0, and corresponds to the one-dimensional velocity +dispersion of dark matter particles, namely +σ2 = GM(Cmr0) +Cmr0 += 4πGρ0r2 +0 +g(Cm) +Cm +. +(8) +As a result, a relation between ρ0, r0, and MSMBH can +be established via Eqs. (5) and (8). Based on the results +of N-body simulations, the concentration parameter is a +decreasing function of halo mass and is a function of red- +shift at constant mass (e.g. Prada et al. (2012); Dutton +& Macci`o (2014); Ludlow et al. (2016); Okoli & Afshordi +(2016)), which is consistent with the dynamics expected +from the evolution of dark matter haloes. In this work, +to calculate the merger rate of compact binaries in the +present-time Universe, we utilize the concentration pa- +rameter presented in Ludlow et al. (2016) for spherical- +collapse dark matter halo models, and we employ the +corresponding one obtained in Okoli & Afshordi (2016) +for ellipsoidal-collapse dark matter halo models. +2.3. The mass function of SMBHs +Having sufficient knowledge about how SMBHs grow +and evolve is one of the most fundamental challenges +of extragalactic astronomy. +Accordingly, the SMBH +mass function provides comprehensive information on +the mass of SMBHs and their evolution at the center of +galactic halos. Therefore, the SMBH mass function can +be considered a powerful and available tool to investigate +the growth of SMBHs and constrain related theoretical +models. On the other hand, the SMBH mass function +might play a significant role in the structuring of up- +coming surveys because it provides an estimate of the +mass classification of SMBHs Kelly & Merloni (2012). It +should be noted that obtaining an accurate mass func- +tion for SMBHs is a relatively difficult task. For this rea- +son, the current estimates of the SMBH mass function +include many theoretical uncertainties, which in turn +may affect the accuracy of calculating the merger rate +of compact binaries in dark-matter spikes. A reasonable +approach for managing this uncertainty is to compare +the results from several different empirical SMBH mass +functions. +In Benson et al. (2007), by employing the Galactica +code, a sample of 8839 SDSS galaxies was employed to +extrapolate the luminosity functions of spheroid and disc +galaxies, and a mass function of SMBHs was obtained +as follows +φ(MSMBH) = 109 +�φ0M α +SMBH +M α+1 +∗ +� +exp +� +− +�MSMBH +M∗ +�β� +, +(9) +in which α = −0.65, β = 0.6, φ0 = 2.9 × 10−3 h3Mpc−3, +and M∗ = 4.07 × 107 h−2M⊙. + +5 +In addition, in Vika et al. (2009), a convenient mass +function for SMBHs has been obtained via the Mil- +lenium Galaxy Catalogue (Liske et al. 2003) for 1743 +galaxies. This mass function is based on the experimen- +tal relation between the mass SMBH and the luminosity +of the host spheroid, which has the following form +φ(MSMBH) = φ∗ +�MSMBH +M∗ +�α+1 +exp +� +1 − +�MSMBH +M∗ +�� +, +(10) +where log φ∗ = −3.15, log M∗/M = 8.71, and α = 1.20. +This mass function is valid for the masse range 106M⊙ < +MSMBH < 1010M⊙. +Also, in Shankar et al. (2004), another suitable mass +function was derived for SMBHs according to the obser- +vational relation between the SMBH mass and the halo +velocity dispersion and using kinematic and photometric +data, which takes the following formula +φ(MSMBH) = φ∗ +�MSMBH +M∗ +�α+1 +exp +� +1 − +�MSMBH +M∗ +�β� +, +(11) +where φ∗ = 7.7 × 10−3 Mpc−3, M∗ = 6.4 × 107 M⊙, +β = 0.49, and α = −1.11. This mass function is valid +for the mass range 106M⊙ ≤ MSMBH ≤ 5 × 109M⊙. +3. COMPACT BINARY MERGER RATE +Assume in dark-matter spike, a compact object with +mass m1 suddenly encounters another one with mass +m2 on a hyperbolic orbit, and their relative velocity at +large separation is vrel = |v1 − v2|. +Hence, based on +two-body scattering, highly significant gravitational ra- +diation emits at the periastron ra. Keplerian mechanics +states that such a system is gravitationally bound when +the emitted gravitational energy dominates the kinetic +energy of the system. Under these conditions, a maxi- +mum value for periastron can be obtained as follows +rmp = +� 85π +6 +√ +2 +G7/2m1m2(m1 + m2)3/2 +c5v2 +rel +�2/7 +. +(12) +Furthermore, in the Newtonian limit, the impact param- +eter is determined by the periastron as follows: +b2(rp) = 2G(m1 + m2)rp +v2 +rel ++ r2 +p. +(13) +For the regions of dark-matter spikes that are gravita- +tionally active, a strong limit of gravitational focusing, +i.e., rp ≪ b, can be considered in such a way that the +relevant distortions of surrounding compact objects on +the formed binaries can be ignored. +Thus, the cross- +section for the binary formation can be obtained via the +following equation +ξ(m1, m2, vrel) = πb2(rmp) ≃ 2πG(m1 + m2)rmp +v2 +rel +. (14) +Hence, by Substituting Eq. (12) into Eq. (14), the cross +section for the binary formation can be derived as +ξ ≃ 2π +� 85π +6 +√ +2 +�2/7 G2(m1 + m2)10/7(m1m2)2/7 +c10/7v18/7 +rel +. +(15) +Therefore, the merger rate of compact binaries within +each region of the dark-matter spike is determined as +follows +Nsp = 4π +� rsp +4rs +T (ρ, m1, m2)⟨σvrel⟩ r2dr, +(16) +where for the PBH-PBH events: +T = +� +1 +2 +[fPBH ρsp(r)]2 +m1m2 +� +, +(17) +and for the PBH-NS events: +T = +�fPBH ρsp(r) +m1 +� �ρNS(r) +m2 +� +. +(18) +In the above relation, 0 < fPBH ≤ 1 represents the +fraction of PBHs that specifies their contribution to dark +matter, and the angle bracket denotes an average over +the relative velocity distribution at the vicinity of the +central SMBH. Furthermore, ρNS(r) is the NS density +profile that we define through the spherically symmetric +form: +ρNS = ρ∗ +NS exp +� +− r +r∗ +NS +� +, +(19) +in which r∗ +NS and ρ∗ +NS are respectively characteristic ra- +dius and density of NSs that need to be determined. For +the characteristic radius of NSs, an approximative value +has been proposed as r∗ +NS ≃ 0.1 rs (Sasaki et al. 2022), +which we use in our calculations. +Furthermore, the characteristic density of NSs must +be obtained by normalizing the distribution of NSs to +their estimated population in an arbitrary galaxy. To +accomplish this, we utilize the time-independent form +of the initial Salpeter stellar mass function, which is in +the form χ(m∗) ≈ m−2.35 +∗ +. +Our main assumption is +based on the fact that the entire population of stars in +the mass range of 8M⊙-20M⊙ will eventually yield a +supernova explosion, and their outcome will be an NS. +Accordingly, the number of NSs in a single galaxy with +stellar mass M∗ is given by +nNS = M∗ +� mmax +∗ +mmin +∗ +χ(m∗)dm∗, +(20) + +6 +10-10 +10-8 +10-6 +10-4 +10-2 +100 +106 +107 +108 +109 +Merger Rate Per Spike [Yr-1] +MSMBH [Msun] +γ=2.0 +γ=1.4 +γ=0.8 +γ=0.2 +(a) PBH-PBH – Spherical Model +10-10 +10-8 +10-6 +10-4 +10-2 +100 +106 +107 +108 +109 +Merger Rate Per Spike [Yr-1] +MSMBH [Msun] +γ=2.0 +γ=1.4 +γ=0.8 +γ=0.2 +(b) PBH-PBH – Ellipsoidal Model +10-21 +10-20 +10-19 +10-18 +106 +107 +108 +109 +Merger Rate Per Spike [Yr-1] +MSMBH [Msun] + γ=2.0 + γ=1.4 + γ=0.8 + γ=0.2 +(c) PBH-NS – Spherical Model +10-20 +10-19 +10-18 +106 +107 +108 +109 +Merger Rate Per Spike [Yr-1] +MSMBH [Msun] + γ=2.0 + γ=1.4 + γ=0.8 + γ=0.2 +(d) PBH-NS – Ellipsoidal Model +Figure 2. The merger rate of compact binaries in a single dark-matter spike as a function of SMBH mass for different values of +γ. The top panels demonstrate this relation for PBH-PBH events in spherical- and ellipsoidal-collapse dark matter halo models, +while the bottom panels display the corresponding results for PBH-NS events. +where χ(m∗)m∗ is normalized to unity. +It should be +noted that to characterize the galactic stellar mass M∗, +the stellar mass–halo mass relation M∗(Mhalo) must be +determined. +For this purpose, the stellar mass–halo +mass relation obtained in Behroozi et al. (2013) can +be used, the basic assumption of which is the presence +of the maximum number of NSs at the center of the +galactic halo. We must note that in the present anal- +ysis, for the relative velocity near the central SMBH, +we use the circular velocity v(r) = +� +GMSMBH/r at +each radius bounded by the dark-matter spike. +This +is a reasonable choice since the total mass enclosed by +the region of dark-matter spike is negligible versus the +mass of the central SMBH. We consider the mass of in- +volving PBHs in PBH-PBH events as MPBH = 30 M⊙ +and fix the masses of PBHs and NSs participating in +PBH-NS events as MPBH = 5 M⊙ and MNS = 1.4 M⊙. +Also, in the present analysis, we consider the contribu- +tion of PBHs in dark matter to be fPBH = 1. +It is +obvious from Eqs. (16), (17) and (18) that the merger +rate of PBH-PBH binaries is straightly proportional to +the f 2 +PBH, while it changes directly with fPBH for the +PBH-NS events. +In Fig. 2, we have plotted the merger rate of compact +binaries within a single spike as a function of SMBH +mass for several values of power-law index γ while ac- +counting for dark matter halo models with spherical and +ellipsoidal collapses. +As can be seen from the figure, +the merger rate of PBH-PBH binaries changes inversely +with the mass of the SMBH for both spherical- and + +7 +10-10 +10-8 +10-6 +10-4 +10-2 +100 +102 +104 +106 +106 +107 +108 +109 +Merger Rate [Gpc-3yr-1] +MSMBH [Msun] +γ=2.0 +γ=1.4 +γ=0.8 +γ=0.2 +(a) PBH-PBH – Spherical – Shankar M.F. +10-10 +10-8 +10-6 +10-4 +10-2 +100 +102 +104 +106 +106 +107 +108 +109 +Merger Rate [Gpc-3yr-1] +MSMBH [Msun] + γ=2.0 + γ=1.4 + γ=0.8 + γ=0.2 +(b) PBH-PBH – Spherical – Vika M.F. +10-10 +10-8 +10-6 +10-4 +10-2 +100 +102 +104 +106 +106 +107 +108 +109 +Merger Rate [Gpc-3yr-1] +MSMBH [Msun] + γ=2.0 + γ=1.4 + γ=0.8 + γ=0.2 +(c) PBH-PBH – Spherical – Benson M.F. +10-10 +10-8 +10-6 +10-4 +10-2 +100 +102 +104 +106 +106 +107 +108 +109 +Merger Rate [Gpc-3yr-1] +MSMBH [Msun] +γ=2.0 +γ=1.4 +γ=0.8 +γ=0.2 +(d) PBH-PBH – Ellipsoidal – Shankar M.F. +10-10 +10-8 +10-6 +10-4 +10-2 +100 +102 +104 +106 +106 +107 +108 +109 +Merger Rate [Gpc-3yr-1] +MSMBH [Msun] + γ=2.0 + γ=1.4 + γ=0.8 + γ=0.2 +(e) PBH-PBH – Ellipsoidal – Vika M.F. +10-10 +10-8 +10-6 +10-4 +10-2 +100 +102 +104 +106 +106 +107 +108 +109 +Merger Rate [Gpc-3yr-1] +MSMBH [Msun] + γ=2.0 + γ=1.4 + γ=0.8 + γ=0.2 +(f) PBH-PBH – Ellipsoidal – Benson M.F. +Figure 3. The merger rate of PBH-PBH binaries per unit time and volume as a function of SMBH mass for different values +of γ. The top panels show this relation for spherical-collapse dark matter halo models while considering three different SMBH +mass functions, whereas the bottom panels exhibit the corresponding results for ellipsoidal-collapse dark matter halo models. +ellipsoidal-collapse dark matter halo models. It is also +evident that the merger rate of PBH-PBH binaries is di- +rectly proportional to the value of the power-law index +γ. However, comparing PBH-PBH events for spherical- +and ellipsoidal-collapse dark matter halo models, it can +be inferred that the merger rate per spike for ellipsoidal- +collapse dark matter halo models is higher than the +corresponding one derived from spherical-collapse dark +matter halo models. +However, for the merger rate of PBH-NS binaries, the +situation is slightly different. Interestingly, unlike the +previous case, the merger rate of PBH-NS binaries per +spike reaches the maximum value for the SMBH mass +of MSMBH ≃ 107M⊙ for both spherical- and ellipsoidal- +collapse dark matter halo models. Also, the merger rate +of PBH-NS binaries per spike for a power-law index of +γ = 1.7 has a maximum value that is completely dif- +ferent from the relevant results of PBH-PBH events. +In other words, the merger rate of PBH-NS binaries +increases monotonically up to γ = 1.7 and decreases +for the larger values. +In addition, it can be inferred +from the results that the merger rate of PBH-NS bina- +ries for ellipsoidal-collapse dark matter halo models is +much higher than the corresponding one derived from +spherical-collapse dark matter halo models. +On the other hand, the cumulative merger rate of +compact binaries is considered the main quantity to be +recorded and processed through the LIGO-Virgo detec- +tors. Therefore, the overall merger rate of compact bi- +naries per unit volume and per unit time needs to be +specified. To perform this task, one has to convolve the +mas function of SMBH, φ(MSMBH), with the merger rate +of compact binaries per spike, Nsp(MSMBH): +R = +� Mmax +Mmin +Nsp(MSMBH)φ(MSMBH)dMSMBH. +(21) +According to Eqs. (9), (10) and (11), the mentioned +mass functions have a decreasing exponential term with +respect to the mass of SMBHs. Therefore, it can be con- +cluded that Mmax does not have a significant effect on +the final result. In contrast, the introduced mass func- +tions indicate that the maximum abundance belongs to +the smallest central black holes in the Universe. Conse- +quently, Mmin can have a significant contribution to the +merger rate of compact binaries in dark-matter spikes. + +8 +10-18 +10-17 +10-16 +10-15 +10-14 +10-13 +10-12 +10-11 +106 +107 +108 +109 +Merger Rate [Gpc-3yr-1] +MSMBH [Msun] + γ=2.0 + γ=1.4 + γ=0.8 + γ=0.2 +(a) PBH-NS – Spherical – Shankar M.F. +10-18 +10-17 +10-16 +10-15 +10-14 +10-13 +10-12 +106 +107 +108 +109 +Merger Rate [Gpc-3yr-1] +MSMBH [Msun] + γ=2.0 + γ=1.4 + γ=0.8 + γ=0.2 +(b) PBH-NS – Spherical – Vika M.F. +10-19 +10-18 +10-17 +10-16 +10-15 +10-14 +10-13 +106 +107 +108 +109 +Merger Rate [Gpc-3yr-1] +MSMBH [Msun] + γ=2.0 + γ=1.4 + γ=0.8 + γ=0.2 +(c) PBH-NS – Spherical – Benson M.F. +10-18 +10-17 +10-16 +10-15 +10-14 +10-13 +10-12 +10-11 +106 +107 +108 +109 +Merger Rate [Gpc-3yr-1] +MSMBH [Msun] + γ=2.0 + γ=1.4 + γ=0.8 + γ=0.2 +(d) PBH-NS – Ellipsoidal – Shankar M.F. +10-18 +10-17 +10-16 +10-15 +10-14 +10-13 +10-12 +107 +108 +109 +Merger Rate [Gpc-3yr-1] +MSMBH [Msun] + γ=2.0 + γ=1.4 + γ=0.8 + γ=0.2 +(e) PBH-NS – Ellipsoidal – Vika M.F. +10-19 +10-18 +10-17 +10-16 +10-15 +10-14 +10-13 +106 +107 +108 +109 +Merger Rate [Gpc-3yr-1] +MSMBH [Msun] + γ=2.0 + γ=1.4 + γ=0.8 + γ=0.2 +(f) PBH-NS – Ellipsoidal – Benson M.F. +Figure 4. The merger rate of PBH-NS binaries per unit time and volume as a function of SMBH mass for different values of γ. +The top panels indicate this relation for spherical-collapse dark matter halo models while accounting for three different SMBH +mass functions, whereas the bottom panels show the corresponding results for ellipsoidal-collapse dark matter halo models. +In Fig. 3, we have depicted the merger rate of PBH- +PBH binaries per unit time and volume as a function +of SMBH mass for several values of power-law index γ +while considering dark matter halo models with spher- +ical and ellipsoidal collapses. +We have provided the +results for three mass functions Shankar et al. (2004), +Benson et al. (2007), and Vika et al. (2009) to jus- +tify the possible uncertainties in the present analysis as +much as possible. +The merger rate of PBH-PBH bi- +naries in both dark matter halo models with spherical +and ellipsoidal collapses decreases monotonically with +increasing the mass of SMBHs. Given the classification +of the mass functions of SMBHs for their abundance, +this result seems reasonable. +As it is clear from the +figures, the merger rate of PBH-PBH binaries for all +three mass functions and both dark matter halo models +reaches the maximum value as MSMBH = 106M⊙. Also, +the direct proportionality of the merger rate of PBH- +PBH binaries to the values of the power-law index is ev- +ident. Moreover, the merger rate of PBH-PBH binaries +in ellipsoidal-collapse dark matter halo models is slightly +higher than that obtained from spherical-collapse dark +matter halo models. In the best case, which can be re- +alized at the minimum value of the power-law index, +e.g. γ = 0.05, the amplification of the overall merger +rate is 62%. +Additionally, it can be concluded that +the merger rate of PBH-PBH binaries yields the high- +est, middle, and lowest values while considering Shankar +et al. (2004), Vika et al. (2009), and Benson et al. (2007) +mass functions respectively. +As previously discussed the merger rate per spike, +Fig. 4 exhibits that the overall merger rate of PBH-NS +binaries is also obtained differently from that of PBH- +PBH binaries. +As a common point of whole applied +dark matter models and mass functions, it is deduced +that the merger rate of PBH-NS binaries has a plateau +at γ = 1.7 and experiences a decreasing behavior for +values greater than that. +It is also evident that the +merger rate of PBH-NS binaries for ellipsoidal-collapse +dark matter halo models is in turn higher than that ob- +tained for spherical-collapse dark matter halo models. +Such a relative advantage is because by using more real- +istic dark matter halo models, one can hope to improve +the theoretical predictions of the merger rate of compact +matter in galactic halos. Interestingly, for PBH-NS bi- +naries, the inverse proportionality of the overall merger + +9 +10-2 +10-1 +100 +101 +102 +103 +104 +105 +106 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 1.6 + 1.8 + 2 +Total Merger Rate [Gpc-3yr-1] +γ +Ellipsoidal Model +Spherical Model +(a) PBH-PBH – Shankar M.F. +10-2 +10-1 +100 +101 +102 +103 +104 +105 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 1.6 + 1.8 + 2 +Total Merger Rate [Gpc-3yr-1] +γ +Ellipsoidal Model +Spherical Model +(b) PBH-PBH – Vika M.F. +10-3 +10-2 +10-1 +100 +101 +102 +103 +104 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 1.6 + 1.8 + 2 +Total Merger Rate [Gpc-3yr-1] +γ +Ellipsoidal Model +Spherical Model +(c) PBH-PBH – Benson M.F. +10-12 +10-11 +10-10 +10-9 + 0.2 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 1.6 1.8 + 2 +Total Merger Rate [Gpc-3yr-1] +γ +Ellipsoidal Model +Spherical Model +(d) PBH-NS – Shankar M.F. +10-13 +10-12 +10-11 +10-10 + 0.2 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 1.6 1.8 + 2 +Total Merger Rate [Gpc-3yr-1] +γ +Ellipsoidal Model +Spherical Model +(e) PBH-NS – Vika M.F. +10-14 +10-13 +10-12 +10-11 + 0.2 0.4 + 0.6 + 0.8 + 1 + 1.2 + 1.4 + 1.6 1.8 + 2 +Total Merger Rate [Gpc-3yr-1] +γ +Ellipsoidal Model +Spherical Model +(f) PBH-NS – Benson M.F. +Figure 5. The overall merger rate of compact binaries as a function of power-law index γ for both spherical- and ellipsoidal- +collapse dark matter halo models. The top panels demonstrate this relation for PBH-PBH events for three different SMBH +mass functions, while the bottom panels show the corresponding results for PBH-NS events. The shaded cyan bands represent +the BH-BH mergers estimated by the LIGO-Virgo detectors during the latest observing run, i.e., (17.9-44) Gpc−3Yr−1. +rate with the mass of SMBHs is not monotonic and +has a plateau in some cases. If spherical-collapse dark +matter halo models are reliable, the results obtained for +Shankar et al. (2004) and Vika et al. (2009) mass func- +tions for γ < 1.7 decrease monotonically with the mass +of PBHs, while for γ > 1.7 it has a plateau around +MSMBH = 107M⊙. However, the result obtained from +the Benson et al. (2007) mass function has a plateau +around the same SMBH mass for all values of γ. On the +other hand, if the ellipsoidal-collapse dark matter halo +models are plausible, the existence of the plateau is evi- +dent for all the mass functions used and for all values of +the power-law index. These results suggest that dark- +matter spikes structured around central SMBHs with +mass MSMBH = 107M⊙ may contain a non-standard +abundance of NSs that disturbs the monotonic depen- +dence of the merger rate on the mass SMBHs. +In Fig. 5, to quantitatively compare the results ob- +tained from ellipsoidal-collapse dark matter halo models +with those obtained from spherical-collapse dark matter +halo models, we have displayed the merger rate of com- +pact binaries in terms of power-law index γ while taking +into account Shankar et al. (2004), Vika et al. (2009), +and Benson et al. (2007) mass functions. To assess the +theoretical predictions via the experimental data, we +have also included the relevant mergers estimated by +the GW detectors in the case of PBH-PBH events, while +such an action has not been taken in the case of PBH- +NS events. This is because the prediction of the present +analysis from the merger rate of PBH-PBH binaries is +capable of justifying the data recorded by GW detec- +tors, i.e., (17.9-44) Gpc−3Yr−1 (Abbott et al. 2021b), +whereas the merger rate of PBH-NS binaries cannot in +any way perform this task. +Specifically, the estimate +of the LIGO-Virgo detectors from the total merger rate +of BH-NS binaries is presented as (7.8-140) Gpc−3Yr−1 +(Abbott et al. 2021d), while the relevant prediction of +both dark matter halo models in the current analysis is +extremely far from this range. +As it is evident from the figures related to PBH- +PBH events, and of course as mentioned earlier, the +merger rate of PBH-PBH binaries while accounting for +ellipsoidal-collapse dark matter halo models is higher +than that extracted from spherical-collapse dark mat- + +10 +ter halo models. In addition, such enhancement in the +merger rate is greater at lower power-law index values +than that at higher ones. In this regard, the merger rate +of PBH-PBH binaries in spherical-collapse dark mat- +ter halo models while considering Shankar et al. (2004), +Vika et al. (2009), and Benson et al. (2007) mass func- +tions will be consistent with the BH-BH mergers esti- +mated by GW detectors if the value of power-law index +lies in the interval γ = (1.05-1.15), γ = (1.10-1.20), +and γ = (1.40-1.50), respectively. However, the corre- +sponding results obtained from ellipsoidal-collapse dark +matter halo models can potentially modify these values +as γ = (0.95-1.05), γ = (1.0-1.20), and γ = (1.30-1.40), +respectively. +On the other hand, the results of PBH-NS events in- +dicate that despite the mismatch of the outcome of the +present analysis with GW data, the effect of ellipsoidal- +collapse dark matter halo models in the amplification +of the merger rate of such binaries is significant. It is +also obvious that the merger rate of PBH-NS binaries +in both dark matter halo models increases monotonical +with the power-law index reaches a maximum at γ = 1.7, +and decreases for higher values of γ. +4. CONCLUSIONS +In this work, we have calculated the merger rate of +compact binaries in dark-matter spikes, which are ex- +pected to be structured around SMBHs at the center +of galactic halos. +For this purpose, we have initially +described theoretical models, which suit dark matter +spikes. +We have also discussed crucial quantities for +dark-matter spikes, such as the density profile, concen- +tration parameter, and MSMBH-σ relation, which can +specify the distribution of dark matter particles in the +region of spikes. On the other hand, the strong corre- +lation between the growth of central SMBHs and halo +parameters suggests that another quantity called the +mass function of SMBHs also plays a prominent role +in the present analysis. +However, the insufficiency of +our knowledge of the exact distribution of dark mat- +ter particles in the central regions of galactic halos and +the abundance of SMBHs in the Universe may lead to +uncertainties in the results. Hence, to manage this un- +certainty, we consider three empirical SMBH mass func- +tions to compare their results. +In the following, relying on the PBH scenario and with +the assumption that PBHs are capable of contributing to +the structure of dark matter, we have discussed the con- +ditions of encountering compact objects such as PBHs +and NSs in dark-matter spikes. Accordingly, we have +calculated the merger rate of compact binaries in a single +dark-matter spike for spherical- and ellipsoidal-collapse +dark matter halo models. Our results confirm that the +merger rate of PBH-PBH binaries within each spike for +ellipsoidal-collapse dark matter halo models is slightly +higher than that derived from spherical-collapse dark +matter halo models. It is concluded that the merger rate +of PBH-PBH binaries changes directly with the value +of the power-law index γ. +The results obtained from +the analysis of PBH-NS events show that the maximum +value of the merger rate in each spike takes place around +the central SMBH with a mass MSMBH = 107 M⊙. +Moreover, it turned out that the merger rate of PBH- +NS binaries per spike has a maximum value at γ = 1.7. +Also, the results indicate that the merger rate of PBH- +NS binaries per spike for ellipsoidal-collapse dark mat- +ter halo models is much higher than that derived from +spherical-collapse dark matter halo models. +As the main measurable factor in GW detectors, we +have calculated the total merger rate of compact bina- +ries in dark-matter spikes around central SMBHs with +masses of MSMBH = (106-109)M⊙. As mentioned ear- +lier, to account for possible uncertainties in our analysis, +we have used three different mass functions for calculat- +ing the total merger rate of compact binaries. Our find- +ings indicate that the merger rate of PBH-PBH binaries +in spherical- and ellipsoidal-collapse dark matter halo +models has the maximum value at MSMBH = 106M⊙ +and decreases monotonically with increasing the mass +of SMBHs. +Also, the results exhibit that the overall +merger rate of PBH-PBH binaries for ellipsoidal-collapse +dark matter halo models is slightly higher than that ob- +tained from spherical-collapse dark matter halo mod- +els, in such a way that in the best case, e.g. at about +γ = 0.05, the amplification of the overall merger rate +is about 62%. +Moreover, among the considered mass +functions, the highest, middle, and lowest values of the +merger rate of PBH-PBH binaries have been extracted +from Shankar et al. (2004), Vika et al. (2009), and Ben- +son et al. (2007) mass functions, respectively. However, +the results of the merger rate of PBH-NS binaries were +obtained slightly different. Interestingly, the inverse pro- +portionality of the overall merger rate PBH-NS bina- +ries with the mass of SMBHs is not monotonic and has +a plateau at MSMBH = 107M⊙ in almost all consid- +ered models. This may be due to a non-standard abun- +dance of NSs in dark-matter spikes clustered around the +central SMBH with of mass MSMBH = 107M⊙, which +should be validated with informative observational data. +Finally, we have calculated the merger rate of com- +pact binaries according to the power-law index γ for +ellipsoidal-collapse dark matter halo models and com- +pared them with the corresponding results of spherical- +collapse dark matter halo models. To compare our find- + +11 +ings with the experimental data of GWs, we have also +included the LIGO-Virgo sensitivity band for the merger +rate of BH-BH binaries. This task was not possible to do +for PBH-NS events, as our results were far from those +estimated via the LIGO-Virgo detectors. +Our results +show that the inclusion of ellipsoidal-collapse dark mat- +ter halo models in the calculations of the merger rate of +PBH-PBH binaries can reduce the range of the power- +law index obtained from spherical-collapse dark matter +halo models by 0.1. This result comes from the fact that +the dark-matter spikes in ellipsoidal-collapse halo mod- +els are denser (and naturally smaller in radius) than +those in spherical-collapse halo models. +Additionally, +the results show that, unlike PBH-PBH events, the ef- +fect of ellipsoidal-collapse dark matter halo models in +the amplification of the merger rate of PBH-NS binaries +is significant. A maximum at γ = 1.7 is also confirmed +for such events. +REFERENCES +Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016a, +Phys. Rev. Lett., 116, 061102, +doi: 10.1103/PhysRevLett.116.061102 +—. 2016b, Phys. Rev. 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D. 1967, Soviet Astron. +AJ., 10, 602 + diff --git a/udE0T4oBgHgl3EQfbgBl/content/tmp_files/load_file.txt b/udE0T4oBgHgl3EQfbgBl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..af1601cefbd28a4d4b45f9d403bf0bcbca04afad --- /dev/null +++ b/udE0T4oBgHgl3EQfbgBl/content/tmp_files/load_file.txt @@ -0,0 +1,1507 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf,len=1506 +page_content='Draft version January 9, 2023 Typeset using LATEX twocolumn style in AASTeX631 Compact Binary Merger Rate in Dark-Matter Spikes Saeed Fakhry,1, 2 Zahra Salehnia,3, 2 Azin Shirmohammadi,3, 2 Mina Ghodsi Yengejeh,1, 2 and Javad T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Firouzjaee3, 4, 2 1Department of Physics, Shahid Beheshti University, Evin, Tehran 19839, Iran 2PDAT Laboratory, Department of Physics, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Toosi University of Technology, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Box 15875-4416, Tehran, Iran 3Department of Physics, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Toosi University of Technology, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Box 15875-4416, Tehran, Iran 4School of Physics, Institute for Research in Fundamental Sciences (IPM), P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Box 19395-5531, Tehran, Iran ABSTRACT Nowadays, the existence of supermassive black holes (SMBHs) in the center of galactic halos is almost confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It is expected that in case of adiabatic growth of SMBHs in the center of galactic halos, one can expect to form extremely dense regions known as dark-matter spikes around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In this work, we calculate the merger rate of compact binaries in dark-matter spikes while considering halo models with spherical and ellipsoidal collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Our findings exhibit that ellipsoidal-collapse dark matter halo models can potentially yield the enhancement of the merger rate of compact binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Finally, our results confirm that the merger rate of primordial black hole binaries is consistent with the results estimated by the LIGO-Virgo detectors, while such results can not be realized for primordial black hole-neutron star binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Keywords: Dark-Matter Spike — Primordial Black Hole — Neutron Star — Ellipsoidal Collapse 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' INTRODUCTION Over the last few decades, gravitational waves (GWs) have been studied as an interesting cosmological observ- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Regarding this, a large part of astrophysical and cosmological phenomena have been evaluated through the study of GWs and their direct detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Mean- while, compact binary mergers are always considered potential sources for the propagation of GWs (Mandel & Broekgaarden 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' During recent years, dozens of compact binary mergers have been recorded by the LIGO-Virgo detectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2016a,b,c, 2020a,b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In this regard, three main classes of com- pact binaries as binary black holes (BBHs), black hole- neutron star (BH-NS) binaries, and binary neutron stars (BNSs) can potentially be captured by the LIGO- Virgo detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Meanwhile, most of the recorded GW s fakhry@sbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='ir zahra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='salehnia@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='kntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='ir azinshr@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='kntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='ir m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='ghodsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='y@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='com firouzjaee@kntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='ir events belong to BBH merger events in the mass range 10 M⊙- 100 M⊙ (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2019, 2021a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' There have been several investigations into the origin of such BHs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', Raidal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Bouffanais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Nitz & Wang (2021a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Mandel & Farmer (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' However, our information on this matter is still insuf- ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' There is a possibility that they have formed during stellar collapse (likely via different channels) or that they have formed at the beginning of the Universe as a result of gravitational collapse of density fluctua- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Surprisingly, most of the BBH merger events reported by the LIGO-Virgo collaboration are consistent with the primordial black hole (PBH) scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In the cosmologi- cal perturbation theory, PBHs are predicted to form due to nonlinear peaks in initial density fluctuations during horizon reentry (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', Zel’dovich & Novikov (1967);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Hawking (1971);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Carr & Hawking (1974)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Several stud- ies show that the existence of a critical state in primor- dial density fluctuations seems necessary for the forma- tion of PBHs, which is achieved through exceeding a threshold value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', Shibata & Sasaki (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Polnarev & Musco (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Musco & Miller (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Young et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Bloomfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Allahyari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' According to this argument, exceeding a certain thresh- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='02349v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='CO] 6 Jan 2023 2 old value is equivalent to the direct collapse of primor- dial density fluctuations and consequently the formation of PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In addition, PBHs are characterized by their broad range of masses, which makes them distinct from astrophysical black holes (Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Mean- while, the standard model of cosmology tries to explain the nature of two basic components of the dark sector of the Universe: dark matter and dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In this regard, dark energy is considered a type of energy that governs the accelerating expansion of the late-time Uni- verse, which corresponds to the cosmological constant in general relativity (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', Sherwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Yang & Xu (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Huterer & Shafer (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Di Valentino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Ghodsi Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Ghodsi Yengejeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2023)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Dark matter is also attributed to an in- visible matter that is thought to make up approximately 85% of the matter in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Nowadays, it is be- lieved that PBHs can be regarded as one of the poten- tial candidates for dark matter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', Bird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Kashlinsky (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Blinnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Clesse & Garc´ıa-Bellido (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Bellomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Villanueva-Domingo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Carr & Kuh- nel (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' However, other candidates for dark mat- ter are being discussed seriously (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', Spergel & Stein- hardt (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Alcock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Liebling & Palenzuela (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Roszkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Braine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Di Giovanni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2020, 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Efficient observational methods have been used to constrain the abundance of PBHs in different mass ranges, which in turn are com- pelling contexts for studying the early Universe at small scales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Lehmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Carr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' However, it is possible to obtain strong constraints on the contribution of PBHs to dark matter by calculating their merger rate and validating the results via GW data (Bird et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Clesse & Garc´ıa-Bellido 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Furthermore, the random distribution of BHs in the Universe allows them to form binaries with NSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Inter- estingly, BH-NS mergers can provide important infor- mation about multi-messenger astronomy (Ruiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In addition to the emitting of GWs, they can propagate electromagnetic signals during the merger process (Barbieri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In such events, residual matter from the NS is usually accreted by the BH and results in a luminous event (Fragione 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Moreover, recording and processing data from BH-NS binary merg- ers by GW detectors can include individual information about the NS nuclear equation of state and accretion processes of BHs, as well as constraining their spin and abundance (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', Hinderer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Zappa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Fragione et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In recent years, two merger events have been cap- tured by the LIGO-Virgo detectors, which are attributed to BH-NS binaries and their component masses are (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='9+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='5M⊙, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2M⊙) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='1M⊙, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='3M⊙), respectively Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2021c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' There are many un- certainties surrounding the formation of BH-NS binaries and their merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Nevertheless, describing this class of merger events using the evolution of the field bina- ries can be a viable approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It is expected that a significant number of BH-NS merger events can be de- tected by GW detectors in the upcoming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Due to these circumstances, it seems imperative to fully un- derstand how compact objects involved in these events are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2022), taking into account spherical-collapse dark matter halo models and in the framework of the PBH scenario, the merger rate of BH- NS binaries has been calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Also, by comparing the results with the GW observations it has been claimed that the BH components participating in such events can not have primordial origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' However, by considering more realistic halo models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', those with ellipsoidal- collapse) a more accurate picture of galactic halos can be obtained (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', Fakhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2021, 2022a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Fakhry & Del Popolo (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' With this argument, in Fakhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2022c), we have indicated that by considering ellipsoidal-collapse dark matter halo models and vali- dating the results with relevant events estimated by the LIGO-Virgo detectors, the BHs contributing to the BH- NS events are most likely PBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' On the other hand, there is convincing evidence of the presence of supermassive black holes (SMBHs) in the center of galactic halos (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', Volonteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Prokhorenko & Sazonov (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Shapiro & Heg- gie (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Diana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' This can be deduced from the Keplerian behavior of the velocity dispersion of the stars in the inner regions of galactic halos (Ghez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It is believed that the central SMBHe is capable of amplifying the density of surrounding dark matter particles to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In this regard, (Gondolo & Silk 1999) has proposed that if the central SMBH evolves adiabatically and initially has a power- law cusp, a dense region called the dark-matter spike will form around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Given that the dark matter den- sity is very high in the spike regions, it is expected that the number density of PBHs is also significant in such regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It is noteworthy that the structure of dark- matter spikes is determined by the growth of the cen- tral SMBH and dark matter halo model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In Nishikawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2019), the merger rate of PBH-PBH binaries has been calculated in dark-matter spikes while accounting for spherical-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' How- ever, as mentioned earlier, more realistic halo models 3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', those with ellipsoidal collapse) can potentially af- fect the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In this work, we propose to employ the ellipsoidal- collapse dark matter halo models to calculate the merger rate of compact binaries within dark-matter spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In this regard, the outline of this work is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2, we introduce a convenient model for dark-matter spikes and discuss some crucial quantities such as spike density profile, SMBH mass function, and concentra- tion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Next, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 3, we calculate the merger rate of compact binaries in the framework of ellipsoidal- collapse dark matter halo models and compare it with that obtained from spherical-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' We also compare the relevant results of the present analysis with the corresponding data estimated by the LIGO-Virgo detectors and constrain the value of the power-law index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Finally, we scrutinize the results and summarize the findings in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' MODEL OF DARK-MATTER SPIKE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The density profile According to the standard model of cosmology, dark matter halos are known as nonlinear structures that have been formed hierarchically and distributed in the Universe as a result of the collapse of linear cosmological fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Based on indirect observations from the rotation curve of galaxies, it can be argued that dark matter particles should not be uniformly distributed in galactic halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In this regard, special attention should be paid to the supermassive black holes (SMBHs) struc- tured at the galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Although ordinary black holes might be born from a stellar collapse, SMBHs are difficult to accommodate in standard astrophysical sce- narios at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It is suggested that SMBH mass can be related to the mass of the dark matter halo, implying that SMBHs could coevolve with their host halos (Volonteri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Bansal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Ac- cordingly, self-interacting dark matter halo models pre- dict early seeds for supermassive black holes through the gravothermal catastrophe (Choquette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Consequently, the SMBH is expected to be surrounded by a highly dense spike of dark matter at the center of the galactic halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Assume MSMBH is the mass of SMBH at the galactic center, which takes a density profile of the form ρ(r) ≃ ρ0(r0/r)γ, where ρ0 and r0 are characteristic parameters of halo, and γ represents the power-law index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In light of this reasoning, one can expect to form a dark-matter spike whose radius is specified by the following relation (Gondolo & Silk 1999) rsp = aγr0 �MSMBH ρ0r3 0 �1/(3−γ) , (1) where aγ is determined through numerical suggestions for each power-law index γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Accordingly, the density profile of the dark-matter spike for r in the range of 4rs < r < rsp can be ob- tained as follows (Gondolo & Silk 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Nishikawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2019) ρsp(r) = ρ0 � r0 rsp �γ � 1 − 4rs r �3 �rsp r �γsp , (2) where γsp = (9 − 2γ)/(4 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Furthermore, rs indicates the Schwarzchild radius of SMBH which is in the form of rs = 2GMSMBH c2 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='95 km �MSMBH M⊙ � , (3) in which G is the gravitational constant and c is the velocity of light in a vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The results of numeri- cal simulations and analytical approaches exhibit that the density profile in small radii has a power-law form (Merritt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Stadel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' However, there are different predictions about the value of the power-law index γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In this work, we consider a range for the power-law index as 0 < γ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' On the other hand, to describe dark matter distribution in galactic halos, a convenient density profile was pre- sented by Navarro, Frenk, and White (NFW) (Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 1996), which is specified by the following formula ρNFW(r) = ρ0 (r/r0) (1 + r/r0)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (4) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 1, we have depicted the difference in the be- havior of the density profile corresponding to the dark- matter spike with the NFW density profile while consid- ering the mass of SMBH as MSMBH = 106M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' As can be seen from the figure, the density profile in the regions specific to the dark-matter spike is extremely higher than the NFW density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Therefore, it seems inter- esting to calculate the merger rate of compact binaries in dark-matter spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In addition, it should be noted that r = rsp defines the radius within which the merger rate of compact objects must be calculated as dark-matter spike profiles cross the NFW density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The MSMBH-σ relation Nowadays, there are several lines of evidence indicat- ing that the growth of SMBHs and the evolution of their host halos are intricately linked (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', McLure & Dun- lop (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Tremaine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' H¨aring & Rix (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Graham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2007)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Regard- ing this, it is proposed that the mass of SMBH can be strongly correlated with the velocity dispersion of dark matter particles in a galactic halo, σ, which is called the 4 γ = 2 γ = 1 NFW 10-8 10-5 10-2 101 104 1013 1016 1019 1022 1025 1028 1031 1034 1037 1040 r [pc] ρ [M☉ Mpc-3] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Comparison of the NFW density profile with the profile of dark-matter spike with γ = 1 and 2 structured around the SMBH with a mass MSMBH = 106M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' MSMBH-σ relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In other words, the halo characteris- tic parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' ρ0 and r0, can be related to MSMBH by employing such a relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' A convenient form of the MSMBH-σ relation has been obtained as (Ferrarese & Merritt 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2000) log �MSMBH M⊙ � = a + b log � σ 200 kms−1 � , (5) where a and b are determined via empirical situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In G¨ultekin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2009), values a = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='08 and b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='41 have been suggested, which fits reasonably well with the various types of galactic halos (Kormendy & Ho 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The NFW profile is assumed to manage the density profile outside of spike regions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' r ≫ rsp, up to the virial radius rvir > r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In fact, rvir is attributed to a radius that includes a volume in which the average halo density reaches 200 to 500 times the critical density of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Under such assumptions, the total mass enclosed by a volume of radius r is determined as follows M(r) = 4πρ0r0 � r 0 rdr (1 + r/r0)2 = 4πρ0r3 0g(r/r0), (6) where g(x) = log(1+x)−x/(1+x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It needs to be men- tioned that the contribution of dark-matter spike and central SMBH are negligible in comparison to the total mass of dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Moreover, the concentration parameter determines the central density of dark matter halos, which is defined as C ≡ rvir/r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Hence, the virial mass takes the following form Mvir = 4πρ0r3 0g(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (7) Also, the circular velocity of dark matter particles reaches the maximum value at a distance rm = Cmr0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='16 r0, and corresponds to the one-dimensional velocity dispersion of dark matter particles, namely σ2 = GM(Cmr0) Cmr0 = 4πGρ0r2 0 g(Cm) Cm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (8) As a result, a relation between ρ0, r0, and MSMBH can be established via Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (5) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Based on the results of N-body simulations, the concentration parameter is a decreasing function of halo mass and is a function of red- shift at constant mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Prada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Dutton & Macci`o (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Ludlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Okoli & Afshordi (2016)), which is consistent with the dynamics expected from the evolution of dark matter haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In this work, to calculate the merger rate of compact binaries in the present-time Universe, we utilize the concentration pa- rameter presented in Ludlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2016) for spherical- collapse dark matter halo models, and we employ the corresponding one obtained in Okoli & Afshordi (2016) for ellipsoidal-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The mass function of SMBHs Having sufficient knowledge about how SMBHs grow and evolve is one of the most fundamental challenges of extragalactic astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Accordingly, the SMBH mass function provides comprehensive information on the mass of SMBHs and their evolution at the center of galactic halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Therefore, the SMBH mass function can be considered a powerful and available tool to investigate the growth of SMBHs and constrain related theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' On the other hand, the SMBH mass function might play a significant role in the structuring of up- coming surveys because it provides an estimate of the mass classification of SMBHs Kelly & Merloni (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It should be noted that obtaining an accurate mass func- tion for SMBHs is a relatively difficult task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' For this rea- son, the current estimates of the SMBH mass function include many theoretical uncertainties, which in turn may affect the accuracy of calculating the merger rate of compact binaries in dark-matter spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' A reasonable approach for managing this uncertainty is to compare the results from several different empirical SMBH mass functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In Benson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2007), by employing the Galactica code, a sample of 8839 SDSS galaxies was employed to extrapolate the luminosity functions of spheroid and disc galaxies, and a mass function of SMBHs was obtained as follows φ(MSMBH) = 109 �φ0M α SMBH M α+1 ∗ � exp � − �MSMBH M∗ �β� , (9) in which α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='65, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='6, φ0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='9 × 10−3 h3Mpc−3, and M∗ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='07 × 107 h−2M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 5 In addition, in Vika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2009), a convenient mass function for SMBHs has been obtained via the Mil- lenium Galaxy Catalogue (Liske et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2003) for 1743 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' This mass function is based on the experimen- tal relation between the mass SMBH and the luminosity of the host spheroid, which has the following form φ(MSMBH) = φ∗ �MSMBH M∗ �α+1 exp � 1 − �MSMBH M∗ �� , (10) where log φ∗ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='15, log M∗/M = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='71, and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' This mass function is valid for the masse range 106M⊙ < MSMBH < 1010M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Also, in Shankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2004), another suitable mass function was derived for SMBHs according to the obser- vational relation between the SMBH mass and the halo velocity dispersion and using kinematic and photometric data, which takes the following formula φ(MSMBH) = φ∗ �MSMBH M∗ �α+1 exp � 1 − �MSMBH M∗ �β� , (11) where φ∗ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='7 × 10−3 Mpc−3, M∗ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 × 107 M⊙, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='49, and α = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' This mass function is valid for the mass range 106M⊙ ≤ MSMBH ≤ 5 × 109M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' COMPACT BINARY MERGER RATE Assume in dark-matter spike, a compact object with mass m1 suddenly encounters another one with mass m2 on a hyperbolic orbit, and their relative velocity at large separation is vrel = |v1 − v2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Hence, based on two-body scattering, highly significant gravitational ra- diation emits at the periastron ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Keplerian mechanics states that such a system is gravitationally bound when the emitted gravitational energy dominates the kinetic energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Under these conditions, a maxi- mum value for periastron can be obtained as follows rmp = � 85π 6 √ 2 G7/2m1m2(m1 + m2)3/2 c5v2 rel �2/7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (12) Furthermore, in the Newtonian limit, the impact param- eter is determined by the periastron as follows: b2(rp) = 2G(m1 + m2)rp v2 rel + r2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (13) For the regions of dark-matter spikes that are gravita- tionally active, a strong limit of gravitational focusing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', rp ≪ b, can be considered in such a way that the relevant distortions of surrounding compact objects on the formed binaries can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Thus, the cross- section for the binary formation can be obtained via the following equation ξ(m1, m2, vrel) = πb2(rmp) ≃ 2πG(m1 + m2)rmp v2 rel .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (14) Hence, by Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (12) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (14), the cross section for the binary formation can be derived as ξ ≃ 2π � 85π 6 √ 2 �2/7 G2(m1 + m2)10/7(m1m2)2/7 c10/7v18/7 rel .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (15) Therefore, the merger rate of compact binaries within each region of the dark-matter spike is determined as follows Nsp = 4π � rsp 4rs T (ρ, m1, m2)⟨σvrel⟩ r2dr, (16) where for the PBH-PBH events: T = � 1 2 [fPBH ρsp(r)]2 m1m2 � , (17) and for the PBH-NS events: T = �fPBH ρsp(r) m1 � �ρNS(r) m2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (18) In the above relation, 0 < fPBH ≤ 1 represents the fraction of PBHs that specifies their contribution to dark matter, and the angle bracket denotes an average over the relative velocity distribution at the vicinity of the central SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Furthermore, ρNS(r) is the NS density profile that we define through the spherically symmetric form: ρNS = ρ∗ NS exp � − r r∗ NS � , (19) in which r∗ NS and ρ∗ NS are respectively characteristic ra- dius and density of NSs that need to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' For the characteristic radius of NSs, an approximative value has been proposed as r∗ NS ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='1 rs (Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2022), which we use in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Furthermore, the characteristic density of NSs must be obtained by normalizing the distribution of NSs to their estimated population in an arbitrary galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' To accomplish this, we utilize the time-independent form of the initial Salpeter stellar mass function, which is in the form χ(m∗) ≈ m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='35 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Our main assumption is based on the fact that the entire population of stars in the mass range of 8M⊙-20M⊙ will eventually yield a supernova explosion, and their outcome will be an NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Accordingly, the number of NSs in a single galaxy with stellar mass M∗ is given by nNS = M∗ � mmax ∗ mmin ∗ χ(m∗)dm∗, (20) 6 10-10 10-8 10-6 10-4 10-2 100 106 107 108 109 Merger Rate Per Spike [Yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (a) PBH-PBH – Spherical Model 10-10 10-8 10-6 10-4 10-2 100 106 107 108 109 Merger Rate Per Spike [Yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (b) PBH-PBH – Ellipsoidal Model 10-21 10-20 10-19 10-18 106 107 108 109 Merger Rate Per Spike [Yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (c) PBH-NS – Spherical Model 10-20 10-19 10-18 106 107 108 109 Merger Rate Per Spike [Yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (d) PBH-NS – Ellipsoidal Model Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The merger rate of compact binaries in a single dark-matter spike as a function of SMBH mass for different values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The top panels demonstrate this relation for PBH-PBH events in spherical- and ellipsoidal-collapse dark matter halo models, while the bottom panels display the corresponding results for PBH-NS events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' where χ(m∗)m∗ is normalized to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It should be noted that to characterize the galactic stellar mass M∗, the stellar mass–halo mass relation M∗(Mhalo) must be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' For this purpose, the stellar mass–halo mass relation obtained in Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2013) can be used, the basic assumption of which is the presence of the maximum number of NSs at the center of the galactic halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' We must note that in the present anal- ysis, for the relative velocity near the central SMBH, we use the circular velocity v(r) = � GMSMBH/r at each radius bounded by the dark-matter spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' This is a reasonable choice since the total mass enclosed by the region of dark-matter spike is negligible versus the mass of the central SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' We consider the mass of in- volving PBHs in PBH-PBH events as MPBH = 30 M⊙ and fix the masses of PBHs and NSs participating in PBH-NS events as MPBH = 5 M⊙ and MNS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Also, in the present analysis, we consider the contribu- tion of PBHs in dark matter to be fPBH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It is obvious from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (16), (17) and (18) that the merger rate of PBH-PBH binaries is straightly proportional to the f 2 PBH, while it changes directly with fPBH for the PBH-NS events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2, we have plotted the merger rate of compact binaries within a single spike as a function of SMBH mass for several values of power-law index γ while ac- counting for dark matter halo models with spherical and ellipsoidal collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' As can be seen from the figure, the merger rate of PBH-PBH binaries changes inversely with the mass of the SMBH for both spherical- and 7 10-10 10-8 10-6 10-4 10-2 100 102 104 106 106 107 108 109 Merger Rate [Gpc-3yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (a) PBH-PBH – Spherical – Shankar M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 10-10 10-8 10-6 10-4 10-2 100 102 104 106 106 107 108 109 Merger Rate [Gpc-3yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (b) PBH-PBH – Spherical – Vika M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 10-10 10-8 10-6 10-4 10-2 100 102 104 106 106 107 108 109 Merger Rate [Gpc-3yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (c) PBH-PBH – Spherical – Benson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 10-10 10-8 10-6 10-4 10-2 100 102 104 106 106 107 108 109 Merger Rate [Gpc-3yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (d) PBH-PBH – Ellipsoidal – Shankar M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 10-10 10-8 10-6 10-4 10-2 100 102 104 106 106 107 108 109 Merger Rate [Gpc-3yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (e) PBH-PBH – Ellipsoidal – Vika M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 10-10 10-8 10-6 10-4 10-2 100 102 104 106 106 107 108 109 Merger Rate [Gpc-3yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (f) PBH-PBH – Ellipsoidal – Benson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The merger rate of PBH-PBH binaries per unit time and volume as a function of SMBH mass for different values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The top panels show this relation for spherical-collapse dark matter halo models while considering three different SMBH mass functions, whereas the bottom panels exhibit the corresponding results for ellipsoidal-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' ellipsoidal-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It is also evident that the merger rate of PBH-PBH binaries is di- rectly proportional to the value of the power-law index γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' However, comparing PBH-PBH events for spherical- and ellipsoidal-collapse dark matter halo models, it can be inferred that the merger rate per spike for ellipsoidal- collapse dark matter halo models is higher than the corresponding one derived from spherical-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' However, for the merger rate of PBH-NS binaries, the situation is slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Interestingly, unlike the previous case, the merger rate of PBH-NS binaries per spike reaches the maximum value for the SMBH mass of MSMBH ≃ 107M⊙ for both spherical- and ellipsoidal- collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Also, the merger rate of PBH-NS binaries per spike for a power-law index of γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='7 has a maximum value that is completely dif- ferent from the relevant results of PBH-PBH events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In other words, the merger rate of PBH-NS binaries increases monotonically up to γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='7 and decreases for the larger values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In addition, it can be inferred from the results that the merger rate of PBH-NS bina- ries for ellipsoidal-collapse dark matter halo models is much higher than the corresponding one derived from spherical-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' On the other hand, the cumulative merger rate of compact binaries is considered the main quantity to be recorded and processed through the LIGO-Virgo detec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Therefore, the overall merger rate of compact bi- naries per unit volume and per unit time needs to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' To perform this task, one has to convolve the mas function of SMBH, φ(MSMBH), with the merger rate of compact binaries per spike, Nsp(MSMBH): R = � Mmax Mmin Nsp(MSMBH)φ(MSMBH)dMSMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (21) According to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (9), (10) and (11), the mentioned mass functions have a decreasing exponential term with respect to the mass of SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Therefore, it can be con- cluded that Mmax does not have a significant effect on the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In contrast, the introduced mass func- tions indicate that the maximum abundance belongs to the smallest central black holes in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Conse- quently, Mmin can have a significant contribution to the merger rate of compact binaries in dark-matter spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 8 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 106 107 108 109 Merger Rate [Gpc-3yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (a) PBH-NS – Spherical – Shankar M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 10-18 10-17 10-16 10-15 10-14 10-13 10-12 106 107 108 109 Merger Rate [Gpc-3yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (b) PBH-NS – Spherical – Vika M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 10-19 10-18 10-17 10-16 10-15 10-14 10-13 106 107 108 109 Merger Rate [Gpc-3yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (c) PBH-NS – Spherical – Benson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 106 107 108 109 Merger Rate [Gpc-3yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (d) PBH-NS – Ellipsoidal – Shankar M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 10-18 10-17 10-16 10-15 10-14 10-13 10-12 107 108 109 Merger Rate [Gpc-3yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (e) PBH-NS – Ellipsoidal – Vika M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 10-19 10-18 10-17 10-16 10-15 10-14 10-13 106 107 108 109 Merger Rate [Gpc-3yr-1] MSMBH [Msun] γ=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0 γ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 γ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 (f) PBH-NS – Ellipsoidal – Benson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The merger rate of PBH-NS binaries per unit time and volume as a function of SMBH mass for different values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The top panels indicate this relation for spherical-collapse dark matter halo models while accounting for three different SMBH mass functions, whereas the bottom panels show the corresponding results for ellipsoidal-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 3, we have depicted the merger rate of PBH- PBH binaries per unit time and volume as a function of SMBH mass for several values of power-law index γ while considering dark matter halo models with spher- ical and ellipsoidal collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' We have provided the results for three mass functions Shankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2004), Benson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2007), and Vika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2009) to jus- tify the possible uncertainties in the present analysis as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The merger rate of PBH-PBH bi- naries in both dark matter halo models with spherical and ellipsoidal collapses decreases monotonically with increasing the mass of SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Given the classification of the mass functions of SMBHs for their abundance, this result seems reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' As it is clear from the figures, the merger rate of PBH-PBH binaries for all three mass functions and both dark matter halo models reaches the maximum value as MSMBH = 106M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Also, the direct proportionality of the merger rate of PBH- PBH binaries to the values of the power-law index is ev- ident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Moreover, the merger rate of PBH-PBH binaries in ellipsoidal-collapse dark matter halo models is slightly higher than that obtained from spherical-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In the best case, which can be re- alized at the minimum value of the power-law index, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='05, the amplification of the overall merger rate is 62%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Additionally, it can be concluded that the merger rate of PBH-PBH binaries yields the high- est, middle, and lowest values while considering Shankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2004), Vika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2009), and Benson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2007) mass functions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' As previously discussed the merger rate per spike, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 4 exhibits that the overall merger rate of PBH-NS binaries is also obtained differently from that of PBH- PBH binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' As a common point of whole applied dark matter models and mass functions, it is deduced that the merger rate of PBH-NS binaries has a plateau at γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='7 and experiences a decreasing behavior for values greater than that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It is also evident that the merger rate of PBH-NS binaries for ellipsoidal-collapse dark matter halo models is in turn higher than that ob- tained for spherical-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Such a relative advantage is because by using more real- istic dark matter halo models, one can hope to improve the theoretical predictions of the merger rate of compact matter in galactic halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Interestingly, for PBH-NS bi- naries, the inverse proportionality of the overall merger 9 10-2 10-1 100 101 102 103 104 105 106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 2 Total Merger Rate [Gpc-3yr-1] γ Ellipsoidal Model Spherical Model (e) PBH-NS – Vika M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8 2 Total Merger Rate [Gpc-3yr-1] γ Ellipsoidal Model Spherical Model (f) PBH-NS – Benson M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The overall merger rate of compact binaries as a function of power-law index γ for both spherical- and ellipsoidal- collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The top panels demonstrate this relation for PBH-PBH events for three different SMBH mass functions, while the bottom panels show the corresponding results for PBH-NS events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The shaded cyan bands represent the BH-BH mergers estimated by the LIGO-Virgo detectors during the latest observing run, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='9-44) Gpc−3Yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' rate with the mass of SMBHs is not monotonic and has a plateau in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' If spherical-collapse dark matter halo models are reliable, the results obtained for Shankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2004) and Vika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2009) mass func- tions for γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='7 decrease monotonically with the mass of PBHs, while for γ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='7 it has a plateau around MSMBH = 107M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' However, the result obtained from the Benson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2007) mass function has a plateau around the same SMBH mass for all values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' On the other hand, if the ellipsoidal-collapse dark matter halo models are plausible, the existence of the plateau is evi- dent for all the mass functions used and for all values of the power-law index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' These results suggest that dark- matter spikes structured around central SMBHs with mass MSMBH = 107M⊙ may contain a non-standard abundance of NSs that disturbs the monotonic depen- dence of the merger rate on the mass SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 5, to quantitatively compare the results ob- tained from ellipsoidal-collapse dark matter halo models with those obtained from spherical-collapse dark matter halo models, we have displayed the merger rate of com- pact binaries in terms of power-law index γ while taking into account Shankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2004), Vika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2009), and Benson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2007) mass functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' To assess the theoretical predictions via the experimental data, we have also included the relevant mergers estimated by the GW detectors in the case of PBH-PBH events, while such an action has not been taken in the case of PBH- NS events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' This is because the prediction of the present analysis from the merger rate of PBH-PBH binaries is capable of justifying the data recorded by GW detec- tors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='9-44) Gpc−3Yr−1 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2021b), whereas the merger rate of PBH-NS binaries cannot in any way perform this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Specifically, the estimate of the LIGO-Virgo detectors from the total merger rate of BH-NS binaries is presented as (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='8-140) Gpc−3Yr−1 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2021d), while the relevant prediction of both dark matter halo models in the current analysis is extremely far from this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' As it is evident from the figures related to PBH- PBH events, and of course as mentioned earlier, the merger rate of PBH-PBH binaries while accounting for ellipsoidal-collapse dark matter halo models is higher than that extracted from spherical-collapse dark mat- 10 ter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In addition, such enhancement in the merger rate is greater at lower power-law index values than that at higher ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In this regard, the merger rate of PBH-PBH binaries in spherical-collapse dark mat- ter halo models while considering Shankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2004), Vika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2009), and Benson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2007) mass func- tions will be consistent with the BH-BH mergers esti- mated by GW detectors if the value of power-law index lies in the interval γ = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='05-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='15), γ = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='20), and γ = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='40-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='50), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' However, the corre- sponding results obtained from ellipsoidal-collapse dark matter halo models can potentially modify these values as γ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='95-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='05), γ = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='20), and γ = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='30-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='40), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' On the other hand, the results of PBH-NS events in- dicate that despite the mismatch of the outcome of the present analysis with GW data, the effect of ellipsoidal- collapse dark matter halo models in the amplification of the merger rate of such binaries is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It is also obvious that the merger rate of PBH-NS binaries in both dark matter halo models increases monotonical with the power-law index reaches a maximum at γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='7, and decreases for higher values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' CONCLUSIONS In this work, we have calculated the merger rate of compact binaries in dark-matter spikes, which are ex- pected to be structured around SMBHs at the center of galactic halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' For this purpose, we have initially described theoretical models, which suit dark matter spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' We have also discussed crucial quantities for dark-matter spikes, such as the density profile, concen- tration parameter, and MSMBH-σ relation, which can specify the distribution of dark matter particles in the region of spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' On the other hand, the strong corre- lation between the growth of central SMBHs and halo parameters suggests that another quantity called the mass function of SMBHs also plays a prominent role in the present analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' However, the insufficiency of our knowledge of the exact distribution of dark mat- ter particles in the central regions of galactic halos and the abundance of SMBHs in the Universe may lead to uncertainties in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Hence, to manage this un- certainty, we consider three empirical SMBH mass func- tions to compare their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' In the following, relying on the PBH scenario and with the assumption that PBHs are capable of contributing to the structure of dark matter, we have discussed the con- ditions of encountering compact objects such as PBHs and NSs in dark-matter spikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Accordingly, we have calculated the merger rate of compact binaries in a single dark-matter spike for spherical- and ellipsoidal-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Our results confirm that the merger rate of PBH-PBH binaries within each spike for ellipsoidal-collapse dark matter halo models is slightly higher than that derived from spherical-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' It is concluded that the merger rate of PBH-PBH binaries changes directly with the value of the power-law index γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' The results obtained from the analysis of PBH-NS events show that the maximum value of the merger rate in each spike takes place around the central SMBH with a mass MSMBH = 107 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Moreover, it turned out that the merger rate of PBH- NS binaries per spike has a maximum value at γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Also, the results indicate that the merger rate of PBH- NS binaries per spike for ellipsoidal-collapse dark mat- ter halo models is much higher than that derived from spherical-collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' As the main measurable factor in GW detectors, we have calculated the total merger rate of compact bina- ries in dark-matter spikes around central SMBHs with masses of MSMBH = (106-109)M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' As mentioned ear- lier, to account for possible uncertainties in our analysis, we have used three different mass functions for calculat- ing the total merger rate of compact binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Our find- ings indicate that the merger rate of PBH-PBH binaries in spherical- and ellipsoidal-collapse dark matter halo models has the maximum value at MSMBH = 106M⊙ and decreases monotonically with increasing the mass of SMBHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Also, the results exhibit that the overall merger rate of PBH-PBH binaries for ellipsoidal-collapse dark matter halo models is slightly higher than that ob- tained from spherical-collapse dark matter halo mod- els, in such a way that in the best case, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' at about γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='05, the amplification of the overall merger rate is about 62%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Moreover, among the considered mass functions, the highest, middle, and lowest values of the merger rate of PBH-PBH binaries have been extracted from Shankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2004), Vika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2009), and Ben- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' (2007) mass functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' However, the results of the merger rate of PBH-NS binaries were obtained slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Interestingly, the inverse pro- portionality of the overall merger rate PBH-NS bina- ries with the mass of SMBHs is not monotonic and has a plateau at MSMBH = 107M⊙ in almost all consid- ered models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' This may be due to a non-standard abun- dance of NSs in dark-matter spikes clustered around the central SMBH with of mass MSMBH = 107M⊙, which should be validated with informative observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Finally, we have calculated the merger rate of com- pact binaries according to the power-law index γ for ellipsoidal-collapse dark matter halo models and com- pared them with the corresponding results of spherical- collapse dark matter halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' To compare our find- 11 ings with the experimental data of GWs, we have also included the LIGO-Virgo sensitivity band for the merger rate of BH-BH binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' This task was not possible to do for PBH-NS events, as our results were far from those estimated via the LIGO-Virgo detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Our results show that the inclusion of ellipsoidal-collapse dark mat- ter halo models in the calculations of the merger rate of PBH-PBH binaries can reduce the range of the power- law index obtained from spherical-collapse dark matter halo models by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' This result comes from the fact that the dark-matter spikes in ellipsoidal-collapse halo mod- els are denser (and naturally smaller in radius) than those in spherical-collapse halo models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' Additionally, the results show that, unlike PBH-PBH events, the ef- fect of ellipsoidal-collapse dark matter halo models in the amplification of the merger rate of PBH-NS binaries is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' A maximum at γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='7 is also confirmed for such events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' REFERENCES Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', Abbott, R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' 2006, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', 132, 2685, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content='1086/508988 Musco, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=', & Miller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udE0T4oBgHgl3EQfbgBl/content/2301.02349v1.pdf'} 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0000000000000000000000000000000000000000..50ebd7d1170c141af9add8dc68c0292ed506064d --- /dev/null +++ b/udFKT4oBgHgl3EQf3i5N/content/tmp_files/2301.11928v1.pdf.txt @@ -0,0 +1,2191 @@ +arXiv:2301.11928v1 [math.NA] 5 Dec 2022 +Introduction to the Virtual Element Method for +2D Elasticity +L. L. Yaw 1,* +1Engineering Department, Walla Walla University, 100 SW 4th St, College Place, WA 99324, +USA +Summary +An introductory exposition of the virtual element method (VEM) is provided. The intent +is to make this method more accessible to those unfamiliar with VEM. Familiarity with the +finite element method for solving 2D linear elasticity problems is assumed. Derivations rele- +vant to successful implementation are covered. Some theory is covered, but the focus here is +on implementation and results. Examples are given that illustrate the utility of the method. +Numerical results are provided to help researchers implement and verify their own results. +KEY WORDS: virtual element method, VEM, consistency, stability, polynomial base, poly- +gon, vertices, elasticity, polymesher +1. +Introduction +The virtual element method (VEM) originated around 2013 [2]. It is yet another numerical +method to solve partial differential equations. VEM has many similarities with the finite +element method (FEM). One key difference is that VEM allows the problem domain to +be discretized by a collection of arbitrary polygons. The polygons need not all have the +same number of sides. +Hence, one can have triangles, quadrilaterals, pentagons, and so +on. This is attractive as it makes meshing the problem domain easier using, for example, a +Voronoi tesselation. Convex and concave polygons are allowed. Linear, quadratic, and higher +polynomial consistency is allowed within the method if implemented. In this introductory +exposition the focus is on solving 2D linear elasticity using linear order (k = 1) polynomial +interpolation. Additionally, VEM has the ability to handle non-conforming discretizations +(see Mengolini [5]). For this document the goal is to provide implementation details and +example results. An attempt is made to present the information in a logical and meaningful +order and thus provide the rationale for the method. However, it is almost certainly not +*Correspondence to: L. L. Yaw, Engineering Department, Walla Walla University, 100 SW 4th St, +College Place, WA, 99324 USA. E-mail: louie.yaw@wallawalla.edu +1 + +Introduction to the Virtual Element Method for 2D Elasticity +the order in which the method was discovered or rationalized originally. For attributes not +covered the interested reader is referred to the provided references. Derivations and notation +closely follows the paper by Mengolini et al. [5], with some exceptions. +2. +The Continuous 2D Linear Elasticity Problem +The goal is to solve 2D elasticity problems using VEM. The weak form for the elasticity +problem is: Find u ∈ V such that +a(u, v) = L(v) +∀v ∈ V, +(2.1) +where the bilinear form is +a(u, v) = +� +Ω +σ(u) : ǫ(v)dΩ, +(2.2) +and the linear form is +L(v) = +� +Ω +v · fdΩ + +� +∂Ωt +v · ¯td∂Ω. +(2.3) +Remarks +(i) The vector-valued function space V has components v1 and v2 that belong to the first- +order Sobolev space H1(Ω) with zero values on displacement boundaries. +(ii) The function u ∈ V is a trial solution, v ∈ V is a weight function. +3. +Discretization of the problem domain +For 2D elasticity the domain is the geometric region, Figure 1a, for which stresses, strains, +and displacements are calculated. +In VEM, the domain is discretized with an arbitrary +number of polygons, Figure 1b. Unlike FEM, polygon elements, convex or non-convex, with +an arbitrary number of sides are used in VEM. Due to the expectation that arbitrary polygon +elements are used, it is necessary to imagine a space of interpolation functions that include +polynomials but may also include non-polynomial functions. This is necessary because the +polygon elements must interconnect compatibly along their sides. Hence, along the edges the +interpolation functions are polynomials, but on the polygon interior the functions are possibly +non-polynomial. It turns out that it is not necessary to know the interpolation functions on +the interior of the polygon elements, rather it is sufficient to know the polynomial functions +along the polygon edges only. The order of the polynomials along the edges are chosen at +the beginning of the formulation. As already indicated, this document chooses first order +polynomials. +4. +VEM Functions +The discrete space of VEM functions, Vh, over individual elements are a subset of the space +of functions, V. The functions contained in V can satisfy the weak form of the continuous + +L. L. Yaw +x1 +x2 +n +¯u +¯t +Ω +∂Ω +f +(a) +x1 +x2 +E +vertex +u1 +u4 +u3 +u2 +(b) +Fig. 1: 2D Solid Domain: (a) Elasticity problem with boundary conditions, (b) Virtual el- +ement method domain discretization and example polygonal element with vector of +nodal displacements labeling each vertex. +2D elasticity problem. The superscript h indicates a space of functions used as part of a +discretization. Mathematically, Vh ⊂ V. +A typical VEM function vh ∈ Vh in 2D is a vector-valued displacement function of +spatial dimensions (x1, x2). For example, vh = [v1(x1, x2), v2(x1, x2)] is a two component +vector. To represent such a function, basis (or shape) functions are needed for each element +of the discretization. A basis for the VEM space of functions within an element, along one +spatial dimension, is represented as {ϕi}i=1,...,nd, with nd degrees of freedom. To organize +this more clearly along both spatial directions (2D case) a vector-valued form for the basis +is written as ϕ1 = [ϕ1, 0], ϕ2 = [0, ϕ1],..., ϕ2i−1 = [ϕi, 0], ϕ2i = [0, ϕi],..., ϕ2nd−1 = [ϕnd, 0], +ϕ2nd = [0, ϕnd]. Consequently, a VEM displacement function within an element written in +terms of the basis functions is +vh = +2nd +� +j=1 +dofj(vh)ϕj +(4.1) +where here the operator dofj extracts the value of vh at the jth degree of freedom. As +expected (4.1) is a linear combination of basis functions. +Remarks +(i) VEM basis (shape) functions have the following characteristics: +• continuous polynomial components of degree k along element edges, +• composed of polynomial functions and possibly non-polynomial functions on the +interior of the element, this is why they are said to be unknown, and is why the +word virtual is used in the name of the method, +• square integrable up to and including first derivatives, +• Laplacian, ∆vh|E, is made of polynomials of degree k − 2 in the interior of ele- +ment E, + +Introduction to the Virtual Element Method for 2D Elasticity +• Kronecker delta property +(ii) VEM basis functions on the interior of the element can be found numerically by solving +a PDE ∆vh = f, with f being a polynomial and prescribing values along the element’s +boundary. This is not necessary, is costly, and is avoided to accomplish VEM. +(iii) dofi(ϕj) = δij, this is enforced by how assumed characteristics of the basis functions +are implemented in the calculations along with the operator dofi. +(iv) The dof operator can refer to different components of the function, in the case of vector +valued functions. +(v) The equation (4.1) is similar to how interpolation between nodal values is accomplished +in FEM. Importantly, the basis functions are not actually known. Although (4.1) is +a familiar form, it cannot be used. Instead the VEM functions are projected onto a +space of polynomial functions with a projection operator. This is done with appropriate +restrictions and adjustments in place to account for the fact that the ’correct’ VEM +functions aren’t being used directly. +(vi) In this document, element degrees of freedom are only considered at element vertices. +Hence, for 2D elasticity the vertices (or nodes) have 2 degrees of freedom (one in +each coordinate direction (x1, x2)). This is due to the choice of only using first order +polynomials along the element boundaries. More degrees of freedom per element and +higher order polynomials are possible (see [5]). +5. +Polynomial Functions +With the concept of VEM functions realized, but knowing that they are not actually in +hand, a clever strategy is to imagine the projection of the VEM functions onto a space of +polynomial functions. As it unfolds, in later sections, the strategy proves to be useful. First, +it is useful, since a polynomial basis for the space of polynomial functions is easily created. +Second, because a very specific condition allows a projection operator to be found. Last, +because conforming polynomials along the boundary are in line with VEM function assumed +behavior and experience with FEM interpolation functions. +A space of scalar-valued polynomials of order equal to k or less on an element E is denoted +as Pk(E). This is extended to a 2D vector space of polynomials in two variables Pk ≡ [Pk]2. +The polynomial space has a basis Pk = {pα}α=1,...,nk. An example case for polynomials of +order k = 1, clarifies the meaning. +P1 = [p1, p2, p3, p4, p5, p6] +(5.1) +or +P1 = +�� 1 +0 +� +, +� 0 +1 +� +, +� −η +ξ +� +, +� η +ξ +� +, +� ξ +0 +� +, +� 0 +η +�� +. +(5.2) +In the preceding equations, scaled monomials are used to construct the components of the +polynomial basis of order k = 1. They are defined as +ξ = +�x1 − ¯x1 +hE +� +, +η = +�x2 − ¯x2 +hE +� +, +(5.3) + +L. L. Yaw +where ¯x = (¯x1, ¯x2) is the centroid location of element E and hE is its diameter (i.e., diameter +of smallest circle that encloses all vertices of the element). +Remarks +(i) In this document only polynomials of order k = 1 are considered. Higher order poly- +nomials are possible (see [5]). +(ii) With a choice of degrees of freedom the polynomials are unambiguously defined. For +example, two points make a line (a first order polynomial). That is, nodal values are +at vertices and first order polynomials interpolate between vertices of a given element +edge. +(iii) Just like R2 represents the space of 2D vectors, [Pk]2 represents a 2D vector polynomial +with components in two variables. +(iv) The number of terms (cardinality) in a polynomial base is calculated as nk = (k + +1)(k + 2). For the case of k = 1, nk = 6, which matches the number of terms in the +polynomial base P1. This arises by the requirement that all monomials of Pascal’s +triangle of order less than or equal to k are included. +(v) Infinitesimal rigid body motions are represented in the first three monomials p1, p2, p3 +of (5.2). +(vi) Based on α = 1, 2, 3 of the polynomial base the infinitesimal strain equals zero, +ǫ(pα) = 0, since these terms are associated with rigid body motion. To see this, recall +that the strain displacement relations are often represented as ǫ = BuE = ∂NuE. +Here, vh = [v1 v2]T is used as the displacement vector and uE as the nodal(vertex) val- +ues of a typical polygon element. Then (with Voigt notation in mind) the differential +operator, vector of shape functions, strain displacement matrix, and vector of nodal +values are as follows: +∂ = + + +∂x1 +0 +0 +∂x2 +∂x2 +∂x1 + + +(5.4) +N = +� +ϕ1 +ϕ2 +... +ϕnd +� +(5.5) +B = ∂N = +� +∂ϕ1 +∂ϕ2 +... +∂ϕnd +� +(5.6) +uE = +� +u1 +1 +u1 +2 +... +u2nd +1 +u2nd +2 +�T +(5.7) + +Introduction to the Virtual Element Method for 2D Elasticity +Finally, with the above in hand, the engineering strains are written with the strain +operator as +ǫ = + + +∂x1v1 +∂x2v2 +∂x2v1 + ∂x1v2 + + = BuE = [∂N] uE = +� +∂ϕ1 +∂ϕ2 +... +∂ϕnd +� +uE +(5.8) +(vii) It is important to note that in the preceding equations, (5.4) to (5.8), VEM basis +functions are used conceptually. +Yet, these functions are not known. +In fact, it is +necessary to insert the projection of VEM basis functions. +In later sections this is +discussed further. Toward this end, the projection operator is determined next. +6. +The Projector +A descretization using polygons is the goal. Functions that fit the necessary conditions on +polygons are called VEM functions. These functions are not known in a form that allows im- +plementation in a numerical formulation. However, it is possible to recover an approximation +of the VEM functions by projecting them onto a polynomial basis. Importantly, the VEM +functions projected onto the polynomial basis create polynomials along the element edges +and are able to exactly reproduce polynomials up to order k. This is called k-consistency. +Consistency and stability are both required for the success of a numerical discretization. +Stability is addressed later. Nevertheless, to achieve consistency and for the projection to +provide the best approximation of the VEM functions, the following orthogonality criteria +using the projection operator, Π, in each polygon is enforced: +aE(uh − Πuh, p) = 0, +∀p ∈ Pk(E), +(6.1) +where trial solution uh ∈ Vh. Recall the bilinear form (2.2). The terms in aE are inserted in +(2.2) and the integration takes place over an individual element E. The result is a measure +of strain energy. In (6.1) uh − Πuh is the error (or difference) between the VEM function +and the projection. In the ensuing derivations the projector is solved for by using (6.1) +so that the error is orthogonal to each polynomial basis in the polynomial space Pk(E). +In essence, this implies that the energy error is not captured by the polynomial basis. In +other words, the polynomial basis is forced to not include any of the energy error caused +by using the projection. This is exactly how k-consistency is enforced. The equation (6.1) +is the starting point for finding the projector matrix. Furthermore, unless explicitly stated, +the simplification Π ≡ ΠE,k is implied. The symbol ΠE,k projects element functions from +the VEM space onto the space of polynomials of order k, mathematically, ΠE,k : Vh(E) → +Pk(E). +To solve for the projector begin by rearranging (6.1). +aE(uh, p) − aE(Πuh, p) = 0 +⇒ +aE(uh, p) = aE(Πuh, p), +∀p ∈ Pk(E). +(6.2) + +L. L. Yaw +Then substituting terms into (6.2), and noting nodal displacements cancel from both +sides and that the strain operator is linear, yields +� +E +ǫ(ϕi)TCǫ(pα)dE = +� +E +ǫ(Π(ϕi))TCǫ(pα)dE. +(6.3) +Since the projection is onto the space of polynomials, it is reasonable to replace it with a +linear combination of polynomial basis functions. To this end, observe +Π(ϕi) = +nk +� +β=1 +si,βpβ +i = 1, ..., 2nd. +(6.4) +Inserting (6.4) into (6.3) yields +� +E +ǫ(ϕi)TCǫ(pα)dE = +nk +� +β=1 +si,β +� +E +ǫ(pβ)TCǫ(pα)dE. +(6.5) +The above equation, for a particular value of VEM shape function i, gives α = 1, ..., nk +simultaneous linear equations with nk unknowns si,β. This is written as +bi,α = +nk +� +β=1 +si,β ˜Gαβ. +(6.6) +In matrix form (6.6) becomes +bi = ˜Gsi, +(6.7) +where +bi = + + +aE(p1, ϕi) +... +aE(pnk, ϕi) + + , +si = + + +si,1 +... +si,nk + + +(6.8) +and +˜Gαβ = +� +E +ǫ(pβ)TCǫ(pα)dE +⇒ +˜G = [nk × nk], +˜G = ˜GT. +(6.9) +Then recognizing that i = 1, ..., 2nd the above equations are repeated for all values of i so +that +˜B = ˜G ˜Π +∗ +(6.10) +˜B = +� +b1 +b2 +... +b2nd +� +, +˜B = [nk × 2nd] +(6.11) +˜Π +∗ = +� +s1 +s2 +... +s2nd +� +, +˜Π +∗ = [nk × 2nd]. +(6.12) +Yet, (6.10) needs modification. Observe, for α = 1, 2, 3 equation (6.5) results in 0 = 0. This +is because the strain terms evaluate to zero for the rigid body modes of the polynomial base. + +Introduction to the Virtual Element Method for 2D Elasticity +Hence, (6.10) is an undetermined system for ˜Π +∗. Three additional equations are obtained +by requiring that +1 +nv +2nv +� +i=1 +dofi(vh)dofi(pα) = 1 +nv +2nv +� +i=1 +dofi(Π(vh))dofi(pα), +for α = 1, 2, 3 +⇒ 1 +nv +2nv +� +i=1 +vh +i dofi(ϕI)dofi(pα) = 1 +nv +2nv +� +i=1 +vh +i dofi +� nk +� +β=1 +sI,βpβ +� +dofi(pα) +⇒ 1 +nv +2nv +� +i=1 +dofi(ϕI)dofi(pα) = 1 +nv +2nv +� +i=1 +dofi +� nk +� +β=1 +sI,βpβ +� +dofi(pα) +⇒ 1 +nv +2nv +� +i=1 +dofi(ϕI)dofi(pα) = 1 +nv +2nv +� +i=1 +nk +� +β=1 +sI,βdofi(pβ)dofi(pα) +(6.13) +The last line of (6.13) in matrix form becomes +˘bI = ˘G˘sI, +(6.14) +where +˘bI = + + +1 +nv +2nv +� +i=1 +dofi(ϕI)dofi(p1) +1 +nv +2nv +� +i=1 +dofi(ϕI)dofi(p2) +1 +nv +2nv +� +i=1 +dofi(ϕI)dofi(p3) + + +, +˘sI = + + +˘sI,1 +... +˘sI,nk + + , +(6.15) +˘B = +� ˘b1 +˘b2 +... +˘b2nd +� +, +˘B = [3 × 2nd], +(6.16) +˘Π = +� ˘s1 +˘s2 +... +˘s2nd +� +, +˘Π = [nk × 2nd], +(6.17) +and +˘Gαβ = 1 +nv +2nv +� +i=1 +dofi(pα)dofi(pβ) +⇒ +˘G = [3 × nk]. +(6.18) +Consequently, the three equations in matrix form are +˘B = ˘G ˘Π. +(6.19) +Finally, (6.10) is modified so that (6.19) occupies the first three rows. The final modified +form of the equations is denoted as +¯B = G ˜Π. +(6.20) +Hence, the projector is +˜Π = G−1 ¯B. +(6.21) +Remarks + +L. L. Yaw +(i) The matrix ¯B, defined in (6.20), should not be confused with the ¯B matrix [4] used in +the finite element method for incompressibility problems. +(ii) The terms in (6.15) for ˘bI simplify further to +˘bI = 1 +nv +δiIdofi(pα), +(6.22) +where δiI is the Kronecker delta. +7. +Element Stiffness +Recall the objective is to discretize the domain with polygon elements. +VEM functions +with all the requisite characteristics are necessary to interpolate over the domain of each +individual element. The strain energy for an element is expressed in the discrete bilinear +form as +aE(uh, vh) = +� +E +σ(uh) : ǫ(vh)dE = +� +E +ǫ(vh)TCǫ(uh)dE. +(7.1) +Yet the VEM functions are not known, and it is preferable to write the bilinear form in terms +of the projection [6]. With this motivation the bilinear form is written as +aE(uh, vh) = aE(Πuh + (uh − Πuh), Πvh + (vh − Πvh)) += aE(Πuh, Πvh) + aE(uh − Πuh, Πvh) ++ aE(Πuh, vh − Πvh) + aE(uh − Πuh, vh − Πvh) += aE(Πuh, Πvh) +� +�� +� +1st part ++ aE(uh − Πuh, vh − Πvh) +� +�� +� +2nd part +. +(7.2) +In the last step of (7.2) the other terms vanish due to the enforcement of equation (6.1). +Partial success is now achieved in the last line of (7.2). +The first part is expressed +entirely in terms of the projected VEM functions and leads to the consistent part of the +stiffness matrix. The second part leads to stiffness stability, which ‘corrects’ for what is lost +of the VEM functions due to the projection. Each of the parts are dealt with in turn. +7.1. +Stiffness providing consistency +It is possible to obtain the consistent part of the stiffness exactly since it is projected onto +the known polynomial base functions. The first part of (7.2), similar to (7.1), leads to +aE(Πuh, Πvh) = +� +E +ǫ(Πvh)TCǫ(Πuh)dE. +(7.3) +Then using (4.1) and focusing on specific dofs i and j +aE(Πuh, Πvh)i,j = dofi(vh) +� +E +ǫ(Π(ϕi))TCǫ(Π(ϕj))dE +� +�� +� +(kc +E)ij +dofj(uh). +(7.4) + +Introduction to the Virtual Element Method for 2D Elasticity +Now, taking the ij component of the element stiffness from (7.4), equation (6.4), and linearity +of the strain operator, observe +(kc +E)ij = +� +E +ǫ(Π(ϕi))TCǫ(Π(ϕj))dE += +� +E +ǫ +� nk +� +α=1 +si,αpα +�T +Cǫ +� nk +� +β=1 +sj,βpβ +� +dE += +nk +� +α=1 +nk +� +β=1 +si,αsj,β +� +E +ǫ(pα)TCǫ(pβ)dE += +nk +� +α=1 +nk +� +β=1 +si,αsj,βaE(pα, pβ) += +nk +� +α=1 +nk +� +β=1 +˜Πα,i ˜Πβ,j ˜Gαβ += +� +˜Π +T ˜G ˜Π +� +ij , +(7.5) +where +˜G = aE(pα, pβ) = +� +E +ǫ(pα)TCǫ(pβ)dE. +(7.6) +Consequently, the consistent part of the stiffness matrix is represented as +kc +E = ˜Π +T ˜G ˜Π. +(7.7) +7.2. +Stiffness providing stability +To deal with the stability stiffness several constructions need to be set in place, motivated +by the approach of Sukumar and Tupek [7], yet with some matrices ordered to match the +approach of Mengolini et al. [5], as used herein. First, define a matrix of VEM basis functions +as +ϕ = +� +ϕ1 +0 +ϕ2 +0 +· · · +ϕi +0 +· · · +ϕnd +0 +0 +ϕ1 +0 +ϕ2 +· · · +0 +ϕi +· · · +0 +ϕnd +� += [ϕ1 ϕ2 · · ·ϕ2nd]. +(7.8) +Then, considering (6.4), which relates the projector to the polynomial basis for a single basis +function, ϕi, the corresponding matrix expression is +Π(ϕ) = Π{ϕ1 ϕ2 ... ϕ2nd} = +nk +� +β=1 +pβ{s1,β s2,β · · · s2nd,β} = P1 ˜Π. +(7.9) +Equation (7.9) is an expression in matrix form that represents the projection of the VEM +basis functions onto the polynomial basis. +Next, a D matrix is defined as +Diα = dofi(pα). +(7.10) + +L. L. Yaw +Observe that constant and linear reproducing conditions of the VEM basis provide the +following relations: +nd +� +i=1 +ϕi(x) = 1, +nd +� +i=1 +ϕi(x)ξi = ξ, +nd +� +i=1 +ϕi(x)ηi = η. +(7.11) +The preceding reproducing conditions are used to express the polynomial base in terms of +ϕ and D as follows: +P1 = ϕD. +(7.12) +To see how (7.12) comes about, it is instructive to write out the matrices ϕ and D with +internal components. Observing how the components multiply together and sum, reveals +the reproducing conditions. Finally, using (7.12) in (7.9) yields +Π(ϕ) = P1 ˜Π = ϕD ˜Π = ϕΠ. +(7.13) +Note that equation (7.13) provides the matrix representation of the projection in two ways. +The first way is the projection onto the polynomial basis as found in (7.9). The second way +is the projection on the ϕ basis set, from which the projection matrix, Π is defined as +Π = D ˜Π. +(7.14) +This last form of the projection proves useful to determine the stability stiffness. +From the second part of (7.2) and using (4.1) it follows that +aE(uh − Πuh, vh − Πvh)ij += aE(ϕjdofj(uh) − Π(ϕj)dofj(uh), ϕidofi(vh) − Π(ϕi)dofi(vh)) += dofi(vh) aE(ϕj − Π(ϕj), ϕi − Π(ϕi)) +� +�� +� +(ks +E)ij +dofj(uh). +(7.15) +Hence, the stability part of the stiffness for dofs i and j is +(ks +E)ij = aE(ϕj − Π(ϕj), ϕi − Π(ϕi)) += aE((1 − Π)ϕj, (1 − Π)ϕi). +(7.16) +In light of (7.13) and equation (7.16) the complete stability stiffness is +ks +E = aE((1 − Π)ϕ, (1 − Π)ϕ) += aE(ϕ(I − Π), ϕ(I − Π)). +(7.17) +Observing the similarity to (7.5) the terms (I − Π) are moved outside the bilinear form, so +that (7.17) becomes +ks +E = (I − Π)TaE(ϕ, ϕ)(I − Π). +(7.18) +It is not possible to evaluate the term aE(ϕ, ϕ) because it contains VEM shape functions, +which are not known. Hence, the effect of this term is approximated [2] [5] [1] using the + +Introduction to the Virtual Element Method for 2D Elasticity +scaling, τ htr(kc +E), where τ h is a user-defined parameter which is taken as 1/2 for linear +elasticity. The stability stiffness then is written as +ks +E = τ htr(kc +E)(I − Π)T(I − Π). +(7.19) +In the above expression, (7.19), I is the 2nd by 2nd identity matrix. +An alternative approach advocated by [7] is to approximate aE(ϕ, ϕ) with a 2nd by +2nd diagonal matrix, Sd +E, scaled appropriately. The terms along the diagonal are taken as: +(Sd +E)ii = max(α0 tr(C)/m, (kc +E)ii), where m = 3 in 2D, C is the 2D modular matrix for plane +strain or plain stress, tr denotes the trace operator, and α0 = 1 since the formulation uses +scaled monomials associated with the elements whose diameters are on the order of 1. With +this in hand the stability stiffness is represented as +ks +E = (I − Π)TSd +E(I − Π). +(7.20) +7.3. +Final form of element stiffness +With the stiffnesses in hand, the total stiffness for element E is +kE = kc +E + ks +E +(7.21) +7.4. +Numerical Evaluation of Various Terms +In this subsection derivations and details are provided to assist in the numerical implementa- +tion of VEM. Useful simplified expressions are given. In particular, this subsection focuses on +specific matrices necessary to construct the VEM element stiffness matrices. Other specific +implementation details of a VEM computer program are discussed by Mengolini et al. [5]. +7.4.1 +The ˜B matrix +Some discussion is necessary to illustrate how certain terms are calculated. First, consider +calculation of a typical term in the ˜B matrix. A typical term is (see (6.5)) +˜Bαi = aE(ϕi, pα) = +� +E +ǫ(ϕi)TCǫ(pα)dE +� +�� +� +Voigt notation += +� +E +ǫ(ϕi) : σ(pα)dE +� +�� +� +tensor notation +(7.22) +The symmetric part of ∇ϕi is ǫ(ϕi). Then, since σ is symmetric, it follows that +∇ϕi : σ = ǫ(ϕi) : σ. +(7.23) +Therefore, (7.22) becomes (continuing with tensor notation) +˜Bαi = +� +E +∇ϕi : σ(pα)dE. +(7.24) +Next, observe that +∇(ϕi · σ) = ∇ϕi : σ + ϕi · ∇σ +⇒ ∇ϕi : σ = −ϕi · ∇σ + ∇(ϕi · σ) +⇒ +� +E +∇ϕi : σdE = − +� +E +ϕi · ∇σdE + +� +E +∇(ϕi · σ)dE +(7.25) + +L. L. Yaw +x1 +x2 +vertex +nej +ej +ej−1 +nej−1 +j +j − 1 +Fig. 2: Single five sided element edges, normals, and nodes labeled. +Using the divergence theorem on the last line of (7.25) and substituting the result into (7.24) +˜Bαi = − +� +E +ϕi · ∇σ(pα)dE + +� +∂E +ϕi · σ(pα)nede, +(7.26) +where ne is the outward unit normal to the element edge and e denotes an element edge. It +is equation (7.26) that is numerically integrated to determine the entries in the ˜B matrix. +Realize now that for k = 1 only the boundary integral of (7.26) is nonzero. As a result, +˜Bαi = +� +∂E +ϕi · σ(pα)nede. +(7.27) +Numerically, (7.27) is calculated by integrating around the boundary of the element +(polygon) edges. This is accomplished by using the vertex (node) points as the integration +points and using the outward unit normal along each edge (see Figure 2). In essence, a +trapezoidal rule is used to integrate along each polygon edge. It is convenient to express the +integration around the boundary as a sum over vertices +˜Bαi = +nv +� +j=1 +ϕi · σ(pα) +�|ej−1| +2 +nej−1 + |ej| +2 nej +� +. +(7.28) +where the stress terms are found by matrix multiplication +σ(pα) = Cǫ(pα) + + +σx(pα) +σy(pα) +σxy(pα) + + = C + + +ǫx(pα) +ǫy(pα) +γxy(pα) + + . +(7.29) + +Introduction to the Virtual Element Method for 2D Elasticity +In the (7.29), the appropriate C matrix for plane stress or plain strain is used (see equations +(11.2) and (11.3)). The result of (7.29) is then used to form the stress matrix +σ(pα) = +� +σx(pα) +σxy(pα) +σxy(pα) +σy(pα) +� +. +(7.30) +Then (7.30) is used in (7.28) +˜Bαi = +nv +� +j=1 +ϕi · +� σx(pα) +σxy(pα) +σxy(pα) +σy(pα) +� � +|ej−1| +2 +� ne1 +ne2 +� +j−1 ++ |ej| +2 +� ne1 +ne2 +� +j +� +. +(7.31) +Finally, observe that ϕi is nonzero only at node dofs i = 2j−1 or i = 2j, which correspond +to two columns of the ˜B matrix, so that +˜Bα(2j−1) = +� 1 +0 +� +· +� σx(pα) +σxy(pα) +σxy(pα) +σy(pα) +� � +|ej−1| +2 +� ne1 +ne2 +� +j−1 ++ |ej| +2 +� ne1 +ne2 +� +j +� +and +˜Bα(2j) = +� 0 +1 +� +· +� σx(pα) +σxy(pα) +σxy(pα) +σy(pα) +� � +|ej−1| +2 +� ne1 +ne2 +� +j−1 ++ |ej| +2 +� ne1 +ne2 +� +j +� +. +(7.32) +Remarks +(i) Vertices j range over 1 to nv and (7.32) generates two columns of ˜B at a time for the +rows α = 1 to nk. +(ii) In other references the values at vertices are written in a slightly different form con- +sidering the normal to a line drawn between vertices j − 1 and j + 1. However, for +transparency the formula given above is provided. +(iii) For vertex j = 1 the edge length |ej−1| is taken as the the length between vertex j = nv +and j = 1, where nv is the number of vertices in the element (polygon). Furthermore, +the normal nej−1 is taken as the outward normal to the edge between j = nv and j = 1. +(iv) To be clear, the matrix multiplication of the right hand side of (7.32) results in a vector +that is then dotted with either [1 0]T or [0 1]T, as indicated. +(v) The ˜B matrix is nk by 2nv in size, for k = 1. +(vi) Equality (7.23) is true because, if A is an arbitrary tensor and S is a symmetric tensor, +it can be shown that A : S = Asym : S, where Asym is the symmetric part of A. +(vii) In the above formulas, the more familiar subscripts x, y for stresses and strains are +used, which here refer to coordinate directions x1, x2, respectively. + +L. L. Yaw +7.4.2 +The D matrix +The D matrix is constructed by evaluating polynomial functions at the various degrees of +freedom of polygon E. +The matrix entries are found by a straightforward evaluation of +matrix terms. The result is +D = + + +dof1(p1) +dof1(p2) +· · · +dof1(pnk) +dof2(p1) +dof2(p2) +· · · +dof2(pnk) +... +... +... +... +dof2ndp1) +dof2nd(p2) +· · · +dof2nd(pnk) + + . +(7.33) +7.4.3 +The ˜G matrix +A typical term in the ˜G matrix is expressed as +˜Gαβ = aE(pα, pβ) = +� +E +ǫ(pβ)TCǫ(pα)dE +� +�� +� +Voigt notation += +� +E +ǫ(pβ) : σ(pα)dE +� +�� +� +tensor notation +. +(7.34) +The previous work to find ˜B is modified to find ˜G. From (7.31) recognize that pβ in place +of ϕi yields +˜Gαβ = +nv +� +j=1 +pβ · +� σx(pα) +σxy(pα) +σxy(pα) +σy(pα) +� � +|ej−1| +2 +� ne1 +ne2 +� +j−1 ++ |ej| +2 +� ne1 +ne2 +� +j +� +, +(7.35) +where the quantities pα and pβ are evaluated at vertex j for the jth term in the summation. +The matrix ˜G is nk by nk in size. +Remarks +(i) It is possible to calculate the ˜G matrix as ˜G = ˜BD. Hence, the above formulation for +˜G provides an additional numerical check for verification. +(ii) Observe that the first three rows of ˜G and ˜B need not be calculated because they +contain all zeros. +(iii) It is faster to calculate ˜G by using ˜G = ˜BD, once the algorithm is verified as working. +However, a better approach is to find G = ¯BD and then get ˜G by zeroing the first +three rows of G. The proof that G = ¯BD is shown in [3] for one unknown per dof. +Here, a proof is given using the notation set forth so far and for 2D elasticity wherein +two displacement unknowns per dof are present. +Prove ¯BD = G. +Proof. For α = 1, 2, 3, and making use of (6.22) +2nd +� +i=1 +¯BαiDiβ = +2nd +� +i=1 +1 +nv +δijdofj(pα)dofi(pβ) = +2nd +� +i=1 +1 +nv +dofi(pα)dofi(pβ) = Gαβ. +(7.36) + +Introduction to the Virtual Element Method for 2D Elasticity +For α > 3 +2nd +� +i=1 +¯BαiDiβ = +2nd +� +i=1 +aE(pα, ϕi)dofi(pβ) = aE(pα, +2nd +� +i=1 +dofi(pβ)ϕi) = aE(pα, pβ) = Gαβ. +(7.37) +Consequently, ¯BD = G. +(iv) To clarify, ˜G is needed to obtain the stiffness matrix (see (7.7)). The terms G and ¯B +are needed to calculate the projector in (6.21). +8. +Application of External Forces +External forces are caused by external tractions and body forces as indicated in equation +(2.3). Point loads are also possible. All three forces are expressed for an individual element +in the linear form +LE(vh) = +� +E +vh · fdE + +� +∂E∩Ωt +vh · ¯td∂E + +� +i=1 +vh(xi) · Fi. +(8.1) +The interested reader is directed to the discussion by Mengolini et al. [5]. Herein, only +external point loads are used in the examples shown in later sections. From point loads a +global external force vector is assembled, which is used to solve for the nodal displacements. +In a nonlinear analysis, the global external forces are used in a Newton-Raphson scheme to +enforce equilibrium. +9. +Solving for Unknown Displacements +Once element stiffness matrices are found they are assembled into a global stiffness matrix +similar to FEM. As a result the global stiffness is +K = +nelem +A +i=1 ki +E. +(9.1) +Then with the global external force vector denoted as F the standard set of linear algebraic +equations are +Ku = F. +(9.2) +The global nodal displacements are then found in the typical manner +u = K−1F. +(9.3) +10. +Element Strains +Strains are found by starting with (4.1). Then +vh ≈ Π(vh) = +2nd +� +j=1 +dofj(vh)Π(ϕj) += +� Π(ϕ1) +Π(ϕ2) +... +Π(ϕ2nd) � +uE, +(10.1) + +L. L. Yaw +where the vector of local values at dofs for the given element is +uE = +� +dof1(vh) +dof2(vh) +... +dof2nd(vh) +�T = +� +u1 +1 +u1 +2 +... +u2nd +1 +u2nd +2 +�T . +(10.2) +Next the projection is expressed as +Π(vh) = Π( ¯N)uE +(10.3) +where ¯N ≡ ϕ is the row vector of VEM basis functions +¯N = +� +ϕ1 +ϕ2 +... +ϕ2nd +� +. +(10.4) +Also, observe that (6.4) leads to +Π( ¯N) = +� p1 +p2 +... +pnk +� ˜Π. +(10.5) +Consequently, +Π(vh) = +� p1 +p2 +... +pnk +� ˜ΠuE. +(10.6) +Last, using the strain operator (5.8) +ǫ(vh) ≈ ǫ(Π(vh)) = ǫ +�� p1 +p2 +... +pnk +� ˜ΠuE� += ǫ +� +p1 +p2 +... +pnk +� ˜ΠuE. +(10.7) +To be clear, the strain operator acting on the row vector of polynomial base functions is +ǫ +� +p1 +p2 +... +pnk +� += + + +∂x1p1,1 +∂x1p2,1 +... +∂x1pnk,1 +∂x2p1,2 +∂x2p2,2 +... +∂x2pnk,2 +∂x2p1,1 + ∂x1p1,2 +∂x2p2,1 + ∂x1p2,2 +... +∂x2pnk,1 + ∂x1pnk,2 + + . +(10.8) +Remarks +(i) The strains resulting from the above work are organized using Voigt notation. The +resulting strain vector contains the two-dimensional engineering strains ǫx, ǫy, γxy = +2ǫxy. +(ii) The strains are found for an individual polygonal (VEM) element. Hence, they are +constant within the element. +(iii) In (10.8), pi,j is the jth component of polynomial vector pi. In 2D the vectors pi are +as indicated in (5.1) and (5.2). + +Introduction to the Virtual Element Method for 2D Elasticity +11. +Element Stresses +The element-wise stresses are calculated using the previously found strain vector, ǫ(vh) ≈ +ǫ(Π(vh)). The stress vector is +σ(vh) = Cǫ(Π(vh)), +(11.1) +where for plane stress +C = +E +1 − ν2 + + +1 +ν +0 +ν +1 +0 +0 +0 +1 +2(1 − ν) + + , +(11.2) +and for plain strain +C = +E +(1 + ν)(1 − 2ν) + + +1 − ν +ν +0 +ν +1 − ν +0 +0 +0 +1−2ν +2 + + . +(11.3) +Remarks +(i) The stresses resulting from the above work are organized using Voigt notation. The +resulting stress vector contains the two-dimensional engineering stresses σx, σy, σxy. +(ii) The stresses are found for an individual polygonal (VEM) element. Hence, they are +constant within the element. +(iii) In (11.2) and (11.3), E is the modulus of elasticity and ν is Poisson’s ratio. +12. +Element Internal Forces +With an eye toward applications with nonlinear analysis, element internal forces are calcu- +lated by multiplying the element stiffness matrix times the vector of element displacements. +For example, the internal force vector for a single element i is +qi +int = kEuE. +(12.1) +Then similar to FEM the individual internal force vectors for all elements are assembled into +the global internal force vector using the assembly operator [4]. That is, +Fint = +nelem +A +i=1 qi +int. +(12.2) +13. +Some Relevant Concepts and Terminology +Many concepts are used to construct VEM. It is useful to organize and explain these con- +cepts. Without such an overview it is easy to get lost in the terminology and and lose sight +of the ultimate objective. The objective here is to explain concepts needed for the numer- +ical solution of elasticity problems using VEM. Unless evident otherwise, the definitions of +variables below are for 2D. + +L. L. Yaw +• Ω, the symbol which represents the continuous domain of the 2D elasticity problem to +be solved by VEM (see Figure 1) +• Ω ⊂ R2, the domain is contained in the real 2D coordinate space +• ∂Ω, the boundary of the domain, this can be decomposed into prescribed displacement +boundaries (Dirichlet or essential), ∂Ωu, and prescribed traction boundaries (Neumann +or natural), ∂Ωt. It is true that ∂Ω=∂Ωu ∪ ∂Ωt +• n, used to denote an outward normal vector to the boundary +• nej, used to denote an outward normal vector to the boundary edge ej +• ¯u = 0 on ∂Ωu, prescribed homogeneous displacement boundary condition +• ¯t, prescribed traction boundary condition +• u, the displacement solution to the elasticity problem. In 2D this is just a column vector +with two components (u = [u1 u2]T) that are functions of the coordinates (x1, x2). +• V, defined here as a vector-valued function space, in our case in 2D, with components +v1, v2. The components belong to a first-order Sobolev space H1(Ω) with zero values on +displacement boundaries. The space contains functions that are square-integrable up +to and including first derivatives. Functions that have these characteristics are needed +later. These careful definitions help us know exactly what type of functions we want, +and help us avoid problematic functions (that might give infinite square integrable +results. Such functions would imply infinite strain energy, which is not allowed.) This +function space is defined compactly as V ≡ [H1 +0(Ω)]2. +• a(u, v) = +� +Ω σ(u) : ǫ(v)dΩ, a bilinear form related to internal strain energy used in +problems of linear elasticity. +• L(v) = +� +Ω v · fdΩ + +� +∂Ωt v · ¯td∂Ω, a linear form related to the external energy caused +by external loads applied to the domain. +This could also include external energy +caused by external prescribed displacements. However, in this work external prescribed +displacements are assumed zero for simplicity. +• σ = Cǫ, Hooke’s law for linear elasticity relating stresses to strains. In indicial notation +this is written as σij = Cijklǫkl. In Voigt notation, in 2D, it is a 3x1 column vector. In +tensor notation, in 3D, it has 9 components and it is often expressed as a 3x3 symmetric +matrix. +• ǫ(u) = 1 +2(∇u + ∇uT ), is the linearized (small) strain tensor, in indicial notation this +is written as ǫ = 1 +2(ui,j + uj,i). In Voigt notation, in 2D, it is a 3x1 column vector. In +tensor notation, in 3D, it has 9 components and it is often expressed as a 3x3 symmetric +matrix. + +Introduction to the Virtual Element Method for 2D Elasticity +• Vh, this is the discrete vector-valued function space of VEM trial solutions uh and +weight functions vh. Exact continuous analytical solutions can sometimes (but rarely) +be found for an elasticity problem. Such solutions reside in the space of functions V. +However, for many problems only discrete (numerical) solutions are possible by FEM or +VEM. Hence, the domain is discretized into sub domains (elements) in which discrete +functions uhare used to approximate the elementwise solution. These functions are +piecewise connected, at element boundaries, across the problem domain. The discrete +space of functions is a subset of the space that includes continuous analytical functions +(Vh ⊂ V). This space of functions V(E)h on elements E contains polynomial functions +as well as non-polynomial functions. For a more formal definition, see Appendix A.1 +of Mengolini et al. [5]. +• E, an individual polygon domain +• Ωh, the discretized domain, covered by a collection of elements +• VEM functions, the functions used in the virtual element method are found in the +space of functions Vh. VEM functions include a combination of polynomial and non- +polynomial type functions. +• ϕ, VEM shape function matrix, a 2 × 2nd matrix +• ϕi, VEM shape function vector associated with element degree of freedom i, a 2 × 1 +column vector +• Pk(E), the space of polynomial functions of order less than or equal to k +• ΠE,k, the local projection operator. This operator projects VEM functions onto the +space of polynomials of order k or less. In math terms this is expressed as ΠE,k +: +V(E)h → Pk(E) +• Π, the projector operator, to be understood as a simplified version of ΠE,k, unless +directed otherwise +• nd, number of degrees of freedom along one spatial dimension of an element +• nv, number of polygon vertices for element E. Importantly, for k = 1 the number of +degrees of freedom along one spatial dimension, nd equals the number of vertices, nv. +• nk, the number of terms in in the polynomial base of order k. That is, nk = (k+1)(k+2) +• ∂, the differential operator defined in (5.4), a 3 × 2 operator matrix +• dofi(vh), degree of freedom i of vh for element E. +• uh, discrete trial solution, a 2 × 1 column vector +• vh, discrete weight function, a 2 × 1 column vector +• ∆vh|E = ∆2vh|E = Laplacian of vh|E over element E + +L. L. Yaw +• k, degree of polynomials used to approximate displacement within each element, degree +of the polynomial base +• ej, element edge j +• |ej|, length of element edge j +• |E|, area of element E +• B, strain displacement operator acting on VEM shape functions, a 3 × 2nd matrix +• ¯B, the final modified nk × 2nd “B” matrix used to calculate the projector, ˜Π = G−1 ¯B +• ˜B, the nk × 2nd “B” matrix that results in an undetermined system for the projector, +˜B = ˜G ˜Π +∗, it is the “B” matrix that needs its first three rows modified with ˘B to get +¯B. +• ˘B, this is the 3 × 2nd matrix that is inserted into the first three rows of ˜B to get the +final modified matrix ¯B. +• G, the final nk × nk “G” matrix used to calculate the projector, ˜Π = G−1 ¯B +• ˜G, the nk × nk “G” matrix that is part of the undetermined system ˜B = ˜G ˜Π +∗ +• ˘G, the 3 × nk matrix that is inserted into the first three rows of ˜G to obtain G +• Π, the nk × 2nd projector matrix that is used to construct the stability stiffness, ks +E. +It is the energy projector operator with respect to the ϕ basis set [7]. +• ˜Π, the nk × 2nd projector matrix that is used to construct the consistency stiffness, +kc +E. It is the energy projector operator with respect to the polynomial basis set [7]. +• ˜Π +∗, the nk × 2nd projector matrix that is part of the undetermined system, ˜B = ˜G ˜Π +∗ +• ˘Π, the 3 × 2nd matrix that relates ˘G and ˘B +• D, this 2nd × nk matrix is “used to express the projection of a VEM function as a +linear combination of the VEM functions themselves” [5]. +• P1, the 2 × nk polynomial basis matrix of order k = 1 +• pi, an individual scaled vector monomial in the polynomial basis set, a 2 × 1 column +vector +• kc +E, the 2nd × 2nd stiffness matrix for element E that provides consistency +• ks +E, the 2nd × 2nd stiffness matrix for element E that provides stability + +Introduction to the Virtual Element Method for 2D Elasticity +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +x +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +y +1 +2 +3 +4 +5 +Fig. 3: Single five sided element with vertex labels shown. +14. +Example – A single 5 sided element +Various terms in the VEM formulation are calculated for a single 5 sided element. The nu- +merical results are provided so that readers implementing VEM can verify that calculations +are correct. Element geometry is provided in figure 3. +Input +• Polynomial degree on polygon edges: k = 1 +• Modulus of Elasticity: E = 1000 +• Poisson’s ratio: ν = 0.3 +• Plane stress problem +• Element Node Numbers: [1,2,3,4,5] +• Domain Thickness: t=1 +• Specified zero displacements: Node 1 (ux = 0, uy = 0), Node 5 (ux = 0) +• Specified point loads: Node 2 (Fx = 40), Node 3 (Fx = 80), Node 4 (Fx = 40) +Output + +L. L. Yaw +• Centroid location: ¯x = 1.3571, ¯y = 1.8095 +• Number of vertices: nv = 5 +• Number of dofs: 2nd = 10 +• Polygon Diameter: hE = 5 +• Polygon Area: |E| = 10.5 +• ¯B matrix: +0.2000 +0 +0.2000 +0 +0.2000 +0 +0.2000 +0 +0.2000 +0 +0 +0.2000 +0 +0.2000 +0 +0.2000 +0 +0.2000 +0 +0.2000 +0.0724 +-0.0543 +0.0724 +0.0657 +-0.0076 +0.0657 +-0.0876 +0.0057 +-0.0876 +-0.0543 +-230.7692 -307.6923 -230.7692 +153.8462 +115.3846 +307.6923 +230.7692 +153.8462 +115.3846 -307.6923 +-439.5604 +-98.9011 +219.7802 +-98.9011 +439.5604 +49.4505 +219.7802 +98.9011 -439.5604 +49.4505 +-131.8681 -329.6703 +65.9341 -329.6703 +131.8681 +164.8352 +65.9341 +329.6703 -131.8681 +164.8352 +• D matrix: +1.0000 +0 +0.3619 +-0.3619 +-0.2714 +0 +0 +1.0000 +-0.2714 +-0.2714 +0 +-0.3619 +1.0000 +0 +0.3619 +-0.3619 +0.3286 +0 +0 +1.0000 +0.3286 +0.3286 +0 +-0.3619 +1.0000 +0 +-0.0381 +0.0381 +0.3286 +0 +0 +1.0000 +0.3286 +0.3286 +0 +0.0381 +1.0000 +0 +-0.4381 +0.4381 +0.0286 +0 +0 +1.0000 +0.0286 +0.0286 +0 +0.4381 +1.0000 +0 +-0.4381 +0.4381 +-0.2714 +0 +0 +1.0000 +-0.2714 +-0.2714 +0 +0.4381 +• G matrix: +1.0000 +0 +-0.0381 +0.0381 +0.0286 +0 +0 +1.0000 +0.0286 +0.0286 +0 +0.0381 +-0.0381 +0.0286 +0.2023 +-0.0566 +0.0229 +-0.0229 +0.0000 +0 +-0.0000 +646.1538 +0 +0 +0 +0 +-0.0000 +-0.0000 +461.5385 +138.4615 +0.0000 +0 +0.0000 +0.0000 +138.4615 +461.5385 +• ˜Π matrix: +0.2566 +-0.0016 +0.2093 +0.0016 +0.1635 +-0.0000 +0.1592 +-0.0033 +0.2114 +0.0033 +-0.0016 +0.2556 +0.0033 +0.2124 +-0.0033 +0.1592 +0.0000 +0.1616 +0.0016 +0.2112 +0.4143 +-0.5190 +0.2429 +0.2810 +-0.0643 +0.4762 +-0.3571 +0.1524 +-0.2357 +-0.3905 +-0.3571 +-0.4762 +-0.3571 +0.2381 +0.1786 +0.4762 +0.3571 +0.2381 +0.1786 +-0.4762 +-0.9524 +0 +0.4762 +0.0000 +0.9524 +0.0000 +0.4762 +0 +-0.9524 +-0.0000 +0 +-0.7143 +0.0000 +-0.7143 +0.0000 +0.3571 +-0.0000 +0.7143 +0 +0.3571 +• Π matrix: + +Introduction to the Virtual Element Method for 2D Elasticity +0.7943 +-0.0171 +0.2971 +0.0171 +-0.1829 +-0.0000 +-0.2286 +-0.0343 +0.3200 +0.0343 +-0.0171 +0.7843 +0.0343 +0.3300 +-0.0343 +-0.2286 +0.0000 +-0.2029 +0.0171 +0.3171 +0.2229 +-0.0171 +0.5829 +0.0171 +0.3886 +0.0000 +0.0571 +-0.0343 +-0.2514 +0.0343 +0.0171 +0.1871 +-0.0343 +0.6414 +0.0343 +0.3429 +-0.0000 +0.0314 +-0.0171 +-0.2029 +-0.0857 +-0.0000 +0.3429 +0.0000 +0.4857 +0.0000 +0.3429 +-0.0000 +-0.0857 +-0.0000 +0.0171 +-0.0986 +-0.0343 +0.3557 +0.0343 +0.4857 +-0.0000 +0.3171 +-0.0171 +-0.0600 +-0.1086 +0.0171 +-0.0400 +-0.0171 +0.2971 +0.0000 +0.4857 +0.0343 +0.3657 +-0.0343 +0.0000 +-0.0857 +0.0000 +-0.0857 +0.0000 +0.3429 +-0.0000 +0.4857 +-0.0000 +0.3429 +0.1771 +0.0171 +-0.1829 +-0.0171 +0.0114 +-0.0000 +0.3429 +0.0343 +0.6514 +-0.0343 +-0.0171 +0.2129 +0.0343 +-0.2414 +-0.0343 +0.0571 +-0.0000 +0.3686 +0.0171 +0.6029 +• kE, element stiffness matrix: +523.2489 +204.4601 -159.9480 +38.8680 -438.1401 -156.9859 -269.0252 -148.3797 +343.8645 +62.0375 +204.4601 +404.4220 +62.0375 +128.4422 -148.3797 -241.5527 -156.9859 -286.5997 +38.8680 +-4.7119 +-159.9480 +62.0375 +251.9156 -101.2839 +104.5264 +-86.3422 +19.7167 +-9.3631 -216.2107 +134.9518 +38.8680 +128.4422 -101.2839 +338.6842 +-67.4759 -110.0770 +7.8493 -200.8041 +122.0425 -156.2453 +-438.1401 -148.3797 +104.5264 +-67.4759 +522.9966 +102.0408 +210.1555 +123.1778 -399.5384 +-9.3631 +-156.9859 -241.5527 +-86.3422 -110.0770 +102.0408 +291.1714 +133.4380 +150.6317 +7.8493 +-90.1734 +-269.0252 -156.9859 +19.7167 +7.8493 +210.1555 +133.4380 +272.8564 +102.0408 -233.7034 +-86.3422 +-148.3797 -286.5997 +-9.3631 -200.8041 +123.1778 +150.6317 +102.0408 +356.7551 +-67.4759 +-19.9830 +343.8645 +38.8680 -216.2107 +122.0425 -399.5384 +7.8493 -233.7034 +-67.4759 +505.5879 -101.2839 +62.0375 +-4.7119 +134.9518 -156.2453 +-9.3631 +-90.1734 +-86.3422 +-19.9830 -101.2839 +271.1137 +• ux, uy, nodal displacements: +0.00 +0.000 +0.12 +0.000 +0.12 -0.024 +0.06 -0.048 +0.00 -0.048 +• ǫ, strains (ǫx, ǫy, γxy = 2ǫxy): +0.0400 +-0.0120 +-0.0000 +• σ, stresses (σx, σy, σxy): +40.0000 +-0.0000 +-0.0000 +15. +Example – Cantilever +A 12 inch long cantilever is loaded with a point load at its free end. The cantilever is 1 +inch deep and 1 inch thick into the page. The load at the end is 0.1 kips. The modulus of +elasticity is 1000 ksi and Poisson’s ratio is 0.3. The deflected shape is shown in Figure 4a. +The bending stresses, σx, are shown in Figure 3b. It is evident that stresses are constant +over each polygon element according to the VEM formulation. The cantilever has all nodes + +L. L. Yaw +pinned in the x and y direction at the support for this example. The maximum bending +stress is 6.19 ksi compared to the theoretical prediction of 7.2 ksi. A finer discretization +of the domain would provided better results. In this example, polymesher [8] was used to +randomly discretize the domain with 200 polygons. The tip displacement for this example +is 0.71 inches and the predicted value is 0.691 inches, according to the simple beam theory +formula, ∆ = P L3 +3EI . +16. +Example – Plate with hole +A plate with a hole is loaded in tension. Due to symmetry only one quadrant of the plate is +analyzed. The modulus of elasticity is 1000 ksi and Poisson’s ratio is 0.3. The deflected shape +is shown in Figure 5a. The σx stresses are shown in Figures 5b,c with 500 and 5000 polygons +respectively. It is evident that stresses are constant over each polygon element according to +the VEM formulation. The plate has zero displacement supports in the x-direction at the +left vertical edge, and y-direction at the bottom horizontal edge for this example. +17. +Conclusion +An explanation of the 2D virtual element method (VEM) is provided. Detailed derivations +and numerical examples are given. It is shown that VEM is a viable alternative to standard +FEM formulations. +18. +Acknowledgments +The author would like to thank Professor N. Sukumar for helpful discussions regarding +the Virtual Element Method. Yet, any errors or conceptual shortcomings are entirely the +responsibility of the author. +References +[1] E. Artioli, L. Beir˜ao da Veiga, C. Lovadina, and E. Sacco, Arbitrary order 2D +virtual elements for polygonal meshes: Part I, elastic problem, Computational Mechanics, +60 (2017), pp. 355–377. +[2] L. Beir˜ao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and +A. Russo, Basic principles of virtual element methods, Mathematical Models and Meth- +ods in Applied Sciences, 23 (2013), pp. 199–214. +[3] L. Beir˜ao da Veiga, F. Brezzi, L. D. Marini, and A. Russo, The hitchhiker’s +guide to the virtual element method, Mathematical Models and Methods in Applied Sci- +ences, 24 (2014), pp. 1541–1573. +[4] T. J. R. Hughes, The Finite Element Method - Linear Static and Dynamic Finite +Element Analysis, Dover, Mineola, NY, 1st ed., 2000. + +Introduction to the Virtual Element Method for 2D Elasticity +0 +2 +4 +6 +8 +10 +12 +x +-5 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +y +(a) +0 +2 +4 +6 +8 +10 +12 +x +-4 +-2 +0 +2 +4 +6 +y +VEM stress plot ( +x) +-6 +-4 +-2 +0 +2 +4 +(b) +Fig. 4: End Loaded Cantilever: (a) Original and deformed shape, (b) Bending stresses, σx. + +L. L. Yaw +(a) +0 +1 +2 +3 +4 +5 +x +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +y +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +(b) +0 +1 +2 +3 +4 +5 +x +-1 +-0.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +y +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +(c) +Fig. 5: Quadrant of plate with hole in tension: (a) Original and deformed shape, (b) σx +stresses with 500 polygon discretization, (c) σx stresses with 5000 polygon discretiza- +tion. + +3 +2.5 +24 +51.5 +0.5 +0 +-0.5 +-1 +0 +1 +2 +3 +XIntroduction to the Virtual Element Method for 2D Elasticity +[5] M. Mengolini, M. F. Benedetto, and A. M. Arag´on, An engineering perspective +to the virtual element method and its interplay with the standard finite element method, +Computer Methods in Applied Mechanics and Engineering, 350 (2019), pp. 995–1023. +[6] V. M. Nguyen-Thanh, X. Zhuang, H. Nguyen-Xuan, T. Tabczuk, and +P. Wriggers, A virtual element method for 2D linear elastic fracture analysis, Com- +puter Methods in Applied Mechanics and Engineering, 340 (2018), pp. 366–395. +[7] N. Sukumar and M. R. Tupek, Virtual elements on agglomerated finite elements +to increase the critical time step in elastodynamic simulations, 2021, arXiv:2110.00514 +[math.NA]. +[8] C. Talischi, G. H. Paulino, A. Pereira, and I. F. M. Menezes, Polymesher: a +general-purpose mesh generator for polygonal elements written in matlab, Structural and +Multidisciplinary Optimization, 45 (2012), pp. 309–328. + diff --git a/udFKT4oBgHgl3EQf3i5N/content/tmp_files/load_file.txt b/udFKT4oBgHgl3EQf3i5N/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..521425d0339bfac9c5e206dff19b63611334a618 --- /dev/null +++ b/udFKT4oBgHgl3EQf3i5N/content/tmp_files/load_file.txt @@ -0,0 +1,1154 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf,len=1153 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='11928v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='NA] 5 Dec 2022 Introduction to the Virtual Element Method for 2D Elasticity L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw 1,* 1Engineering Department, Walla Walla University, 100 SW 4th St, College Place, WA 99324, USA Summary An introductory exposition of the virtual element method (VEM) is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The intent is to make this method more accessible to those unfamiliar with VEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Familiarity with the finite element method for solving 2D linear elasticity problems is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Derivations rele- vant to successful implementation are covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Some theory is covered, but the focus here is on implementation and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Examples are given that illustrate the utility of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Numerical results are provided to help researchers implement and verify their own results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' KEY WORDS: virtual element method, VEM, consistency, stability, polynomial base, poly- gon, vertices, elasticity, polymesher 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Introduction The virtual element method (VEM) originated around 2013 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' It is yet another numerical method to solve partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' VEM has many similarities with the finite element method (FEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' One key difference is that VEM allows the problem domain to be discretized by a collection of arbitrary polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The polygons need not all have the same number of sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Hence, one can have triangles, quadrilaterals, pentagons, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This is attractive as it makes meshing the problem domain easier using, for example, a Voronoi tesselation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Convex and concave polygons are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Linear, quadratic, and higher polynomial consistency is allowed within the method if implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In this introductory exposition the focus is on solving 2D linear elasticity using linear order (k = 1) polynomial interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Additionally, VEM has the ability to handle non-conforming discretizations (see Mengolini [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' For this document the goal is to provide implementation details and example results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' An attempt is made to present the information in a logical and meaningful order and thus provide the rationale for the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' However, it is almost certainly not Correspondence to: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw, Engineering Department, Walla Walla University, 100 SW 4th St, College Place, WA, 99324 USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' E-mail: louie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='yaw@wallawalla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='edu 1 Introduction to the Virtual Element Method for 2D Elasticity the order in which the method was discovered or rationalized originally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' For attributes not covered the interested reader is referred to the provided references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Derivations and notation closely follows the paper by Mengolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' [5], with some exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The Continuous 2D Linear Elasticity Problem The goal is to solve 2D elasticity problems using VEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The weak form for the elasticity problem is: Find u ∈ V such that a(u, v) = L(v) ∀v ∈ V, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) where the bilinear form is a(u, v) = � Ω σ(u) : ǫ(v)dΩ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) and the linear form is L(v) = � Ω v · fdΩ + � ∂Ωt v · ¯td∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3) Remarks (i) The vector-valued function space V has components v1 and v2 that belong to the first- order Sobolev space H1(Ω) with zero values on displacement boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (ii) The function u ∈ V is a trial solution, v ∈ V is a weight function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Discretization of the problem domain For 2D elasticity the domain is the geometric region, Figure 1a, for which stresses, strains, and displacements are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In VEM, the domain is discretized with an arbitrary number of polygons, Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Unlike FEM, polygon elements, convex or non-convex, with an arbitrary number of sides are used in VEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Due to the expectation that arbitrary polygon elements are used, it is necessary to imagine a space of interpolation functions that include polynomials but may also include non-polynomial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This is necessary because the polygon elements must interconnect compatibly along their sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Hence, along the edges the interpolation functions are polynomials, but on the polygon interior the functions are possibly non-polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' It turns out that it is not necessary to know the interpolation functions on the interior of the polygon elements, rather it is sufficient to know the polynomial functions along the polygon edges only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The order of the polynomials along the edges are chosen at the beginning of the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' As already indicated, this document chooses first order polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' VEM Functions The discrete space of VEM functions, Vh, over individual elements are a subset of the space of functions, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The functions contained in V can satisfy the weak form of the continuous L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw x1 x2 n ¯u ¯t Ω ∂Ω f (a) x1 x2 E vertex u1 u4 u3 u2 (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 1: 2D Solid Domain: (a) Elasticity problem with boundary conditions, (b) Virtual el- ement method domain discretization and example polygonal element with vector of nodal displacements labeling each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 2D elasticity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The superscript h indicates a space of functions used as part of a discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Mathematically, Vh ⊂ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' A typical VEM function vh ∈ Vh in 2D is a vector-valued displacement function of spatial dimensions (x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' For example, vh = [v1(x1, x2), v2(x1, x2)] is a two component vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' To represent such a function, basis (or shape) functions are needed for each element of the discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' A basis for the VEM space of functions within an element, along one spatial dimension, is represented as {ϕi}i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=',nd, with nd degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' To organize this more clearly along both spatial directions (2D case) a vector-valued form for the basis is written as ϕ1 = [ϕ1, 0], ϕ2 = [0, ϕ1],.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=', ϕ2i−1 = [ϕi, 0], ϕ2i = [0, ϕi],.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=', ϕ2nd−1 = [ϕnd, 0], ϕ2nd = [0, ϕnd].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Consequently, a VEM displacement function within an element written in terms of the basis functions is vh = 2nd � j=1 dofj(vh)ϕj (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) where here the operator dofj extracts the value of vh at the jth degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' As expected (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) is a linear combination of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Remarks (i) VEM basis (shape) functions have the following characteristics: continuous polynomial components of degree k along element edges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' composed of polynomial functions and possibly non-polynomial functions on the interior of the element,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' this is why they are said to be unknown,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' and is why the word virtual is used in the name of the method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' square integrable up to and including first derivatives,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Laplacian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ∆vh|E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' is made of polynomials of degree k − 2 in the interior of ele- ment E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Introduction to the Virtual Element Method for 2D Elasticity Kronecker delta property (ii) VEM basis functions on the interior of the element can be found numerically by solving a PDE ∆vh = f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' with f being a polynomial and prescribing values along the element’s boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This is not necessary, is costly, and is avoided to accomplish VEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (iii) dofi(ϕj) = δij, this is enforced by how assumed characteristics of the basis functions are implemented in the calculations along with the operator dofi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (iv) The dof operator can refer to different components of the function, in the case of vector valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (v) The equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) is similar to how interpolation between nodal values is accomplished in FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Importantly, the basis functions are not actually known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Although (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) is a familiar form, it cannot be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Instead the VEM functions are projected onto a space of polynomial functions with a projection operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This is done with appropriate restrictions and adjustments in place to account for the fact that the ’correct’ VEM functions aren’t being used directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (vi) In this document, element degrees of freedom are only considered at element vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Hence, for 2D elasticity the vertices (or nodes) have 2 degrees of freedom (one in each coordinate direction (x1, x2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This is due to the choice of only using first order polynomials along the element boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' More degrees of freedom per element and higher order polynomials are possible (see [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Polynomial Functions With the concept of VEM functions realized, but knowing that they are not actually in hand, a clever strategy is to imagine the projection of the VEM functions onto a space of polynomial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' As it unfolds, in later sections, the strategy proves to be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' First, it is useful, since a polynomial basis for the space of polynomial functions is easily created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Second, because a very specific condition allows a projection operator to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Last, because conforming polynomials along the boundary are in line with VEM function assumed behavior and experience with FEM interpolation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' A space of scalar-valued polynomials of order equal to k or less on an element E is denoted as Pk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This is extended to a 2D vector space of polynomials in two variables Pk ≡ [Pk]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The polynomial space has a basis Pk = {pα}α=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=',nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' An example case for polynomials of order k = 1, clarifies the meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' P1 = [p1, p2, p3, p4, p5, p6] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) or P1 = �� 1 0 � , � 0 1 � , � −η ξ � , � η ξ � , � ξ 0 � , � 0 η �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) In the preceding equations, scaled monomials are used to construct the components of the polynomial basis of order k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' They are defined as ξ = �x1 − ¯x1 hE � , η = �x2 − ¯x2 hE � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw where ¯x = (¯x1, ¯x2) is the centroid location of element E and hE is its diameter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=', diameter of smallest circle that encloses all vertices of the element).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Remarks (i) In this document only polynomials of order k = 1 are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Higher order poly- nomials are possible (see [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (ii) With a choice of degrees of freedom the polynomials are unambiguously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' For example, two points make a line (a first order polynomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' That is, nodal values are at vertices and first order polynomials interpolate between vertices of a given element edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (iii) Just like R2 represents the space of 2D vectors, [Pk]2 represents a 2D vector polynomial with components in two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (iv) The number of terms (cardinality) in a polynomial base is calculated as nk = (k + 1)(k + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' For the case of k = 1, nk = 6, which matches the number of terms in the polynomial base P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This arises by the requirement that all monomials of Pascal’s triangle of order less than or equal to k are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (v) Infinitesimal rigid body motions are represented in the first three monomials p1, p2, p3 of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (vi) Based on α = 1, 2, 3 of the polynomial base the infinitesimal strain equals zero, ǫ(pα) = 0, since these terms are associated with rigid body motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' To see this, recall that the strain displacement relations are often represented as ǫ = BuE = ∂NuE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Here, vh = [v1 v2]T is used as the displacement vector and uE as the nodal(vertex) val- ues of a typical polygon element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Then (with Voigt notation in mind) the differential operator, vector of shape functions, strain displacement matrix, and vector of nodal values are as follows: ∂ = \uf8ee \uf8f0 ∂x1 0 0 ∂x2 ∂x2 ∂x1 \uf8f9 \uf8fb (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4) N = � ϕ1 ϕ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ϕnd � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5) B = ∂N = � ∂ϕ1 ∂ϕ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ∂ϕnd � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='6) uE = � u1 1 u1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' u2nd 1 u2nd 2 �T (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='7) Introduction to the Virtual Element Method for 2D Elasticity Finally, with the above in hand, the engineering strains are written with the strain operator as ǫ = \uf8ee \uf8f0 ∂x1v1 ∂x2v2 ∂x2v1 + ∂x1v2 \uf8f9 \uf8fb = BuE = [∂N] uE = � ∂ϕ1 ∂ϕ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ∂ϕnd � uE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='8) (vii) It is important to note that in the preceding equations, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='8), VEM basis functions are used conceptually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yet, these functions are not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In fact, it is necessary to insert the projection of VEM basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In later sections this is discussed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Toward this end, the projection operator is determined next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The Projector A descretization using polygons is the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Functions that fit the necessary conditions on polygons are called VEM functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' These functions are not known in a form that allows im- plementation in a numerical formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' However, it is possible to recover an approximation of the VEM functions by projecting them onto a polynomial basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Importantly, the VEM functions projected onto the polynomial basis create polynomials along the element edges and are able to exactly reproduce polynomials up to order k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This is called k-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Consistency and stability are both required for the success of a numerical discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Stability is addressed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Nevertheless, to achieve consistency and for the projection to provide the best approximation of the VEM functions, the following orthogonality criteria using the projection operator, Π, in each polygon is enforced: aE(uh − Πuh, p) = 0, ∀p ∈ Pk(E), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) where trial solution uh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Recall the bilinear form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The terms in aE are inserted in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) and the integration takes place over an individual element E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The result is a measure of strain energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) uh − Πuh is the error (or difference) between the VEM function and the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In the ensuing derivations the projector is solved for by using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) so that the error is orthogonal to each polynomial basis in the polynomial space Pk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In essence, this implies that the energy error is not captured by the polynomial basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In other words, the polynomial basis is forced to not include any of the energy error caused by using the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This is exactly how k-consistency is enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) is the starting point for finding the projector matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Furthermore, unless explicitly stated, the simplification Π ≡ ΠE,k is implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The symbol ΠE,k projects element functions from the VEM space onto the space of polynomials of order k, mathematically, ΠE,k : Vh(E) → Pk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' To solve for the projector begin by rearranging (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' aE(uh, p) − aE(Πuh, p) = 0 ⇒ aE(uh, p) = aE(Πuh, p), ∀p ∈ Pk(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw Then substituting terms into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2), and noting nodal displacements cancel from both sides and that the strain operator is linear, yields � E ǫ(ϕi)TCǫ(pα)dE = � E ǫ(Π(ϕi))TCǫ(pα)dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3) Since the projection is onto the space of polynomials, it is reasonable to replace it with a linear combination of polynomial basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' To this end, observe Π(ϕi) = nk � β=1 si,βpβ i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=', 2nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4) Inserting (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4) into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3) yields � E ǫ(ϕi)TCǫ(pα)dE = nk � β=1 si,β � E ǫ(pβ)TCǫ(pα)dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5) The above equation, for a particular value of VEM shape function i, gives α = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=', nk simultaneous linear equations with nk unknowns si,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This is written as bi,α = nk � β=1 si,β ˜Gαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='6) In matrix form (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='6) becomes bi = ˜Gsi, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='7) where bi = \uf8ee \uf8ef\uf8f0 aE(p1, ϕi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' aE(pnk, ϕi) \uf8f9 \uf8fa\uf8fb , si = \uf8ee \uf8ef\uf8f0 si,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' si,nk \uf8f9 \uf8fa\uf8fb (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='8) and ˜Gαβ = � E ǫ(pβ)TCǫ(pα)dE ⇒ ˜G = [nk × nk], ˜G = ˜GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='9) Then recognizing that i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=', 2nd the above equations are repeated for all values of i so that ˜B = ˜G ˜Π ∗ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='10) ˜B = � b1 b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' b2nd � , ˜B = [nk × 2nd] (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='11) ˜Π ∗ = � s1 s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' s2nd � , ˜Π ∗ = [nk × 2nd].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='12) Yet, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='10) needs modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Observe, for α = 1, 2, 3 equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5) results in 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This is because the strain terms evaluate to zero for the rigid body modes of the polynomial base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Introduction to the Virtual Element Method for 2D Elasticity Hence, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='10) is an undetermined system for ˜Π ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Three additional equations are obtained by requiring that 1 nv 2nv � i=1 dofi(vh)dofi(pα) = 1 nv 2nv � i=1 dofi(Π(vh))dofi(pα), for α = 1, 2, 3 ⇒ 1 nv 2nv � i=1 vh i dofi(ϕI)dofi(pα) = 1 nv 2nv � i=1 vh i dofi � nk � β=1 sI,βpβ � dofi(pα) ⇒ 1 nv 2nv � i=1 dofi(ϕI)dofi(pα) = 1 nv 2nv � i=1 dofi � nk � β=1 sI,βpβ � dofi(pα) ⇒ 1 nv 2nv � i=1 dofi(ϕI)dofi(pα) = 1 nv 2nv � i=1 nk � β=1 sI,βdofi(pβ)dofi(pα) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='13) The last line of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='13) in matrix form becomes ˘bI = ˘G˘sI, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='14) where ˘bI = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 nv 2nv � i=1 dofi(ϕI)dofi(p1) 1 nv 2nv � i=1 dofi(ϕI)dofi(p2) 1 nv 2nv � i=1 dofi(ϕI)dofi(p3) \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , ˘sI = \uf8ee \uf8ef\uf8f0 ˘sI,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ˘sI,nk \uf8f9 \uf8fa\uf8fb , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='15) ˘B = � ˘b1 ˘b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ˘b2nd � , ˘B = [3 × 2nd], (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='16) ˘Π = � ˘s1 ˘s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ˘s2nd � , ˘Π = [nk × 2nd], (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='17) and ˘Gαβ = 1 nv 2nv � i=1 dofi(pα)dofi(pβ) ⇒ ˘G = [3 × nk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='18) Consequently, the three equations in matrix form are ˘B = ˘G ˘Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='19) Finally, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='10) is modified so that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='19) occupies the first three rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The final modified form of the equations is denoted as ¯B = G ˜Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='20) Hence, the projector is ˜Π = G−1 ¯B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='21) Remarks L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw (i) The matrix ¯B, defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='20), should not be confused with the ¯B matrix [4] used in the finite element method for incompressibility problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (ii) The terms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='15) for ˘bI simplify further to ˘bI = 1 nv δiIdofi(pα), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='22) where δiI is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Element Stiffness Recall the objective is to discretize the domain with polygon elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' VEM functions with all the requisite characteristics are necessary to interpolate over the domain of each individual element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The strain energy for an element is expressed in the discrete bilinear form as aE(uh, vh) = � E σ(uh) : ǫ(vh)dE = � E ǫ(vh)TCǫ(uh)dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) Yet the VEM functions are not known, and it is preferable to write the bilinear form in terms of the projection [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' With this motivation the bilinear form is written as aE(uh, vh) = aE(Πuh + (uh − Πuh), Πvh + (vh − Πvh)) = aE(Πuh, Πvh) + aE(uh − Πuh, Πvh) + aE(Πuh, vh − Πvh) + aE(uh − Πuh, vh − Πvh) = aE(Πuh, Πvh) � �� � 1st part + aE(uh − Πuh, vh − Πvh) � �� � 2nd part .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) In the last step of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) the other terms vanish due to the enforcement of equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Partial success is now achieved in the last line of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The first part is expressed entirely in terms of the projected VEM functions and leads to the consistent part of the stiffness matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The second part leads to stiffness stability, which ‘corrects’ for what is lost of the VEM functions due to the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Each of the parts are dealt with in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Stiffness providing consistency It is possible to obtain the consistent part of the stiffness exactly since it is projected onto the known polynomial base functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The first part of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2), similar to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1), leads to aE(Πuh, Πvh) = � E ǫ(Πvh)TCǫ(Πuh)dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3) Then using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) and focusing on specific dofs i and j aE(Πuh, Πvh)i,j = dofi(vh) � E ǫ(Π(ϕi))TCǫ(Π(ϕj))dE � �� � (kc E)ij dofj(uh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4) Introduction to the Virtual Element Method for 2D Elasticity Now, taking the ij component of the element stiffness from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4), equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4), and linearity of the strain operator, observe (kc E)ij = � E ǫ(Π(ϕi))TCǫ(Π(ϕj))dE = � E ǫ � nk � α=1 si,αpα �T Cǫ � nk � β=1 sj,βpβ � dE = nk � α=1 nk � β=1 si,αsj,β � E ǫ(pα)TCǫ(pβ)dE = nk � α=1 nk � β=1 si,αsj,βaE(pα, pβ) = nk � α=1 nk � β=1 ˜Πα,i ˜Πβ,j ˜Gαβ = � ˜Π T ˜G ˜Π � ij , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5) where ˜G = aE(pα, pβ) = � E ǫ(pα)TCǫ(pβ)dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='6) Consequently, the consistent part of the stiffness matrix is represented as kc E = ˜Π T ˜G ˜Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='7) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Stiffness providing stability To deal with the stability stiffness several constructions need to be set in place, motivated by the approach of Sukumar and Tupek [7], yet with some matrices ordered to match the approach of Mengolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' [5], as used herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' First, define a matrix of VEM basis functions as ϕ = � ϕ1 0 ϕ2 0 · · ϕi 0 · · ϕnd 0 0 ϕ1 0 ϕ2 · · 0 ϕi · · 0 ϕnd � = [ϕ1 ϕ2 · · ·ϕ2nd].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='8) Then, considering (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4), which relates the projector to the polynomial basis for a single basis function, ϕi, the corresponding matrix expression is Π(ϕ) = Π{ϕ1 ϕ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ϕ2nd} = nk � β=1 pβ{s1,β s2,β · · · s2nd,β} = P1 ˜Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='9) Equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='9) is an expression in matrix form that represents the projection of the VEM basis functions onto the polynomial basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Next, a D matrix is defined as Diα = dofi(pα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='10) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw Observe that constant and linear reproducing conditions of the VEM basis provide the following relations: nd � i=1 ϕi(x) = 1, nd � i=1 ϕi(x)ξi = ξ, nd � i=1 ϕi(x)ηi = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='11) The preceding reproducing conditions are used to express the polynomial base in terms of ϕ and D as follows: P1 = ϕD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='12) To see how (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='12) comes about, it is instructive to write out the matrices ϕ and D with internal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Observing how the components multiply together and sum, reveals the reproducing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Finally, using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='12) in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='9) yields Π(ϕ) = P1 ˜Π = ϕD ˜Π = ϕΠ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='13) Note that equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='13) provides the matrix representation of the projection in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The first way is the projection onto the polynomial basis as found in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The second way is the projection on the ϕ basis set, from which the projection matrix, Π is defined as Π = D ˜Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='14) This last form of the projection proves useful to determine the stability stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' From the second part of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) it follows that aE(uh − Πuh, vh − Πvh)ij = aE(ϕjdofj(uh) − Π(ϕj)dofj(uh), ϕidofi(vh) − Π(ϕi)dofi(vh)) = dofi(vh) aE(ϕj − Π(ϕj), ϕi − Π(ϕi)) � �� � (ks E)ij dofj(uh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='15) Hence, the stability part of the stiffness for dofs i and j is (ks E)ij = aE(ϕj − Π(ϕj), ϕi − Π(ϕi)) = aE((1 − Π)ϕj, (1 − Π)ϕi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='16) In light of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='13) and equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='16) the complete stability stiffness is ks E = aE((1 − Π)ϕ, (1 − Π)ϕ) = aE(ϕ(I − Π), ϕ(I − Π)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='17) Observing the similarity to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5) the terms (I − Π) are moved outside the bilinear form, so that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='17) becomes ks E = (I − Π)TaE(ϕ, ϕ)(I − Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='18) It is not possible to evaluate the term aE(ϕ, ϕ) because it contains VEM shape functions, which are not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Hence, the effect of this term is approximated [2] [5] [1] using the Introduction to the Virtual Element Method for 2D Elasticity scaling, τ htr(kc E), where τ h is a user-defined parameter which is taken as 1/2 for linear elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The stability stiffness then is written as ks E = τ htr(kc E)(I − Π)T(I − Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='19) In the above expression, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='19), I is the 2nd by 2nd identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' An alternative approach advocated by [7] is to approximate aE(ϕ, ϕ) with a 2nd by 2nd diagonal matrix, Sd E, scaled appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The terms along the diagonal are taken as: (Sd E)ii = max(α0 tr(C)/m, (kc E)ii), where m = 3 in 2D, C is the 2D modular matrix for plane strain or plain stress, tr denotes the trace operator, and α0 = 1 since the formulation uses scaled monomials associated with the elements whose diameters are on the order of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' With this in hand the stability stiffness is represented as ks E = (I − Π)TSd E(I − Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='20) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Final form of element stiffness With the stiffnesses in hand, the total stiffness for element E is kE = kc E + ks E (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='21) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Numerical Evaluation of Various Terms In this subsection derivations and details are provided to assist in the numerical implementa- tion of VEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Useful simplified expressions are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In particular, this subsection focuses on specific matrices necessary to construct the VEM element stiffness matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Other specific implementation details of a VEM computer program are discussed by Mengolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1 The ˜B matrix Some discussion is necessary to illustrate how certain terms are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' First, consider calculation of a typical term in the ˜B matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' A typical term is (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5)) ˜Bαi = aE(ϕi, pα) = � E ǫ(ϕi)TCǫ(pα)dE � �� � Voigt notation = � E ǫ(ϕi) : σ(pα)dE � �� � tensor notation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='22) The symmetric part of ∇ϕi is ǫ(ϕi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Then, since σ is symmetric, it follows that ∇ϕi : σ = ǫ(ϕi) : σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='23) Therefore, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='22) becomes (continuing with tensor notation) ˜Bαi = � E ∇ϕi : σ(pα)dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='24) Next, observe that ∇(ϕi · σ) = ∇ϕi : σ + ϕi · ∇σ ⇒ ∇ϕi : σ = −ϕi · ∇σ + ∇(ϕi · σ) ⇒ � E ∇ϕi : σdE = − � E ϕi · ∇σdE + � E ∇(ϕi · σ)dE (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='25) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw x1 x2 vertex nej ej ej−1 nej−1 j j − 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 2: Single five sided element edges, normals, and nodes labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Using the divergence theorem on the last line of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='25) and substituting the result into (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='24) ˜Bαi = − � E ϕi · ∇σ(pα)dE + � ∂E ϕi · σ(pα)nede, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='26) where ne is the outward unit normal to the element edge and e denotes an element edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' It is equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='26) that is numerically integrated to determine the entries in the ˜B matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Realize now that for k = 1 only the boundary integral of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='26) is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' As a result, ˜Bαi = � ∂E ϕi · σ(pα)nede.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='27) Numerically, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='27) is calculated by integrating around the boundary of the element (polygon) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This is accomplished by using the vertex (node) points as the integration points and using the outward unit normal along each edge (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In essence, a trapezoidal rule is used to integrate along each polygon edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' It is convenient to express the integration around the boundary as a sum over vertices ˜Bαi = nv � j=1 ϕi · σ(pα) �|ej−1| 2 nej−1 + |ej| 2 nej � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='28) where the stress terms are found by matrix multiplication σ(pα) = Cǫ(pα) \uf8ee \uf8f0 σx(pα) σy(pα) σxy(pα) \uf8f9 \uf8fb = C \uf8ee \uf8f0 ǫx(pα) ǫy(pα) γxy(pα) \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='29) Introduction to the Virtual Element Method for 2D Elasticity In the (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='29), the appropriate C matrix for plane stress or plain strain is used (see equations (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The result of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='29) is then used to form the stress matrix σ(pα) = � σx(pα) σxy(pα) σxy(pα) σy(pα) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='30) Then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='30) is used in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='28) ˜Bαi = nv � j=1 ϕi · � σx(pα) σxy(pα) σxy(pα) σy(pα) � � |ej−1| 2 � ne1 ne2 � j−1 + |ej| 2 � ne1 ne2 � j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='31) Finally, observe that ϕi is nonzero only at node dofs i = 2j−1 or i = 2j, which correspond to two columns of the ˜B matrix, so that ˜Bα(2j−1) = � 1 0 � � σx(pα) σxy(pα) σxy(pα) σy(pα) � � |ej−1| 2 � ne1 ne2 � j−1 + |ej| 2 � ne1 ne2 � j � and ˜Bα(2j) = � 0 1 � � σx(pα) σxy(pα) σxy(pα) σy(pα) � � |ej−1| 2 � ne1 ne2 � j−1 + |ej| 2 � ne1 ne2 � j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='32) Remarks (i) Vertices j range over 1 to nv and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='32) generates two columns of ˜B at a time for the rows α = 1 to nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (ii) In other references the values at vertices are written in a slightly different form con- sidering the normal to a line drawn between vertices j − 1 and j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' However, for transparency the formula given above is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (iii) For vertex j = 1 the edge length |ej−1| is taken as the the length between vertex j = nv and j = 1, where nv is the number of vertices in the element (polygon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Furthermore, the normal nej−1 is taken as the outward normal to the edge between j = nv and j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (iv) To be clear, the matrix multiplication of the right hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='32) results in a vector that is then dotted with either [1 0]T or [0 1]T, as indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (v) The ˜B matrix is nk by 2nv in size, for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (vi) Equality (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='23) is true because, if A is an arbitrary tensor and S is a symmetric tensor, it can be shown that A : S = Asym : S, where Asym is the symmetric part of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (vii) In the above formulas, the more familiar subscripts x, y for stresses and strains are used, which here refer to coordinate directions x1, x2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2 The D matrix The D matrix is constructed by evaluating polynomial functions at the various degrees of freedom of polygon E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The matrix entries are found by a straightforward evaluation of matrix terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The result is D = \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 dof1(p1) dof1(p2) · · dof1(pnk) dof2(p1) dof2(p2) · · dof2(pnk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' dof2ndp1) dof2nd(p2) · · dof2nd(pnk) \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='33) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3 The ˜G matrix A typical term in the ˜G matrix is expressed as ˜Gαβ = aE(pα, pβ) = � E ǫ(pβ)TCǫ(pα)dE � �� � Voigt notation = � E ǫ(pβ) : σ(pα)dE � �� � tensor notation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='34) The previous work to find ˜B is modified to find ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' From (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='31) recognize that pβ in place of ϕi yields ˜Gαβ = nv � j=1 pβ · � σx(pα) σxy(pα) σxy(pα) σy(pα) � � |ej−1| 2 � ne1 ne2 � j−1 + |ej| 2 � ne1 ne2 � j � , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='35) where the quantities pα and pβ are evaluated at vertex j for the jth term in the summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The matrix ˜G is nk by nk in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Remarks (i) It is possible to calculate the ˜G matrix as ˜G = ˜BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Hence, the above formulation for ˜G provides an additional numerical check for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (ii) Observe that the first three rows of ˜G and ˜B need not be calculated because they contain all zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (iii) It is faster to calculate ˜G by using ˜G = ˜BD, once the algorithm is verified as working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' However, a better approach is to find G = ¯BD and then get ˜G by zeroing the first three rows of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The proof that G = ¯BD is shown in [3] for one unknown per dof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Here, a proof is given using the notation set forth so far and for 2D elasticity wherein two displacement unknowns per dof are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Prove ¯BD = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' For α = 1, 2, 3, and making use of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='22) 2nd � i=1 ¯BαiDiβ = 2nd � i=1 1 nv δijdofj(pα)dofi(pβ) = 2nd � i=1 1 nv dofi(pα)dofi(pβ) = Gαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='36) Introduction to the Virtual Element Method for 2D Elasticity For α > 3 2nd � i=1 ¯BαiDiβ = 2nd � i=1 aE(pα, ϕi)dofi(pβ) = aE(pα, 2nd � i=1 dofi(pβ)ϕi) = aE(pα, pβ) = Gαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='37) Consequently, ¯BD = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (iv) To clarify, ˜G is needed to obtain the stiffness matrix (see (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The terms G and ¯B are needed to calculate the projector in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Application of External Forces External forces are caused by external tractions and body forces as indicated in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Point loads are also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' All three forces are expressed for an individual element in the linear form LE(vh) = � E vh · fdE + � ∂E∩Ωt vh · ¯td∂E + � i=1 vh(xi) · Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) The interested reader is directed to the discussion by Mengolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Herein, only external point loads are used in the examples shown in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' From point loads a global external force vector is assembled, which is used to solve for the nodal displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In a nonlinear analysis, the global external forces are used in a Newton-Raphson scheme to enforce equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Solving for Unknown Displacements Once element stiffness matrices are found they are assembled into a global stiffness matrix similar to FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' As a result the global stiffness is K = nelem A i=1 ki E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) Then with the global external force vector denoted as F the standard set of linear algebraic equations are Ku = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) The global nodal displacements are then found in the typical manner u = K−1F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Element Strains Strains are found by starting with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Then vh ≈ Π(vh) = 2nd � j=1 dofj(vh)Π(ϕj) = � Π(ϕ1) Π(ϕ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Π(ϕ2nd) � uE, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw where the vector of local values at dofs for the given element is uE = � dof1(vh) dof2(vh) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' dof2nd(vh) �T = � u1 1 u1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' u2nd 1 u2nd 2 �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) Next the projection is expressed as Π(vh) = Π( ¯N)uE (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3) where ¯N ≡ ϕ is the row vector of VEM basis functions ¯N = � ϕ1 ϕ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ϕ2nd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4) Also, observe that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4) leads to Π( ¯N) = � p1 p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' pnk � ˜Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5) Consequently, Π(vh) = � p1 p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' pnk � ˜ΠuE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='6) Last, using the strain operator (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='8) ǫ(vh) ≈ ǫ(Π(vh)) = ǫ �� p1 p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' pnk � ˜ΠuE� = ǫ � p1 p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' pnk � ˜ΠuE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='7) To be clear, the strain operator acting on the row vector of polynomial base functions is ǫ � p1 p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' pnk � = \uf8ee \uf8f0 ∂x1p1,1 ∂x1p2,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ∂x1pnk,1 ∂x2p1,2 ∂x2p2,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ∂x2pnk,2 ∂x2p1,1 + ∂x1p1,2 ∂x2p2,1 + ∂x1p2,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ∂x2pnk,1 + ∂x1pnk,2 \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='8) Remarks (i) The strains resulting from the above work are organized using Voigt notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The resulting strain vector contains the two-dimensional engineering strains ǫx, ǫy, γxy = 2ǫxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (ii) The strains are found for an individual polygonal (VEM) element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Hence, they are constant within the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (iii) In (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='8), pi,j is the jth component of polynomial vector pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In 2D the vectors pi are as indicated in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Introduction to the Virtual Element Method for 2D Elasticity 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Element Stresses The element-wise stresses are calculated using the previously found strain vector, ǫ(vh) ≈ ǫ(Π(vh)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The stress vector is σ(vh) = Cǫ(Π(vh)), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) where for plane stress C = E 1 − ν2 \uf8ee \uf8f0 1 ν 0 ν 1 0 0 0 1 2(1 − ν) \uf8f9 \uf8fb , (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) and for plain strain C = E (1 + ν)(1 − 2ν) \uf8ee \uf8f0 1 − ν ν 0 ν 1 − ν 0 0 0 1−2ν 2 \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3) Remarks (i) The stresses resulting from the above work are organized using Voigt notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The resulting stress vector contains the two-dimensional engineering stresses σx, σy, σxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (ii) The stresses are found for an individual polygonal (VEM) element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Hence, they are constant within the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (iii) In (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3), E is the modulus of elasticity and ν is Poisson’s ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Element Internal Forces With an eye toward applications with nonlinear analysis, element internal forces are calcu- lated by multiplying the element stiffness matrix times the vector of element displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' For example, the internal force vector for a single element i is qi int = kEuE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1) Then similar to FEM the individual internal force vectors for all elements are assembled into the global internal force vector using the assembly operator [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' That is, Fint = nelem A i=1 qi int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Some Relevant Concepts and Terminology Many concepts are used to construct VEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' It is useful to organize and explain these con- cepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Without such an overview it is easy to get lost in the terminology and and lose sight of the ultimate objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The objective here is to explain concepts needed for the numer- ical solution of elasticity problems using VEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Unless evident otherwise, the definitions of variables below are for 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw Ω, the symbol which represents the continuous domain of the 2D elasticity problem to be solved by VEM (see Figure 1) Ω ⊂ R2, the domain is contained in the real 2D coordinate space ∂Ω, the boundary of the domain, this can be decomposed into prescribed displacement boundaries (Dirichlet or essential), ∂Ωu, and prescribed traction boundaries (Neumann or natural), ∂Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' It is true that ∂Ω=∂Ωu ∪ ∂Ωt n, used to denote an outward normal vector to the boundary nej, used to denote an outward normal vector to the boundary edge ej ¯u = 0 on ∂Ωu, prescribed homogeneous displacement boundary condition ¯t, prescribed traction boundary condition u, the displacement solution to the elasticity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In 2D this is just a column vector with two components (u = [u1 u2]T) that are functions of the coordinates (x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' V, defined here as a vector-valued function space, in our case in 2D, with components v1, v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The components belong to a first-order Sobolev space H1(Ω) with zero values on displacement boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The space contains functions that are square-integrable up to and including first derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Functions that have these characteristics are needed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' These careful definitions help us know exactly what type of functions we want, and help us avoid problematic functions (that might give infinite square integrable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Such functions would imply infinite strain energy, which is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=') This function space is defined compactly as V ≡ [H1 0(Ω)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' a(u, v) = � Ω σ(u) : ǫ(v)dΩ, a bilinear form related to internal strain energy used in problems of linear elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L(v) = � Ω v · fdΩ + � ∂Ωt v · ¯td∂Ω, a linear form related to the external energy caused by external loads applied to the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This could also include external energy caused by external prescribed displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' However, in this work external prescribed displacements are assumed zero for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' σ = Cǫ, Hooke’s law for linear elasticity relating stresses to strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In indicial notation this is written as σij = Cijklǫkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In Voigt notation, in 2D, it is a 3x1 column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In tensor notation, in 3D, it has 9 components and it is often expressed as a 3x3 symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ǫ(u) = 1 2(∇u + ∇uT ), is the linearized (small) strain tensor, in indicial notation this is written as ǫ = 1 2(ui,j + uj,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In Voigt notation, in 2D, it is a 3x1 column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In tensor notation, in 3D, it has 9 components and it is often expressed as a 3x3 symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Introduction to the Virtual Element Method for 2D Elasticity Vh, this is the discrete vector-valued function space of VEM trial solutions uh and weight functions vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Exact continuous analytical solutions can sometimes (but rarely) be found for an elasticity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Such solutions reside in the space of functions V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' However, for many problems only discrete (numerical) solutions are possible by FEM or VEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Hence, the domain is discretized into sub domains (elements) in which discrete functions uhare used to approximate the elementwise solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' These functions are piecewise connected, at element boundaries, across the problem domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The discrete space of functions is a subset of the space that includes continuous analytical functions (Vh ⊂ V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This space of functions V(E)h on elements E contains polynomial functions as well as non-polynomial functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' For a more formal definition, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1 of Mengolini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' E, an individual polygon domain Ωh, the discretized domain, covered by a collection of elements VEM functions, the functions used in the virtual element method are found in the space of functions Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' VEM functions include a combination of polynomial and non- polynomial type functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ϕ, VEM shape function matrix, a 2 × 2nd matrix ϕi, VEM shape function vector associated with element degree of freedom i, a 2 × 1 column vector Pk(E), the space of polynomial functions of order less than or equal to k ΠE,k, the local projection operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' This operator projects VEM functions onto the space of polynomials of order k or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In math terms this is expressed as ΠE,k : V(E)h → Pk(E) Π, the projector operator, to be understood as a simplified version of ΠE,k, unless directed otherwise nd, number of degrees of freedom along one spatial dimension of an element nv, number of polygon vertices for element E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Importantly, for k = 1 the number of degrees of freedom along one spatial dimension, nd equals the number of vertices, nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' nk, the number of terms in in the polynomial base of order k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' That is, nk = (k+1)(k+2) ∂, the differential operator defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='4), a 3 × 2 operator matrix dofi(vh), degree of freedom i of vh for element E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' uh, discrete trial solution, a 2 × 1 column vector vh, discrete weight function, a 2 × 1 column vector ∆vh|E = ∆2vh|E = Laplacian of vh|E over element E L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' degree of polynomials used to approximate displacement within each element,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' degree of the polynomial base ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' element edge j |ej|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' length of element edge j |E|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' area of element E B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' strain displacement operator acting on VEM shape functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' a 3 × 2nd matrix ¯B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' the final modified nk × 2nd “B” matrix used to calculate the projector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ˜Π = G−1 ¯B ˜B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' the nk × 2nd “B” matrix that results in an undetermined system for the projector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ˜B = ˜G ˜Π ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' it is the “B” matrix that needs its first three rows modified with ˘B to get ¯B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ˘B, this is the 3 × 2nd matrix that is inserted into the first three rows of ˜B to get the final modified matrix ¯B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' G, the final nk × nk “G” matrix used to calculate the projector, ˜Π = G−1 ¯B ˜G, the nk × nk “G” matrix that is part of the undetermined system ˜B = ˜G ˜Π ∗ ˘G, the 3 × nk matrix that is inserted into the first three rows of ˜G to obtain G Π, the nk × 2nd projector matrix that is used to construct the stability stiffness, ks E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' It is the energy projector operator with respect to the ϕ basis set [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ˜Π, the nk × 2nd projector matrix that is used to construct the consistency stiffness, kc E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' It is the energy projector operator with respect to the polynomial basis set [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' ˜Π ∗, the nk × 2nd projector matrix that is part of the undetermined system, ˜B = ˜G ˜Π ∗ ˘Π, the 3 × 2nd matrix that relates ˘G and ˘B D, this 2nd × nk matrix is “used to express the projection of a VEM function as a linear combination of the VEM functions themselves” [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' P1, the 2 × nk polynomial basis matrix of order k = 1 pi, an individual scaled vector monomial in the polynomial basis set, a 2 × 1 column vector kc E, the 2nd × 2nd stiffness matrix for element E that provides consistency ks E, the 2nd × 2nd stiffness matrix for element E that provides stability Introduction to the Virtual Element Method for 2D Elasticity 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5 4 x 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='5 4 y 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 3: Single five sided element with vertex labels shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Example – A single 5 sided element Various terms in the VEM formulation are calculated for a single 5 sided element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The nu- merical results are provided so that readers implementing VEM can verify that calculations are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Element geometry is provided in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Input Polynomial degree on polygon edges: k = 1 Modulus of Elasticity: E = 1000 Poisson’s ratio: ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3 Plane stress problem Element Node Numbers: [1,2,3,4,5] Domain Thickness: t=1 Specified zero displacements: Node 1 (ux = 0, uy = 0), Node 5 (ux = 0) Specified point loads: Node 2 (Fx = 40), Node 3 (Fx = 80), 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3422 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='9830 -101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2839 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1137 ux, uy, nodal displacements: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='12 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='06 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='00 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='048 ǫ, strains (ǫx, ǫy, γxy = 2ǫxy): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='0400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='0000 σ, stresses (σx, σy, σxy): 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='0000 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Example – Cantilever A 12 inch long cantilever is loaded with a point load at its free end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The cantilever is 1 inch deep and 1 inch thick into the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The load at the end is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='1 kips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The modulus of elasticity is 1000 ksi and Poisson’s ratio is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The deflected shape is shown in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The bending stresses, σx, are shown in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' It is evident that stresses are constant over each polygon element according to the VEM formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The cantilever has all nodes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yaw pinned in the x and y direction at the support for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The maximum bending stress is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='19 ksi compared to the theoretical prediction of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='2 ksi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' A finer discretization of the domain would provided better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' In this example, polymesher [8] was used to randomly discretize the domain with 200 polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The tip displacement for this example is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='71 inches and the predicted value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='691 inches, according to the simple beam theory formula, ∆ = P L3 3EI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Example – Plate with hole A plate with a hole is loaded in tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Due to symmetry only one quadrant of the plate is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The modulus of elasticity is 1000 ksi and Poisson’s ratio is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The deflected shape is shown in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The σx stresses are shown in Figures 5b,c with 500 and 5000 polygons respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' It is evident that stresses are constant over each polygon element according to the VEM formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' The plate has zero displacement supports in the x-direction at the left vertical edge, and y-direction at the bottom horizontal edge for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Conclusion An explanation of the 2D virtual element method (VEM) is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Detailed derivations and numerical examples are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' It is shown that VEM is a viable alternative to standard FEM formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Acknowledgments The author would like to thank Professor N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Sukumar for helpful discussions regarding the Virtual Element Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Yet, any errors or conceptual shortcomings are entirely the responsibility of the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Artioli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Beir˜ao da Veiga, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Lovadina, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Sacco, Arbitrary order 2D virtual elements for polygonal meshes: Part I, elastic problem, Computational Mechanics, 60 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' 355–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Beir˜ao da Veiga, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Brezzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Cangiani, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/udFKT4oBgHgl3EQf3i5N/content/2301.11928v1.pdf'} +page_content=' Manzini, L.' 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The first part introduces research +organizing with two axes of emotional affect: pleasantness and +arousal. Following this basic of emotional components, the docu- +ment discusses an aspect of emergent properties of emotion, show- +ing interaction studies with human users. With these past author’s +studies, the document concludes that the advantage of the cogni- +tive human-agent interaction approach is in representing human +internal states and processes. +CCS CONCEPTS +• Human-centered computing → HCI theory, concepts and mod- +els. +KEYWORDS +cognitive architecture, emotion, arousal, valence +1 +INTRODUCTION +This document begins by asking the following question: +How can emotion emerge in a computational system? +This is a kind of ultimate question that has attracted enormous +numbers of scientists and engineers, including the author himself. +Toward the complete answer to the question, the author has devel- +oped several models of mental functions (possible components of +emotion process) in ACT-R (Adaptive Control of Thought-Rational +[1]), which is one of the most widely used cognitive architectures +in the world. +Cognitive architectures generally integrate knowledge concern- +ing the human mind in the form of computer programs. The knowl- +edge accumulated in ACT-R ranges from perceptual-motor com- +ponents to abstract and goal-related concepts. Varieties of mental +functions are controlled by symbols stored in modules (correspond- +ing brain regions) and subsymbolic parameters (corresponding neu- +rotransmitters). By utilizing these, this architecture aims to realize +human-level activities in every field of human life. +The author considers that the above characteristic of the archi- +tecture is crucial to answering the question. Thus, this document +Permission to make digital or hard copies of part or all of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for third-party components of this work must be honored. +For all other uses, contact the owner/author(s). +CHAI ’22, December 5, 2022, Christchurch, New Zealand +© 2022 Copyright held by the owner/author(s). +presents the author’s works utilizing ACT-R as ingredients of dis- +cussions on the conditions for enabling emotion in a computational +system. +2 +ISSUES OF EMOTIONAL PROCESS +As the background of the discussion, this section presents the au- +thor’s view on the emotion process. It is assumed that the emotion +process relies on subcortical brain regions, such as the amygdala, +insula, and basal ganglia. The process was considered to be the +product of an adaptation to ancestors’ surrounding environments, +because these regions were formed early in the evolution of the +brain [4, 9]. However, modern human emotion is more complex +than the purely physical processes in the following senses: +(1) Emotion is not a static entity but rather emergent properties +accompanied by dynamic interaction with the environment. +This means a human’s emotional response always fluctuates. +(2) Our environment has drastically changed from the environ- +ment in which our ancestors lived. Therefore, many emo- +tional mechanisms have become maladaptive in the modern +age. +(3) This biological process is somehow modulated by an inten- +tional strategy, as many theories of emotion have suggested +[3, 4]. Therefore, there is room for intervention in the emo- +tion process by our will or technology. +3 +REPRESENTING EMOTION IN ACT-R +Even through such complexities, it is possible to model the basis of +emotion (affect) using the fundamental axes, arousal and pleasant- +ness, as presented in Russel [16]’s circumplex model (Fig 1). The +present author’s approach begins with these simple components. +The following studies represent these components with primitive +cognitive functions, such as activation noise and pattern matching, +implemented in ACT-R. +3.1 +Representation of Emotional Arousal +According to a dictionary in psychology [17], arousal is defined as +follows: +a state of physiological activation or cortical respon- +siveness, associated with sensory stimulation and ac- +tivation of fibers from the reticular activating system. +As indicated by this definition, past researchers have frequently con- +nected arousal with the activation (attentional) process, especially +the degree of concentration (e.g., [8]). In a concentrated situation, +humans can continue monotonous tasks accurately. However, as +such a process long lasts long, people get bored and begin to think +about things outside of the task (i.e., mind-wandering). +arXiv:2301.00003v1 [cs.HC] 28 Dec 2022 + +CHAI ’22, December 5, 2022, Christchurch, New Zealand +Morita +High arousal +Low arousal +Unpleasant +Rewards +Fluctuation +Figure 1: Axes of emotional affect and model parameters. +From this phenomenon and the consideration that emotional +arousal relates to biological fluctuations affecting the cognitive +process, the author and their colleagues have represented arousal +as the noise factor for memory activation in ACT-R. Controlling this +subsymbolic parameter, the authors have demonstrated changes in +memory recollection [10] and task goal switching [14]. This view +is consistent with the discussion in which emotion is a modulator +of cognitive architecture [5, 15]. +3.2 +Representation of Pleasantness +People feel joy when receiving a reward. Thus, the pleasantness axis +can be considered as a reward in reinforcement learning. Various +triggering events for rewards can be assumed in the real world. +Among them, internally generated ones are important to developing +autonomous agents. +Based on the above assumptions, Nagashima et al. [12, 13] devel- +oped a model of intrinsic motivation, assuming a pattern discovery +to be a source of curiosity. In the ACT-R model, patterns in the +data are discovered with the process of pattern matching, and the +experience of the pattern matching is utilized to build procedural- +ized rules (the compilation of production rules). As a task proceeds, +opportunities for pattern matching (internal rewards) gradually +decrease. Thus, the model can explain how intrinsic motivation +decreases with experience and increases with discovering novel +patterns. This pattern-discovery-focused view of intrinsic motiva- +tion is consistent with discussions in the entertainment industry +[7]. In addition, it is supported by the theory emphasizing the role +of pattern-seeking in the history of human civilization [2]. +4 +NEEDS OF INTERACTION +From the discussion so far, the importance of environment for +emotion stands out. Capturing the full complexities of the real- +world dynamics of the two mechanisms (arousal and pleasantness) +requires environmental changes. Among various environmental +factors, the existence of other organisms seems most crucial. In +other words, the author considers that human emotion can be +modeled only when the computational system actually interacts +Figure 2: Framework of interacting human emotion with +machine emotion [11] +with humans. By interacting with humans, the system is able to +learn human’s emotion generation and expression. +Based on this consideration, the author and colleagues have +developed several interactive systems [6, 11], implementing an +emotion model as a component. Those systems receive the user’s +biological signals such as heart rates to automatically modulate +the abovementioned ACT-R parameters (noises and rewards) for +guiding the user’s emotion to an optimal state. Especially, Morita et +al. [11] demonstrated that a web advertisement system containing +an ACT-R memory model could prevent human repetitive thinking +(rumination) when the model behaves in a counterbalanced manner +(Fig 2). The author believes that such a system will eventually lead +to a new human homeostatic process with the help of artificial +emotional systems. +5 +CONCLUSION +This document presents the author’s attempts to answer the ul- +timate question about the computational conditions that enable +emotion. Future work must represent the ideas presented in the +document as a general framework and evaluate it in human exper- +iments. The author considers that such a cognitive-architecture- +based framework is advantageous in constructing trustful human- +agent relations. Academic knowledge implemented in architecture +is the result of continuous endeavors in human history. Including +the formal knowledge agreed in human society is an essential in- +gredient of making common ground between humans and artifacts. +ACKNOWLEDGEMENT +This document summarizes ideas obtained from past collaborative +studies with colleagues at Nagoya University, Shizuoka University, +collaborators from Panasonic Corp. and Mazda Corp., and mem- +bers of the Applied Cognitive Modeling Lab (ACML) at Shizuoka +university. The author thanks everyone for valuable discussions. + +User +Model +Stress +Stress +Activation +HRV +noise +Synchronize +Counterbalance +Relax +RelaxEmotion in Cognitive Architecture: +Emergent Properties from Interactions with Human Emotion +CHAI ’22, December 5, 2022, Christchurch, New Zealand +REFERENCES +[1] John R Anderson. 2007. How Can the Human Mind Occur in the Physical Universe. +Oxford University Press, New York. +[2] Simon Baron-Cohen. 2020. The pattern seekers: How autism drives human invention. +Basic Books. +[3] Lisa Feldman Barrett. 2017. How emotions are made: The secret life of the brain. +Pan Macmillan. +[4] Antonio Damasio. 1994. Descartes’ Error. Random House, New York. +[5] Christopher L Dancy, Frank E Ritter, Keith A Berry, and Laura C Klein. 2015. +Using a cognitive architecture with a physiological substrate to represent ef- +fects of a psychological stressor on cognition. Computational and Mathematical +Organization Theory 21, 1 (2015), 90–114. +[6] K. Itabashi, J. Morita, T. Hirayama, K. Mase, and Yamada K. 2020. Interactive +Model-based Reminiscence Using a Cognitive Model and Physiological Indices. +In Proceedings of the 18th International Conference on Cognitive Modeling. +[7] Raph Koster. 2013. Theory of Fun for Game Design. O’Reilly Media, Sebastopol. +[8] D. M. Landers. 1980. The arousal-performance relationship revisited. Research +Quarterly for Exercise and Sport 51, 1 (1980), 77–90. +[9] Joseph LeDoux. 2020. The deep history of ourselves: The four-billion-year story of +how we got conscious brains. Penguin. +[10] Junya Morita, Takatsugu Hirayama, Kenji Mase, and Kazunori Yamada. 2016. +Model-Based Reminiscence: Guiding Mental Time Travel by Cognitive Modeling. +In Proceedings of the Fourth International Conference on Human Agent Interaction +(Biopolis, Singapore) (HAI ’16). 341–344. +[11] Junya Morita, Thanakit Pitakchokchai, Giri Basanta Raj, Yusuke Yamamoto, +Hiroyasu Yuhashi, and Teppei Koguchi. 2022. Regulating Ruminative Web Brows- +ing Based on the Counterbalance Modeling Approach. Frontiers in Artificial +Intelligence 5 (2022). +[12] Kazuma Nagashima, Junya Morita, and Yugo Takeuchi. 2020. Modeling intrin- +sic motivation in ACT-R : Focusing on the relation between pattern matching +and intellectual curiosity. In Proceedings of the 18th International Conference on +Cognitive Modelling. Applied Cognitive Science Lab, 167–173. +[13] Kazuma Nagashima, Junya Morita, and Yugo Takeuchi. 2021. Curiosity as pattern +matching: Simulating the effects of intrinsic rewards on the levels of processing. +In Proceedings of the 19th International Conference on Cognitive Modelling. 197– +203. +[14] K Nagashima, J Nishikawa, R Yoneda, J Morita, and T Terada. 2022. Model- +ing optimal arousal by integrating basic cognitive components. In International +Conference on Cognitive Modeling 2022. +[15] F. E. Ritter. 2009. Two cognitive modeling frontiers Emotions and usability. +Transactions of the Japanese Society for Artificial Intelligence 24, 2 (2009), 241– +249. +[16] James A Russell. 2003. Core affect and the psychological construction of emotion. +Psychological Review 110, 1 (2003), 145. +[17] Gary R VandenBos. 2007. APA dictionary of psychology. American Psychological +Association. + diff --git a/vtAyT4oBgHgl3EQfOPaQ/content/tmp_files/load_file.txt b/vtAyT4oBgHgl3EQfOPaQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4c4dea5f57ad82cd4126878ec09f030d3971c308 --- /dev/null +++ b/vtAyT4oBgHgl3EQfOPaQ/content/tmp_files/load_file.txt @@ -0,0 +1,159 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf,len=158 +page_content='Emotion in Cognitive Architecture: Emergent Properties from Interactions with Human Emotion Junya Morita j-morita@inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content='shizuoka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content='jp Shizuoka University Hamamatsu, Shizuoka, Japan ABSTRACT This document presents endeavors to represent emotion in a com- putational cognitive architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' The first part introduces research organizing with two axes of emotional affect: pleasantness and arousal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Following this basic of emotional components, the docu- ment discusses an aspect of emergent properties of emotion, show- ing interaction studies with human users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' With these past author’s studies, the document concludes that the advantage of the cogni- tive human-agent interaction approach is in representing human internal states and processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' CCS CONCEPTS Human-centered computing → HCI theory, concepts and mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' KEYWORDS cognitive architecture, emotion, arousal, valence 1 INTRODUCTION This document begins by asking the following question: How can emotion emerge in a computational system?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' This is a kind of ultimate question that has attracted enormous numbers of scientists and engineers, including the author himself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Toward the complete answer to the question, the author has devel- oped several models of mental functions (possible components of emotion process) in ACT-R (Adaptive Control of Thought-Rational [1]), which is one of the most widely used cognitive architectures in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Cognitive architectures generally integrate knowledge concern- ing the human mind in the form of computer programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' The knowl- edge accumulated in ACT-R ranges from perceptual-motor com- ponents to abstract and goal-related concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Varieties of mental functions are controlled by symbols stored in modules (correspond- ing brain regions) and subsymbolic parameters (corresponding neu- rotransmitters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' By utilizing these, this architecture aims to realize human-level activities in every field of human life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' The author considers that the above characteristic of the archi- tecture is crucial to answering the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Thus, this document Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Copyrights for third-party components of this work must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' For all other uses, contact the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' CHAI ’22, December 5, 2022, Christchurch, New Zealand © 2022 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' presents the author’s works utilizing ACT-R as ingredients of dis- cussions on the conditions for enabling emotion in a computational system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' 2 ISSUES OF EMOTIONAL PROCESS As the background of the discussion, this section presents the au- thor’s view on the emotion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' It is assumed that the emotion process relies on subcortical brain regions, such as the amygdala, insula, and basal ganglia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' The process was considered to be the product of an adaptation to ancestors’ surrounding environments, because these regions were formed early in the evolution of the brain [4, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' However, modern human emotion is more complex than the purely physical processes in the following senses: (1) Emotion is not a static entity but rather emergent properties accompanied by dynamic interaction with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' This means a human’s emotional response always fluctuates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' (2) Our environment has drastically changed from the environ- ment in which our ancestors lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Therefore, many emo- tional mechanisms have become maladaptive in the modern age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' (3) This biological process is somehow modulated by an inten- tional strategy, as many theories of emotion have suggested [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Therefore, there is room for intervention in the emo- tion process by our will or technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' 3 REPRESENTING EMOTION IN ACT-R Even through such complexities, it is possible to model the basis of emotion (affect) using the fundamental axes, arousal and pleasant- ness, as presented in Russel [16]’s circumplex model (Fig 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' The present author’s approach begins with these simple components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' The following studies represent these components with primitive cognitive functions, such as activation noise and pattern matching, implemented in ACT-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content='1 Representation of Emotional Arousal According to a dictionary in psychology [17], arousal is defined as follows: a state of physiological activation or cortical respon- siveness, associated with sensory stimulation and ac- tivation of fibers from the reticular activating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' As indicated by this definition, past researchers have frequently con- nected arousal with the activation (attentional) process, especially the degree of concentration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=', [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' In a concentrated situation, humans can continue monotonous tasks accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' However, as such a process long lasts long, people get bored and begin to think about things outside of the task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=', mind-wandering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content='00003v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content='HC] 28 Dec 2022 CHAI ’22, December 5, 2022, Christchurch, New Zealand Morita High arousal Low arousal Unpleasant Rewards Fluctuation Figure 1: Axes of emotional affect and model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' From this phenomenon and the consideration that emotional arousal relates to biological fluctuations affecting the cognitive process, the author and their colleagues have represented arousal as the noise factor for memory activation in ACT-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Controlling this subsymbolic parameter, the authors have demonstrated changes in memory recollection [10] and task goal switching [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' This view is consistent with the discussion in which emotion is a modulator of cognitive architecture [5, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content='2 Representation of Pleasantness People feel joy when receiving a reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Thus, the pleasantness axis can be considered as a reward in reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Various triggering events for rewards can be assumed in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Among them, internally generated ones are important to developing autonomous agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Based on the above assumptions, Nagashima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' [12, 13] devel- oped a model of intrinsic motivation, assuming a pattern discovery to be a source of curiosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' In the ACT-R model, patterns in the data are discovered with the process of pattern matching, and the experience of the pattern matching is utilized to build procedural- ized rules (the compilation of production rules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' As a task proceeds, opportunities for pattern matching (internal rewards) gradually decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Thus, the model can explain how intrinsic motivation decreases with experience and increases with discovering novel patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' This pattern-discovery-focused view of intrinsic motiva- tion is consistent with discussions in the entertainment industry [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' In addition, it is supported by the theory emphasizing the role of pattern-seeking in the history of human civilization [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' 4 NEEDS OF INTERACTION From the discussion so far, the importance of environment for emotion stands out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Capturing the full complexities of the real- world dynamics of the two mechanisms (arousal and pleasantness) requires environmental changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Among various environmental factors, the existence of other organisms seems most crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' In other words, the author considers that human emotion can be modeled only when the computational system actually interacts Figure 2: Framework of interacting human emotion with machine emotion [11] with humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' By interacting with humans, the system is able to learn human’s emotion generation and expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Based on this consideration, the author and colleagues have developed several interactive systems [6, 11], implementing an emotion model as a component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Those systems receive the user’s biological signals such as heart rates to automatically modulate the abovementioned ACT-R parameters (noises and rewards) for guiding the user’s emotion to an optimal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Especially, Morita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' [11] demonstrated that a web advertisement system containing an ACT-R memory model could prevent human repetitive thinking (rumination) when the model behaves in a counterbalanced manner (Fig 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' The author believes that such a system will eventually lead to a new human homeostatic process with the help of artificial emotional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' 5 CONCLUSION This document presents the author’s attempts to answer the ul- timate question about the computational conditions that enable emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Future work must represent the ideas presented in the document as a general framework and evaluate it in human exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' The author considers that such a cognitive-architecture- based framework is advantageous in constructing trustful human- agent relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Academic knowledge implemented in architecture is the result of continuous endeavors in human history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' Including the formal knowledge agreed in human society is an essential in- gredient of making common ground between humans and artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' ACKNOWLEDGEMENT This document summarizes ideas obtained from past collaborative studies with colleagues at Nagoya University, Shizuoka University, collaborators from Panasonic Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' and Mazda Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=', and mem- bers of the Applied Cognitive Modeling Lab (ACML) at Shizuoka university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' The author thanks everyone for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAyT4oBgHgl3EQfOPaQ/content/2301.00003v1.pdf'} +page_content=' User Model Stress Stress Activation HRV noise Synchronize Counterbalance Relax RelaxEmotion in Cognitive Architecture: Emergent Properties from Interactions with Human Emotion CHAI ’22, December 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0000000000000000000000000000000000000000..b613377bd9ff870d4e26c491bcb5be23ea98da04 --- /dev/null +++ b/w9FRT4oBgHgl3EQfhDe7/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:648b7e7650a3f77467fac5eb72396ff0decc9a20fbaf24cfea23768c56c57ca5 +size 392411 diff --git a/wtAyT4oBgHgl3EQfavdq/content/tmp_files/2301.00248v1.pdf.txt b/wtAyT4oBgHgl3EQfavdq/content/tmp_files/2301.00248v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c7c2b534237b064d62cc57209f340defec345a06 --- /dev/null +++ b/wtAyT4oBgHgl3EQfavdq/content/tmp_files/2301.00248v1.pdf.txt @@ -0,0 +1,1110 @@ +Highlights +Nowcasting Stock Implied Volatility with Twitter +Thomas Dierckx +, Jesse Davis +, Wim Schoutens +• Next-day movements in stock implied volatility can be predicted using random forests. +• Attention and sentiment features extracted from Twitter improve predictive performance. +• Predictive performance varies significantly across the 11 traditional stock market sectors. +• Implied volatility regimes identified by hidden Markov models provide actionable insights +on when the proposed approach works best per stock market sector. +arXiv:2301.00248v1 [q-fin.CP] 31 Dec 2022 + +Nowcasting Stock Implied Volatility with Twitter +Thomas Dierckx +a,b,∗, Jesse Davis +b, Wim Schoutens +a +aDepartment of Statistics and Risk, KU Leuven, Celestijnenlaan 200B, Leuven, 3000, Belgium +bDepartment of Computer Science, KU Leuven, Celestijnenlaan 200A, Leuven, 3000, Belgium +Abstract +In this study, we predict next-day movements of stock end-of-day implied volatility using random +forests. Through an ablation study, we examine the usefulness of different sources of predictors +and expose the value of attention and sentiment features extracted from Twitter. We study the +approach on a stock universe comprised of the 165 most liquid US stocks diversified across the 11 +traditional market sectors using a sizeable out-of-sample period spanning over six years. In doing +so, we uncover that stocks in certain sectors, such as Consumer Discretionary, Technology, Real +Estate, and Utilities are easier to predict than others. Further analysis shows that possible reasons +for these discrepancies might be caused by either excess social media attention or low option +liquidity. Lastly, we explore how our proposed approach fares throughout time by identifying +four underlying market regimes in implied volatility using hidden Markov models. We find that +most added value is achieved in regimes associated with lower implied volatility, but optimal +regimes vary per market sector. +1. Introduction +Today’s age is characterized by an ever-increasing connected and opinionated world. The +widespread adoption of social media has caused significant changes in the world across many +domains, and more are probably to follow. In the case of financial markets, participants now +have access to countless online platforms to share their thoughts and feelings on certain events. +Proponents of the Efficient Market Hypothesis (Fama, 1970) ought to be pleased. The advent +of mass media facilitates rapid information diffusion, possibly propelling markets into a higher +tier of price efficiency. However, behavioral economists would argue that this type of media +might very well influence investors and incite herd behavior which in turn induces inefficiency +(e.g. Baker and Wurgler, 2006; Chiang and Zheng, 2010). Theory aside, this new wealth of +information has not escaped the notice of the financial establishment. Indeed, data providers +such as Bloomberg and Refinitiv now offer extensive social media indicators to help financial +institutions navigate this new world. Although the competitive edge that resides in these alterna- +tive data sources remains veiled in secrecy, an abundance of academic studies have already tried +⋆Declarations of interest: none +∗Corresponding author +Email addresses: thdierckx@gmail.com (Thomas Dierckx +), jesse.davis@kuleuven.be (Jesse Davis +), +wim.schoutens@kuleuven.be (Wim Schoutens +) +Preprint submitted to Journal of Empirical Finance +January 3, 2023 + +quantifying the interplay between social media and certain financial variables, providing insight +into the predictive power of the masses. +Existing research has mainly focused on the Twitter platform and its influence on three promi- +nent financial variables: stock price (e.g. Groß-Klußmann et al., 2019; Schnaubelt et al., 2020), +realized volatility (e.g. Karagozoglu and Fabozzi, 2017), trading volume (e.g. Guijarro et al., +2019) or a combination thereof (e.g. Oliveira et al., 2017; Li et al., 2018). Remarkably, cur- +rent literature completely overlooks the interaction between social media and the market implied +volatility of stocks. Derived from option prices, this variable is deemed to be one of the more +important parameters in the world of derivatives. In contrast to historical volatility, this is a +forward-looking metric that indicates how much risk the market expects a certain asset to exhibit +in the coming period. As this variable serves as a proxy for both market sentiment and option +contract prices, the ability to predict its movements would be advantageous for the practice of +asset management and market making alike. +Most prevailing studies in this domain have two important methodological shortcomings. +First, analysis is typically performed on either a handful of arbitrarily chosen stocks or indices +tracking the entire market (e.g. Groß-Klußmann et al., 2019), omitting sector idiosyncrasies in +the process. Second, hypotheses are commonly tested on a relatively small time window ranging +from a month (e.g. Bollen et al., 2011) to a few years (e.g. Schnaubelt et al., 2020). However, the +ever-changing nature of financial markets warrants a closer look into how the interplay between +social media patterns and financial variables evolves over longer periods of time. As patterns +may emerge and dissipate over time, a crucial aspect of analysis is often left out. +The contribution of this study is threefold. First, to the best of our knowledge, we are the +first to investigate to what extent a stock its one-day ahead movement in implied volatility can be +predicted using machine learning on different combinations of feature sources, including Twitter. +Second, instead of arbitrarily choosing a handful of stocks for our study, we diversified our stock +universe across the 11 traditional US stock market sectors yielding 165 stocks in total. This al- +lowed us to measure and explore the variability in predictive performance present among sectors. +Lastly, we examined predictive performance on an out-of-sample period spanning January 1st, +2013 till March 1st, 2019. This period is significantly larger than many other studies and gave us +the opportunity to not only be more robust, but also examine predictive performance throughout +time. Instead of performing a year by year analysis of predictive performance, we used hidden +Markov models to identify four regimes in the implied volatility of a stock and gauged whether +performance varies across them. We argue that this alternative quantification of time yields more +actionable insight as it allows practitioners to better anticipate future performance. +2. Preliminaries +This section presents background information on the key components used in this study. First, +Section 2.1 explains market implied volatility and its relation to the world of derivatives. Sec- +ond, Section 2.2 describes the random forest machine learning model which is used to perform +predictions. Lastly, Section 2.3 describes the hidden Markov model which is used to quantify +regimes in market implied volatility. +2.1. Market Implied Volatility +In the world of derivatives, options are one of the most prominent types of financial instru- +ments. As sellers of options are exposed to risk for the duration of the contract, they want to be +2 + +properly compensated. Measuring this risk requires considering the expected price fluctuations +of the underlying asset over the duration of the contract. This expectation is better known as +implied volatility and it varies with the strike price and duration of an option contract. To obtain +a more general measure, the implied volatility of option contracts that expire on the same date +can be combined into a single implied volatility measure. A famous example of this is the CBOE +Volatility Index, which combines the implied volatility of different option contracts on the SPX +into an index that is better known as the VIX. +More concretely, the VIX is a measure of expected price fluctuations in the S&P 500 Index +over the next 30 days. It is famously known as the fear index and is considered a reflection of +investor sentiment on the condition of the market. Equation 1, taken from the VIX white paper +(CBOE, 2015), shows how to compute the VIX for a given term T: +VIX = 100 × +� +2 +T +� +i +∆Ki +K2 +i +eRT Q(Ki) − 1 +T +� F +K0 +− 1 +�2 +(1) +where: +T += is time to expiration +F += is the forward index level derived from the index option prices +K0 += is the first strike below the forward index level F +Ki += is the first strike price of the ith out-of-the-money option: a call if Ki > K0, a put if +Ki < K0, or both put and call if Ki = K0 +R += is the risk-free rate to expiration +∆Ki += is the interval between strike prices +Q(Ki) = is the midpoint of the bid-ask spread for each option with strike Ki +The equation for computing the VIX is applicable to any asset where option contracts are +available. Although this measure can be calculated for any arbitrary term, the duration of the +option contracts will seldom match the chosen term T. Indeed, option contracts typically have +fixed expiration dates and there is no guarantee that there are option contracts available with a +duration equal to the given term. To overcome this obstacle, the VIX is first calculated for the +option contracts expiring right before and after the desired target date. The VIX for the given +term can then be calculated by linearly interpolating between the two computed measures, as +outlined in (CBOE, 2015). +2.2. Random Forests +Random forests (Breiman, 2001) are a popular machine learning approach for learning a pre- +dictive model. They consist of multiple different decision (or regression) trees whose predictions +are combined into one final prediction. The combination is typically done by taking the mode +(or average) of all outputs. While several variations exist for learning a random forest, all of +them are relatively straightforward. We summarize one of many popular procedures. Given data +D = {(xi, yi)}n +i=1, where each xi has F features and for k = 1 . . . K trees: +1. Obtain subset d by sampling m < n examples with replacement from D. +2. Train a decision tree on d using a random subset features f ⊆ F, i.e. using CART (Breiman +et al., 1984). +3 + +The prediction for a regression problem can then be obtained by: +ˆyi = 1 +K +K +� +k=1 +fk(xi) +(2) +where f is a function in the set of all possible decision trees and K is the total number of trees in +the ensemble. +The advantages of random forests include that they are fast to build, are not affected by +feature scaling, are robust to irrelevant predictors and noisy data (Khoshgoftaar et al., 2011). +Moreover, their method of constructing an ensemble model by randomly subsampling both data +points and features during the learning process helps decorrelate the predictions made by the +individual trees, which in turn reduces overfitting on the training data. +2.3. Hidden Markov Models +Hidden Markov models (HMM) are a generative approach for modeling systems that follow +a Markov process (Rabiner and Juang, 1986). The main assumption is that while this process +Z is hidden, it can be learned from an observable sequence X whose behaviour depends on Z. +More formally, the HMM models the joint distribution of a sequence of hidden states Z and +observations X described by: +P(Z1:K, X1:K) = P(Z1)P(X1|Z1) +K +� +t=2 +P(Zt|Zt−1)P(Xt|Zt) +(3) +Given the number of hidden states K and observed sequence X, the model is fully determined +by its parameters π, A, and B which represent the initial state distribution, state transition model, +and emission probabilities model, respectively. The initial state distribution is a K × 1 vector de- +noting the probabilities that the process is each of the K states in the first timestep. The transition +model is a K × K stochastic matrix where each element Ai, j denotes the probability of transi- +tioning from state Zt−1,i to Zt, j where i, j ∈ {1, . . . , K}. Lastly, the emissions probability model is +a M × K matrix, with M representing the number of distinct observations, whose elements Bk, j +denote the probability of observing Xt,k given state Zt, j. +The three key tasks associated with hidden Markov models are: +1. What is the probability that a sequence of observations X was generated by a given HMM? +2. Given an HMM, what sequence of hidden states Z best explains a given sequence of ob- +servations X? +3. Given a sequence of observations X, learn an HMM with parameters π, A, and B that would +generate them. +The first two tasks can be solved using dynamic programming using the forward-backward +algorithm (Chang and Hancock, 1966) and Viterbi (Forney, 1973) algorithm, respectively. The +third problem is solved by the Baum-Welch algorithm (Baum, 1972) which uses an iterative +expectation-maximization approach. +3. Methodology +The main goal of this study is to explore the following questions: +4 + +1. To what extent are one-day ahead movements in end-of-day implied volatility predictable, +and do features extracted from Twitter improve performance? +2. Does performance vary across the 11 different stock market sectors? If so, are there any +obvious factors that might explain this variability? +3. Can we identify underlying market regimes in implied volatility that influence the perfor- +mance of our proposed approach? +We tackle the first question by performing an ablation study using random forests on feature +configurations including stock price, stock implied volatility, and Twitter features. The study +encompasses a universe of 165 stocks over an out-of-sample period spanning January 1st, 2013 +till March 1st, 2019. To examine the second question, we diversified our stock universe over the +11 traditional stock sectors and grouped predictive performance by stocks belonging to the same +sector. The third and last question was studied by using a hidden Markov model to identify four +distinct implied volatility regimes per stock, after which predictive performance was grouped by +regime. +The next few sections explain our methodology in more detail. First, Section 3.1 outlines +the stock universe we used for our study. Second, Section 3.2 explains how we obtained the +relevant data for each stock and how we constructed features for prediction. Section 3.3 and +Section 3.4 then respectively show how we used machine learning to predict our target variable +and how we evaluated the performance of the approach. Lastly, Section 3.5 explains how we used +hidden Markov models to identify regimes in implied volatility which we later use to evaluate +our prediction performance through time. +3.1. Stock Universe Selection +In order to obtain a diversified universe of stocks, we looked at the popular SPDR and Van- +guard Electronic Traded Funds (ETF) that track the 11 traditional US stock market sectors. For +each sector, we selected the 15 most liquid stocks based on their average daily dollar-weighted +option volume for a total of 165 stocks. Some stocks were excluded due to stock splits (i.e. we +kept GOOG and dropped GOOGL), a late introduction to the stock market (i.e. PYPL, ROKU, +and SNAP only got introduced after 2015), and ambiguous names making it hard to obtain rele- +vant tweets (i.e. DOW is a chemical company but also a common alias for the Dow Jones Index). +Note that we replaced the excluded stocks to maintain 15 stocks per sector for our study. Table 1 +provides a concise overview of our stock universe. Refer to Appendix A for a full overview of +which stocks were selected per market sector. +3.2. Data Acquisition and Feature Generation +We consider data ranging from January 1st, 2011 through March 1st, 2019 for three data +sources: +1. Stock price data which consists of historical end-of-day adjusted closing prices for each +stock in our universe downloaded from Yahoo Finance. +2. Option contract price data which consists of historical end-of-day option chains for each +stock in our universe obtained from IVolatility. +3. Twitter data which consists of all relevant tweets published for each stock. These were +collected by filtering on cashtags, which are popular string identifiers authors use to in- +dicate their message is about a certain stock (i.e. a tweet about the Apple stock typically +contains $AAPL). In contrast to other research, we did not employ additional filtering +5 + +Table 1: This table presents the 11 different stock market sectors together with their corresponding SPDR ETF symbol +and number of stocks considered in this study. The symbols are used to denote sectors throughout this paper, but are not +indicative of stocks only belonging to the SPDR ETF portfolio. +Symbol +Sector +Selected Stocks +XLC +Communication Services +15 +XLY +Consumer Discretionary +15 +XLP +Consumer Staples +15 +XLE +Energy +15 +XLF +Financials +15 +XLV +Healthcare +15 +XLB +Materials +15 +XLI +Industrials +15 +XLK +Technology +15 +XLRE +Real Estate +15 +XLU +Utilities +15 +techniques to discard potential spam. Most additional filtering rules appear arbitrary and +there seems to be no clear evidence of their validity. +In total, four features were extracted per stock for each trading day. First, we simply used +the end-of-day adjusted closing price from the stock price data. Second, we calculated the end- +of-day 30-day implied volatility using the VIX method on the option contract data. Third and +last, we derived two numerical features from our textual Twitter corpus: end-of-day total tweet +publication count and end-of-day average sentiment polarity. The former represents the total +number of published tweets on a given day. The latter was obtained by performing sentiment +analysis using VADER (Hutto and Gilbert, 2014), a lexicon- and rule-based sentiment model that +is specifically well-tailored to social media text, on individual tweets. This yields a sentiment +polarity score s ∈ [−1, 1] for each tweet, which was then used to compute the daily average. +In an effort to capture temporal information residing in the original feature timeseries, we +generated two additional predictors per feature. To this end, the daily difference (or first-order +difference) and the difference between the daily value and its exponential moving average of the +last 10 trading days was taken. Table 2 outlines the different data sources and their features used +in this study. Note that the original adjusted closing price was omitted, as this is typically a +non-stationary variable offering little value to a prediction model. +Table 2: This table provides a summary of the features considered per data source. The first row indicates what original +features were extracted, whereas the last three rows indicate (*) which features were considered for the actual study. +Note that the last two rows denote a specific feature engineering technique applied to the original feature. +Stocks +Options +Twitter +Extracted +Adj. Closing Price +Implied Volatility +Count, Sentiment +Original +* +* +1st Order Diff. +* +* +* +EMA(10) Diff. +* +* +* +6 + +3.3. Predicting Movements in Implied Volatility +This study aims to predict one-day ahead movements in a stock’s 30-day implied volatility. +Concretely, given information at the end of trading day t, we predict whether implied volatility +will have moved up or down by the end of next trading day t + 1. To do so, we construct a binary +target variable for day t as: +yt = +������� +1, +if (ivolatilityt+1 − ivolatilityt) > 0. +0, +otherwise. +(4) +where ivolatilityt denotes the end-of-day implied volatility on day t. +In order to predict our target variable, we used random forest classifiers even though more +powerful models may exist. For example, the highly popular gradient boosted trees (Friedman, +2001) have been shown to generally perform slightly better than random forests (Caruana et al., +2008; Caruana and Niculescu-Mizil, 2006). However, they are very sensitive to hyper-parameter +configurations and require longer runtimes for training. The main goal of this study is not to +maximize predictive performance, but rather probe the feasibility of our proposed approach. In +addition, it has been suggested that random forests might generally work better on noisy data +(Khoshgoftaar et al., 2011), which is especially convenient when working on financial data. +Lastly, we did not consider techniques from the domain of deep learning due to the complexity +of the models and relatively small number of data points in our study. +Ultimately, we used 64 distinctive random forest configurations built using Sklearn. Each +random forest was built with 1000 trees and a unique combination of different hyper-parameters +that control maximum tree depth, the minimum number of samples required to split an internal +node, and the minimum number of samples required to be in a leaf node. Each individual tree was +built by sampling the training dataset (with replacement) and only considering a random number +of +� +f features where f denotes the total amount of features. The models were trained on a +temporally ordered feature matrix X of dimension T ×K, obtained by using any subset of features +K from Section 3.2 and period T. Table 3 specifies the possible random forest configurations +considered in this study. +Table 3: This table presents the different possible values considered for different hyper-parameters available in the random +forest implementation of Sklearn. The default value is used for hyper-parameters not listed. +Hyper-parameter +Values +n estimators +{1000} +max depth +{4, 6, 8, 10} +min samples split +{5, 10, 15, 20} +min samples leaf +{1, 3, 5, 8} +random state +{42} +bootstrap +yes +max features +sqrt +3.4. Experimental Evaluation +We evaluated the different random forest configurations using walk-forward validation, a +cross-validation technique designed specifically for temporally ordered data. Classical cross- +validation methods assume observations to be independent. This assumption does not necessarily +7 + +Table 4: Example of expanding walk forward validation without where ti represents the feature vector of trading day i. +In this example, a training window with an initial size s = 3 is taken together with a testing window of size k = 1. We +therefore consistently use the feature vectors of past trading days to train a model (underlined) and subsequently test said +model on trading day t + k (bold). +Iteration +Variable roles +1 +t1 t2 t3 t4 t5 · · · tn +2 +t1 t2 t3 t4 t5 · · · tn +... +... +m +t1 · · · tn−3 tn−2 tn−1 tn +hold for timeseries data, which inherently contains temporal dependencies among observations. +To this end, the dataset is repeatedly split up in training and test sets where temporal order is +accounted for. In our case, we used an expanding window of initially 504 trading days to train +the models, after which performance was measured on the next out-of-sample 40 trading days. +Table 5 shows an example of this method where ti denotes the feature vector corresponding to +trading day i. Note that in this scenario, when given a total of n observations, an expanding +training window of length t and an out-of-sample test window of length k, you can construct a +maximum of n − t − k different train-test splits. Ultimately, each configuration its performance is +averaged across all folds. We measured performance with the area under the receiver operating +characteristic curve metric (AUC hereafter). +Table 5: Example of walk-forward validation where ti represents the feature vector of trading day i. In this example, a +training window with an initial size s = 3 is taken together with a testing window of size k = 1. We therefore consistently +use the feature vectors of past trading days to train a model (underlined) and subsequently test said model on trading day +t + k (bold). +Iteration +Variable roles +1 +t1 t2 t3 t4 t5 · · · tn +2 +t1 t2 t3 t4 t5 · · · tn +... +... +m +t1 · · · tn−3 tn−2 tn−1 tn +3.5. Analyzing Performance through Time with Hidden Markov Models +The ever-changing nature of financial markets makes it hard to find approaches that con- +sistently work well. The ability to time approaches therefore becomes an interesting perk. In +an effort to evaluate how our proposed approach weathers the evolution of the financial market +through time, we look at its performance under different market regimes. +We quantified market regimes as different states in mean implied volatility using a hidden +Markov model. For each stock, a different HMM model was trained on its end-of-day im- +plied volatility timeseries dating from January 1st, 2007 till December 31st, 2012 and was used +out-of-sample thereafter. Analogous to a study performed by Soci´et´e G´en´erale (Daviaud et al., +2020), four different regimes were identified corresponding to low, medium, high, and very high +mean implied volatility. Table 6 specifies the hyper-parameter configuration used to build hidden +Markov models using the hmmlearn Python package. +8 + +Table 6: This table presents the hyper-parameter configuration used for building hidden Markov models using the hmm- +learn package. The default value is used for hyper-parameters not listed. +Hyper-parameter +Values +n components +4 +n iter +100 +random state +42 +emissions +Gaussian +algorithm +Viterbi +4. Experimental Results and Discussion +In this section, we present and discuss our experimental results. First, Section 4.1 shows the +results of our ablation study where in total seven different feature configurations were consid- +ered. Section 4.2 then builds on these results by looking at the performance of the best feature +configuration per market sector. Lastly, Section 4.3 looks at predictive performance across dif- +ferent implied volatility regimes. Recall that throughout the remainder of this section we may +denote the 11 different stock market sectors by the symbol of their equivalent SPDR ETF tracker. +This is solely done out of convenience and is not indicative of stocks only belonging to said ETF +portfolio. Refer to Table 1 for an overview of the sector symbols. +4.1. Ablation Study +The first objective of this study was to investigate to what extent daily movements in end- +of-day implied volatility can be predicted. To this end, we obtained 11 different features from +three different data sources (Section 3.2) on which we built random forest classifiers to predict +said target variable. We assessed the effectiveness of the different data sources by performing an +ablation study where 7 different scenarios were considered, shown in Table 7. In total, one-day +ahead movements in implied volatility were predicted for 165 stocks (Section 3.1) spanning an +out-of-sample period from January 1st, 2013 till March 1st, 2019. +Table 7: This tables shows the different feature scenarios considered in our ablation study together with their total number +of features. Note that the third column indicates the usage of both original and derived features from the given feature +source. +Scenario +Feature Source +Features +1 +Stock Price +2 +2 +Stock Price, Tweets +8 +3 +Implied Volatility +3 +4 +Implied Volatility, Tweets +9 +5 +Tweets +6 +6 +Stock Price, Implied Volatility +5 +7 +Stock Price, Implied Volatility, Tweets +11 +We compared the predictive performance of different scenarios to that of a stratified dummy +classifier, which makes the comparison more rigid than using a simple random classifier. Indeed, +implied volatility tends to go down more often than it goes up. This causes a stratified dummy +9 + +classifier to achieve a median AUC of 51.8% across all 165 stocks versus a 50.0% achieved by a +fully random one. +Table 8 displays the median AUC achieved for each scenario averaged over the entire selected +stock universe and the difference in AUC between our approach and the stratified dummy clas- +sifier. These results provide empirical evidence that end-of-day movements in implied volatility +can indeed be predicted. All possible feature scenarios perform better than a purely random +classifier that achieves a median of 50.0% AUC. Moreover, 4 out of 7 scenarios outperform the +stratified dummy classifier that achieves a median of 51.8% AUC. The commonality among these +improved scenarios is the use of implied volatility features, indicating that this is an important +source of information. Moreover, including features derived from tweets always yielded a better +median performance (S2 versus S1, S4 versus S3, and S7 versus S6). This implies there is indeed +a predictive interplay between information on Twitter and future implied volatility. Lastly, using +all possible features (S7) yielded the best result overall, suggesting there are predictive patterns +among all three feature sources. +Table 8: This table displays the median predictive performance across all 165 stocks per feature configuration obtained +by predicting daily end-of-day movements in implied volatility over the period of January 1st, 2013 to March 1st, 2019. +Moreover, the second row shows how much the proposed approach does better than the stratified dummy classifier. +S1 +S2 +S3 +S4 +S5 +S6 +S7 +Median AUC +51.1 +51.6 +53.6 +54.2 +50.9 +54.3 +55.1 +Improvement +-0.6 +-0.1 ++1.9 ++2.5 +-0.8 ++2.7 ++3.4 +4.2. Predictive Performance across Sectors +In this section we look at the best performing feature configuration (S7) and its performance +variability across 11 different stock market sectors. Figure 1 shows a box plot where the per- +formance improvement of our proposed approach versus the stratified dummy classifier for each +individual stock is grouped by sector. The results were obtained on an out-of-sample period +spanning January 1st, 2013 till March 1st, 2019. +It is clear that our proposed methodology is generally able to beat the stratified dummy clas- +sifier across all different sectors. The approach beats the dummy classifier on 148 out of 165 +stocks. However, there is a considerable amount of variability present across different sectors. +Indeed, the approach does significantly better on XLRE and XLU, but predictions on XLC, XLY, +and XLK also do better comparatively. In contrast, XLI and XLB seem to lack in performance. +The next two subsections will showcase a preliminary attempt to partially explain this variability +in performance. +4.2.1. The Effect of Option Liquidity +The results from the previous section indicate that predictions on stocks from both XLRE +and XLU do significantly better. Remarkably, it turns out that stocks in these two sectors are +also significantly less liquid compared to other sectors. Figure 2 respectively shows a box plot of +median option liquidity per sector, measured by the average daily dollar amount traded in options, +and a regression plot where the relationship between liquidity and performance improvement +versus the dummy classifier is outlined. +These results seem to suggest that there is indeed a weak negative correlation between pre- +dictive performance and option liquidity, implying that less liquid stocks are easier to predict. +10 + +Figure 1: This box plot shows the performance improvement of our proposed methodology using feature configuration +S7 versus a stratified dummy classifier on individual stocks grouped by sector. The red dotted line represents the min- +imum threshold necessary to beat the stratified dummy classifier. The blue dotted line represents the overall median +improvement of our proposed approach versus the stratified dummy classifier. +Figure 2: This figure presents a box plot (left) where the median of daily stock option liquidity is grouped per sector and +a regression plot (right) where the relationship between stock option liquidity and performance improvement versus the +dummy classifier is outlined. +We hypothesize that the relatively undersized liquidity of both XLRE and XLU in the options +market is possibly accompanied by a less efficient price discovery process. This in turn might +cause these markets to reflect new information more slowly, making them easier to predict with +the information at hand. However, we note that is only one possible explanation and many other +factors might lie at the basis of this phenomenon. +11 + +dummythreshold +0.125 +-medianimprovement +0.100 +(AUC) +0.075 +Improvement +0.050 +0.025 +0.000 +0.025 +-0.050 +XLC +XLY +XLP +XLE +XLF +XLV +XLI +XLB +XLRE +XLK +XLU +Sector16 +median (overall) +0.125 +0.100 +Improvement +0.075 +12 +0.050 +0.025 +10 +Daily +0.000 +8 +0.025 +-0.050 +XLCXLYXLPXLEXLFXLVXLIXLBXLREXLKXLU +8 +10 +12 +14 +16 +Sector +DailyDollarVolume (log)4.2.2. The Effect of Twitter Attention +Lower liquidity might partially explain why sectors such as XLRE and XLU seem easier to +predict, but it certainly does not tell the whole story. Indeed, predictions on stocks from XLC, +XLY, and XLK, examples of very liquid sectors, also do better comparatively. Here, we hypoth- +esize that this might be due to the attention they receive on Twitter. Figure 3 respectively shows +a box plot of the median daily tweets published on stocks grouped per sector and a regression +plot where the relationship between liquidity and daily tweets is outlined. +Figure 3: This figure presents a box plot (left) where the median of daily tweet publication of stocks is grouped per sector +and a regression plot (right) where the relationship between a stock its daily tweets and option liquidity is outlined. +The box plot on the left of Figure 3 shows that stocks in XLC, XLY, and XLK receive signif- +icantly more attention on Twitter than others. Note that the plot demonstrates a striking resem- +blance with the box plot showing liquidity per sector in Figure 2. Indeed, the regression plot on +the right shows a very strong correlation between attention on Twitter and liquidity. These find- +ings seem to suggest that prediction is easier on stocks that are more popular on social media. We +investigated this phenomenon further by looking at the improvement in predictive performance +caused by features extracted from Twitter per sector. More concrete, we looked at the difference +in performance between feature configuration S6, which uses stock and options features, and S7, +which combines features from S6 and Twitter features (Section 4.1). Figure 4 shows the median +improvement of using Twitter features per stock grouped by sector. +With exception of XLE, most sectors seem to have a sizeable number of stocks that benefit +from using social media features. This is no surprise considering the results presented in Section +4.1. However, it seems that sectors that receive more social media attention seem to benefit the +most. For example, the three sectors XLC, XLY, and XLK that are most popular also benefit +more consistently followed closely by XLV. Two reasons might explain these results. First, pre- +vious research has hinted at social media inciting herd behavior and emotional reactions among +investors, possibly driving inefficiency (e.g. Wang and Wang, 2018; Oliveira et al., 2017; Groß- +Klußmann et al., 2019). If this is true, it makes sense that Twitter features provide more added +value for popular stocks. Second, our sentiment extraction technique on tweets is imperfect and +might impact these results as well. Without carefully filtering out tweets that are advertisements +or spam, we rely on the law of large numbers to correctly estimate a stock its sentiment for a +12 + +.median(overall) +16 +(601) +Tweets +5 +12 +4 +10 +Daily +3 +8 +n +: +XLC XLY XLP +XLE +XLE XLY XLI +XLBXLREXLKXLU +2 +3 +4 +5 +6 +7 +Sector +Daily Tweets (log)Figure 4: This figure shows the performance improvement of using Twitter features together with features from S6, +versus only using features from S6. The red dotted line represents the minimum threshold necessary to beat the feature +configuration of S6. +given day. Naturally, daily sentiment estimation will therefore be better on stocks that are more +heavily tweeted about. +4.3. Performance across Implied Volatility Regimes +In this section, we look at the best-performing feature configuration (S7) and its performance +variability across four different market regimes in implied volatility, identified by using a hidden +Markov model. Table 9 shows the median of all results across 165 stocks where, for each of the +four regimes, the columns respectively show how many days were spent in a regime, what the +average implied volatility was, the AUC of a stratified dummy classifier, and how much better +our approach did compared to the latter. +Table 9: This table shows the median number of days a stock resided in one out of four implied volatility regimes, together +with each regime’s mean implied volatility, stratified dummy performance and the improvement of our approach. In total, +165 stocks were considered over a period spanning January 1st, 2013 till March 1st, 2019. +Frequency +Implied Volatility +Dummy AUC +Improvement +Low +392 +18.6 +50.75 ++3.15 +Medium +452 +22.3 +50.65 ++3.83 +High +411 +26.7 +52.12 ++3.84 +Very High +231 +35.3 +54.92 ++1.47 +In general, stocks seem to reside more in the lower implied volatility environments. The +stratified dummy classifier also appears to perform worse here, implying that up and down move- +ments are about equal in occurrence. However, the dummy classifier performance picks up as +implied volatility increases. This makes sense from a theoretical perspective, as implied volatil- +ity is deemed to be a mean reverting process characterized by big up movements after which the +variable slowly trails back down, resulting in more down movements. Although our approach +13 + +0.04 +.--improvementthreshold +0.03 +0.02 +(AUC) +0.01 +Improvement +0.00 +-0.01 +-0.02 +0.03 +0.04 +XLC +XLY +XLP +XLE +XLF +XLV +XLI +XLB +XLRE +XLK +XLU +Sectorseems to outperform the dummy classifier in all regimes, the added value seems to be most sig- +nificant in the low to high regimes. This may suggest that distressed markets are harder to predict +than their calmer counterparts. However, note that the improvement of our approach seems to +be slightly correlated with regime frequency. From a data science perspective, this might imply +that the model performs worse in less frequent regimes because it simply had fewer examples to +learn from. Lastly, Table 10 offers a more finer-grained analysis where the median improvement +for each regime per sector is shown. +Table 10: This table shows the median improvement of our approach compared to a stratified dummy classifier for stocks +grouped per market sector and implied volatility regime. The best and worst regime for each sector are respectively +indicated by bold and underlined text styles. +Low +Medium +High +Very High +XLB +1.87 +3.06 +1.02 +0.77 +XLC +3.57 +4.76 +5.27 +0.26 +XLE +2.55 +2.46 +4.0 +-2.55 +XLF +3.06 +4.08 +2.64 +-1.7 +XLI +0.05 +1.53 +0.17 +-1.5 +XLK +0.60 +2.38 +3.91 +5.70 +XLP +2.72 +3.83 +3.23 +1.53 +XLRE +8.5 +6.38 +4.93 +4.25 +XLU +6.63 +5.95 +5.36 +1.79 +XLV +4.25 +3.23 +1.62 +2.13 +XLY +2.04 +3.66 +5.95 +0.85 +Again, we remark a significant amount of variability in performance across both regimes and +sectors. Optimal implied volatility regimes seem to differ significantly for different sectors. In +contrast, with exception of XLK, sectors seem to comparatively do worse when implied volatility +is very high. As all sectors share roughly the same regime frequency distribution, no clear reason +emerges to explain the performance variability. One potential reason might be due to sector +idiosyncrasies. For example, defensive sectors such as XLRE, XLU, and XLV seem to have +lower optimal regimes than cyclical sectors such as XLC, XLK, and XLY. However, this is not +always the case. Lastly, we have no explanation for the significant difference between XLK +and the other sectors. Here, our approach performs best in the highest implied volatility regime +and worst in the lowest. This is in stark contrast with the other sectors where the opposite is +true. Perhaps the speculative nature of technology stocks is more sensitive to herd behaviour and +therefore investor irrationality. +5. Conclusion +In this study, we presented the first empirical evidence that one-day ahead movements of +end-of-day stock implied volatility can be predicted to a certain extent, and that attention and +sentiment features extracted from Twitter improve the performance of the approach. These alter- +native features were not able to predict implied volatility in isolation, but improved the approach +significantly when combined with predictors extracted from stock and options data. This sug- +gests that the interplay between these sources gives rise to predictive patterns. By conducting +14 + +our experiments on a diversified universe of 165 US stocks, we were able to assess the predic- +tive performance across 11 traditional stock market sectors and found that stocks in real estate, +utilities, consumer discretionary, communications, and technology were easier to predict than +others. Further analysis indicated that these differences could potentially be explained by market +inefficiencies caused by low option liquidity in the real estate and utilities sector, and excessive +Twitter attention in the consumer discretionary, communications, and technology sector. Lastly, +using hidden Markov models, we evaluated the predictive performance of our approach across +four different implied volatility regimes. Although it outperforms the dummy classifier in all +four regimes, we found that it yields the least improvement in the regime associated with the +highest average implied volatility. Moreover, we discovered that different stock market sectors +have different optimal regimes for the application of our approach. By analyzing performance +through the usage of regimes, we showed that this alternative quantification of time provides +additional insight into the performance of models which in turn could help better anticipate their +future performance. +15 + +Appendix A. Selected Stock Universe +This appendix contains more detailed information on the US stock universe used in this study. +Table A.11 outlines which stocks were chosen for each traditional market sector. +Table A.11: This table presents the US stock universe that was used in this study. Note that for each traditional market +sector we chose 15 stocks based on option liquidity. +Materials +FCX +X +NEM +CLF +MOS +DD +IP +CF +AA +NUE +LYB +VMC +SHW +BLL +WRK +Communications +FB +NFLX +GOOG +T +TWTR +VZ +DIS +CMCSA +EA +YELP +ATVI +DISH +CHTR +Z +TTWO +Energy +XOM +OXY +CVX +COP +SLB +HAL +VLO +EOG +APA +KMI +HES +MPC +MRO +RIG +WMB +Financials +BAC +JPM +C +GS +WFC +AIG +BX +MS +AXP +CME +MET +USB +SCHW +COF +BLK +Industrials +GE +BA +CAT +LMT +UPS +UNP +FDX +DE +MMM +HON +CSX +NSC +EMR +NOC +ETN +Technology +AAPL +NVDA +MSFT +INTC +MU +IBM +QCOM +CSCO +CRM +AMD +MA +V +ORCL +ADBE +TXN +C. Staples +PG +WMT +PM +KO +MO +CL +COST +PEP +HLF +WBA +GIS +MNST +TSN +CAG +KR +Real Estate +SPG +WY +AMT +EQIX +IRM +CCI +PSA +AVB +VTR +O +HST +DLR +PLD +MAC +EQR +Utilities +SO +EXC +AEP +DUK +NRG +NEE +FE +D +PPL +ED +ETR +EIX +CNP +NI +SRE +Healthcare +PFE +JNJ +GILD +LLY +MRK +ABT +BMY +AMGN +UNH +CVS +ABBV +ISRG +MDT +CI +DHR +C. Discretionary +AMZN +TSLA +MCD +HD +F +CMG +GM +SBUX +EBAY +NKE +TGT +LOW +BBY +LULU +MGM +16 + +References +Baker, M., Wurgler, J., 2006. Investor sentiment and the cross-section of stock returns. The Journal of Finance 61, +1645–1680. doi:https://doi.org/10.1111/j.1540-6261.2006.00885.x. +Baum, L.E., 1972. An inequality and associated maximization technique in statistical estimation for probabilistic func- +tions of Markov processes, in: Inequalities III: Proceedings of the Third Symposium on Inequalities, Academic Press. +pp. 1–8. +Bollen, J., Mao, H., Zeng, X., 2011. 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The impact of microblogging data for stock market prediction: using twitter to +predict returns, volatility, trading volume and survey sentiment indices. Expert systems with applications 73, 125–144. +doi:https://doi.org/10.1016/j.eswa.2016.12.036. +Rabiner, L., Juang, B., 1986. An introduction to hidden markov models. IEEE ASSP Magazine 3, 4–16. doi:https: +//doi.org/10.1109/MASSP.1986.1165342. +Schnaubelt, M., Fischer, T.G., Krauss, C., 2020. Separating the signal from the noise – financial machine learning +for twitter. Journal of Economic Dynamics and Control 114, 103895. doi:https://doi.org/10.1016/j.jedc. +2020.103895. +Wang, G., Wang, Y., 2018. +Herding, social network and volatility. +Economic Modelling 68, 74–81. +doi:https: +//doi.org/10.1016/j.econmod.2017.04.018. +17 + diff --git a/wtAyT4oBgHgl3EQfavdq/content/tmp_files/load_file.txt b/wtAyT4oBgHgl3EQfavdq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cbc4cc68b5a5e664159fbe5b954335dcf1d9662b --- /dev/null +++ b/wtAyT4oBgHgl3EQfavdq/content/tmp_files/load_file.txt @@ -0,0 +1,866 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf,len=865 +page_content='Highlights Nowcasting Stock Implied Volatility with Twitter Thomas Dierckx , Jesse Davis , Wim Schoutens Next-day movements in stock implied volatility can be predicted using random forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Attention and sentiment features extracted from Twitter improve predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Predictive performance varies significantly across the 11 traditional stock market sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Implied volatility regimes identified by hidden Markov models provide actionable insights on when the proposed approach works best per stock market sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='00248v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='CP] 31 Dec 2022 Nowcasting Stock Implied Volatility with Twitter Thomas Dierckx a,b,∗, Jesse Davis b, Wim Schoutens a aDepartment of Statistics and Risk, KU Leuven, Celestijnenlaan 200B, Leuven, 3000, Belgium bDepartment of Computer Science, KU Leuven, Celestijnenlaan 200A, Leuven, 3000, Belgium Abstract In this study, we predict next-day movements of stock end-of-day implied volatility using random forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Through an ablation study, we examine the usefulness of different sources of predictors and expose the value of attention and sentiment features extracted from Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' We study the approach on a stock universe comprised of the 165 most liquid US stocks diversified across the 11 traditional market sectors using a sizeable out-of-sample period spanning over six years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In doing so, we uncover that stocks in certain sectors, such as Consumer Discretionary, Technology, Real Estate, and Utilities are easier to predict than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Further analysis shows that possible reasons for these discrepancies might be caused by either excess social media attention or low option liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Lastly, we explore how our proposed approach fares throughout time by identifying four underlying market regimes in implied volatility using hidden Markov models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' We find that most added value is achieved in regimes associated with lower implied volatility, but optimal regimes vary per market sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Introduction Today’s age is characterized by an ever-increasing connected and opinionated world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The widespread adoption of social media has caused significant changes in the world across many domains, and more are probably to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In the case of financial markets, participants now have access to countless online platforms to share their thoughts and feelings on certain events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Proponents of the Efficient Market Hypothesis (Fama, 1970) ought to be pleased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The advent of mass media facilitates rapid information diffusion, possibly propelling markets into a higher tier of price efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' However, behavioral economists would argue that this type of media might very well influence investors and incite herd behavior which in turn induces inefficiency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Baker and Wurgler, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Chiang and Zheng, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Theory aside, this new wealth of information has not escaped the notice of the financial establishment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Indeed, data providers such as Bloomberg and Refinitiv now offer extensive social media indicators to help financial institutions navigate this new world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Although the competitive edge that resides in these alterna- tive data sources remains veiled in secrecy, an abundance of academic studies have already tried ⋆Declarations of interest: none ∗Corresponding author Email addresses: thdierckx@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='com (Thomas Dierckx ), jesse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='davis@kuleuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='be (Jesse Davis ), wim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='schoutens@kuleuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='be (Wim Schoutens ) Preprint submitted to Journal of Empirical Finance January 3, 2023 quantifying the interplay between social media and certain financial variables, providing insight into the predictive power of the masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Existing research has mainly focused on the Twitter platform and its influence on three promi- nent financial variables: stock price (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Groß-Klußmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Schnaubelt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2020), realized volatility (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Karagozoglu and Fabozzi, 2017), trading volume (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Guijarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2019) or a combination thereof (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Remarkably, cur- rent literature completely overlooks the interaction between social media and the market implied volatility of stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Derived from option prices, this variable is deemed to be one of the more important parameters in the world of derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In contrast to historical volatility, this is a forward-looking metric that indicates how much risk the market expects a certain asset to exhibit in the coming period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' As this variable serves as a proxy for both market sentiment and option contract prices, the ability to predict its movements would be advantageous for the practice of asset management and market making alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Most prevailing studies in this domain have two important methodological shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' First, analysis is typically performed on either a handful of arbitrarily chosen stocks or indices tracking the entire market (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Groß-Klußmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2019), omitting sector idiosyncrasies in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Second, hypotheses are commonly tested on a relatively small time window ranging from a month (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Bollen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2011) to a few years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Schnaubelt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' However, the ever-changing nature of financial markets warrants a closer look into how the interplay between social media patterns and financial variables evolves over longer periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' As patterns may emerge and dissipate over time, a crucial aspect of analysis is often left out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The contribution of this study is threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' First, to the best of our knowledge, we are the first to investigate to what extent a stock its one-day ahead movement in implied volatility can be predicted using machine learning on different combinations of feature sources, including Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Second, instead of arbitrarily choosing a handful of stocks for our study, we diversified our stock universe across the 11 traditional US stock market sectors yielding 165 stocks in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This al- lowed us to measure and explore the variability in predictive performance present among sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Lastly, we examined predictive performance on an out-of-sample period spanning January 1st, 2013 till March 1st, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This period is significantly larger than many other studies and gave us the opportunity to not only be more robust, but also examine predictive performance throughout time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Instead of performing a year by year analysis of predictive performance, we used hidden Markov models to identify four regimes in the implied volatility of a stock and gauged whether performance varies across them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' We argue that this alternative quantification of time yields more actionable insight as it allows practitioners to better anticipate future performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Preliminaries This section presents background information on the key components used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' First, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1 explains market implied volatility and its relation to the world of derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Sec- ond, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='2 describes the random forest machine learning model which is used to perform predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Lastly, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='3 describes the hidden Markov model which is used to quantify regimes in market implied volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Market Implied Volatility In the world of derivatives, options are one of the most prominent types of financial instru- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' As sellers of options are exposed to risk for the duration of the contract, they want to be 2 properly compensated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Measuring this risk requires considering the expected price fluctuations of the underlying asset over the duration of the contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This expectation is better known as implied volatility and it varies with the strike price and duration of an option contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' To obtain a more general measure, the implied volatility of option contracts that expire on the same date can be combined into a single implied volatility measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' A famous example of this is the CBOE Volatility Index, which combines the implied volatility of different option contracts on the SPX into an index that is better known as the VIX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' More concretely, the VIX is a measure of expected price fluctuations in the S&P 500 Index over the next 30 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' It is famously known as the fear index and is considered a reflection of investor sentiment on the condition of the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Equation 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' taken from the VIX white paper (CBOE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 2015),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' shows how to compute the VIX for a given term T: VIX = 100 × � 2 T � i ∆Ki K2 i eRT Q(Ki) − 1 T � F K0 − 1 �2 (1) where: T = is time to expiration F = is the forward index level derived from the index option prices K0 = is the first strike below the forward index level F Ki = is the first strike price of the ith out-of-the-money option: a call if Ki > K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' a put if Ki < K0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' or both put and call if Ki = K0 R = is the risk-free rate to expiration ∆Ki = is the interval between strike prices Q(Ki) = is the midpoint of the bid-ask spread for each option with strike Ki The equation for computing the VIX is applicable to any asset where option contracts are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Although this measure can be calculated for any arbitrary term, the duration of the option contracts will seldom match the chosen term T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Indeed, option contracts typically have fixed expiration dates and there is no guarantee that there are option contracts available with a duration equal to the given term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' To overcome this obstacle, the VIX is first calculated for the option contracts expiring right before and after the desired target date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The VIX for the given term can then be calculated by linearly interpolating between the two computed measures, as outlined in (CBOE, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Random Forests Random forests (Breiman, 2001) are a popular machine learning approach for learning a pre- dictive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' They consist of multiple different decision (or regression) trees whose predictions are combined into one final prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The combination is typically done by taking the mode (or average) of all outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' While several variations exist for learning a random forest, all of them are relatively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' We summarize one of many popular procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Given data D = {(xi, yi)}n i=1, where each xi has F features and for k = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' K trees: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Obtain subset d by sampling m < n examples with replacement from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Train a decision tree on d using a random subset features f ⊆ F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' using CART (Breiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 3 The prediction for a regression problem can then be obtained by: ˆyi = 1 K K � k=1 fk(xi) (2) where f is a function in the set of all possible decision trees and K is the total number of trees in the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The advantages of random forests include that they are fast to build, are not affected by feature scaling, are robust to irrelevant predictors and noisy data (Khoshgoftaar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Moreover, their method of constructing an ensemble model by randomly subsampling both data points and features during the learning process helps decorrelate the predictions made by the individual trees, which in turn reduces overfitting on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Hidden Markov Models Hidden Markov models (HMM) are a generative approach for modeling systems that follow a Markov process (Rabiner and Juang, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The main assumption is that while this process Z is hidden, it can be learned from an observable sequence X whose behaviour depends on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' More formally, the HMM models the joint distribution of a sequence of hidden states Z and observations X described by: P(Z1:K, X1:K) = P(Z1)P(X1|Z1) K � t=2 P(Zt|Zt−1)P(Xt|Zt) (3) Given the number of hidden states K and observed sequence X, the model is fully determined by its parameters π, A, and B which represent the initial state distribution, state transition model, and emission probabilities model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The initial state distribution is a K × 1 vector de- noting the probabilities that the process is each of the K states in the first timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The transition model is a K × K stochastic matrix where each element Ai, j denotes the probability of transi- tioning from state Zt−1,i to Zt, j where i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Lastly, the emissions probability model is a M × K matrix, with M representing the number of distinct observations, whose elements Bk, j denote the probability of observing Xt,k given state Zt, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The three key tasks associated with hidden Markov models are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' What is the probability that a sequence of observations X was generated by a given HMM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Given an HMM, what sequence of hidden states Z best explains a given sequence of ob- servations X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Given a sequence of observations X, learn an HMM with parameters π, A, and B that would generate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The first two tasks can be solved using dynamic programming using the forward-backward algorithm (Chang and Hancock, 1966) and Viterbi (Forney, 1973) algorithm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The third problem is solved by the Baum-Welch algorithm (Baum, 1972) which uses an iterative expectation-maximization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Methodology The main goal of this study is to explore the following questions: 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' To what extent are one-day ahead movements in end-of-day implied volatility predictable, and do features extracted from Twitter improve performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Does performance vary across the 11 different stock market sectors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' If so, are there any obvious factors that might explain this variability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Can we identify underlying market regimes in implied volatility that influence the perfor- mance of our proposed approach?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' We tackle the first question by performing an ablation study using random forests on feature configurations including stock price, stock implied volatility, and Twitter features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The study encompasses a universe of 165 stocks over an out-of-sample period spanning January 1st, 2013 till March 1st, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' To examine the second question, we diversified our stock universe over the 11 traditional stock sectors and grouped predictive performance by stocks belonging to the same sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The third and last question was studied by using a hidden Markov model to identify four distinct implied volatility regimes per stock, after which predictive performance was grouped by regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The next few sections explain our methodology in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' First, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1 outlines the stock universe we used for our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Second, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='2 explains how we obtained the relevant data for each stock and how we constructed features for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='3 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='4 then respectively show how we used machine learning to predict our target variable and how we evaluated the performance of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Lastly, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='5 explains how we used hidden Markov models to identify regimes in implied volatility which we later use to evaluate our prediction performance through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Stock Universe Selection In order to obtain a diversified universe of stocks, we looked at the popular SPDR and Van- guard Electronic Traded Funds (ETF) that track the 11 traditional US stock market sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' For each sector, we selected the 15 most liquid stocks based on their average daily dollar-weighted option volume for a total of 165 stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Some stocks were excluded due to stock splits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' we kept GOOG and dropped GOOGL), a late introduction to the stock market (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' PYPL, ROKU, and SNAP only got introduced after 2015), and ambiguous names making it hard to obtain rele- vant tweets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' DOW is a chemical company but also a common alias for the Dow Jones Index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Note that we replaced the excluded stocks to maintain 15 stocks per sector for our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 1 provides a concise overview of our stock universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Refer to Appendix A for a full overview of which stocks were selected per market sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Data Acquisition and Feature Generation We consider data ranging from January 1st, 2011 through March 1st, 2019 for three data sources: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Stock price data which consists of historical end-of-day adjusted closing prices for each stock in our universe downloaded from Yahoo Finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Option contract price data which consists of historical end-of-day option chains for each stock in our universe obtained from IVolatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Twitter data which consists of all relevant tweets published for each stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' These were collected by filtering on cashtags, which are popular string identifiers authors use to in- dicate their message is about a certain stock (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' a tweet about the Apple stock typically contains $AAPL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In contrast to other research, we did not employ additional filtering 5 Table 1: This table presents the 11 different stock market sectors together with their corresponding SPDR ETF symbol and number of stocks considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The symbols are used to denote sectors throughout this paper, but are not indicative of stocks only belonging to the SPDR ETF portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Symbol Sector Selected Stocks XLC Communication Services 15 XLY Consumer Discretionary 15 XLP Consumer Staples 15 XLE Energy 15 XLF Financials 15 XLV Healthcare 15 XLB Materials 15 XLI Industrials 15 XLK Technology 15 XLRE Real Estate 15 XLU Utilities 15 techniques to discard potential spam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Most additional filtering rules appear arbitrary and there seems to be no clear evidence of their validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In total, four features were extracted per stock for each trading day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' First, we simply used the end-of-day adjusted closing price from the stock price data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Second, we calculated the end- of-day 30-day implied volatility using the VIX method on the option contract data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Third and last, we derived two numerical features from our textual Twitter corpus: end-of-day total tweet publication count and end-of-day average sentiment polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The former represents the total number of published tweets on a given day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The latter was obtained by performing sentiment analysis using VADER (Hutto and Gilbert, 2014), a lexicon- and rule-based sentiment model that is specifically well-tailored to social media text, on individual tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This yields a sentiment polarity score s ∈ [−1, 1] for each tweet, which was then used to compute the daily average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In an effort to capture temporal information residing in the original feature timeseries, we generated two additional predictors per feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' To this end, the daily difference (or first-order difference) and the difference between the daily value and its exponential moving average of the last 10 trading days was taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 2 outlines the different data sources and their features used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Note that the original adjusted closing price was omitted, as this is typically a non-stationary variable offering little value to a prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 2: This table provides a summary of the features considered per data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The first row indicates what original features were extracted, whereas the last three rows indicate (*) which features were considered for the actual study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Note that the last two rows denote a specific feature engineering technique applied to the original feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Stocks Options Twitter Extracted Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Closing Price Implied Volatility Count, Sentiment Original 1st Order Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' EMA(10) Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Predicting Movements in Implied Volatility This study aims to predict one-day ahead movements in a stock’s 30-day implied volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Concretely, given information at the end of trading day t, we predict whether implied volatility will have moved up or down by the end of next trading day t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' To do so, we construct a binary target variable for day t as: yt = ������� 1, if (ivolatilityt+1 − ivolatilityt) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' (4) where ivolatilityt denotes the end-of-day implied volatility on day t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In order to predict our target variable, we used random forest classifiers even though more powerful models may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' For example, the highly popular gradient boosted trees (Friedman, 2001) have been shown to generally perform slightly better than random forests (Caruana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Caruana and Niculescu-Mizil, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' However, they are very sensitive to hyper-parameter configurations and require longer runtimes for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The main goal of this study is not to maximize predictive performance, but rather probe the feasibility of our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In addition, it has been suggested that random forests might generally work better on noisy data (Khoshgoftaar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2011), which is especially convenient when working on financial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Lastly, we did not consider techniques from the domain of deep learning due to the complexity of the models and relatively small number of data points in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Ultimately, we used 64 distinctive random forest configurations built using Sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Each random forest was built with 1000 trees and a unique combination of different hyper-parameters that control maximum tree depth, the minimum number of samples required to split an internal node, and the minimum number of samples required to be in a leaf node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Each individual tree was built by sampling the training dataset (with replacement) and only considering a random number of � f features where f denotes the total amount of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The models were trained on a temporally ordered feature matrix X of dimension T ×K, obtained by using any subset of features K from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='2 and period T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 3 specifies the possible random forest configurations considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 3: This table presents the different possible values considered for different hyper-parameters available in the random forest implementation of Sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The default value is used for hyper-parameters not listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Hyper-parameter Values n estimators {1000} max depth {4, 6, 8, 10} min samples split {5, 10, 15, 20} min samples leaf {1, 3, 5, 8} random state {42} bootstrap yes max features sqrt 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Experimental Evaluation We evaluated the different random forest configurations using walk-forward validation, a cross-validation technique designed specifically for temporally ordered data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Classical cross- validation methods assume observations to be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This assumption does not necessarily 7 Table 4: Example of expanding walk forward validation without where ti represents the feature vector of trading day i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In this example, a training window with an initial size s = 3 is taken together with a testing window of size k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' We therefore consistently use the feature vectors of past trading days to train a model (underlined) and subsequently test said model on trading day t + k (bold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Iteration Variable roles 1 t1 t2 t3 t4 t5 · · · tn 2 t1 t2 t3 t4 t5 · · · tn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' m t1 · · · tn−3 tn−2 tn−1 tn hold for timeseries data, which inherently contains temporal dependencies among observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' To this end, the dataset is repeatedly split up in training and test sets where temporal order is accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In our case, we used an expanding window of initially 504 trading days to train the models, after which performance was measured on the next out-of-sample 40 trading days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 5 shows an example of this method where ti denotes the feature vector corresponding to trading day i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Note that in this scenario, when given a total of n observations, an expanding training window of length t and an out-of-sample test window of length k, you can construct a maximum of n − t − k different train-test splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Ultimately, each configuration its performance is averaged across all folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' We measured performance with the area under the receiver operating characteristic curve metric (AUC hereafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 5: Example of walk-forward validation where ti represents the feature vector of trading day i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In this example, a training window with an initial size s = 3 is taken together with a testing window of size k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' We therefore consistently use the feature vectors of past trading days to train a model (underlined) and subsequently test said model on trading day t + k (bold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Iteration Variable roles 1 t1 t2 t3 t4 t5 · · · tn 2 t1 t2 t3 t4 t5 · · · tn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' m t1 · · · tn−3 tn−2 tn−1 tn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Analyzing Performance through Time with Hidden Markov Models The ever-changing nature of financial markets makes it hard to find approaches that con- sistently work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The ability to time approaches therefore becomes an interesting perk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In an effort to evaluate how our proposed approach weathers the evolution of the financial market through time, we look at its performance under different market regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' We quantified market regimes as different states in mean implied volatility using a hidden Markov model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' For each stock, a different HMM model was trained on its end-of-day im- plied volatility timeseries dating from January 1st, 2007 till December 31st, 2012 and was used out-of-sample thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Analogous to a study performed by Soci´et´e G´en´erale (Daviaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2020), four different regimes were identified corresponding to low, medium, high, and very high mean implied volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 6 specifies the hyper-parameter configuration used to build hidden Markov models using the hmmlearn Python package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 8 Table 6: This table presents the hyper-parameter configuration used for building hidden Markov models using the hmm- learn package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The default value is used for hyper-parameters not listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Hyper-parameter Values n components 4 n iter 100 random state 42 emissions Gaussian algorithm Viterbi 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Experimental Results and Discussion In this section, we present and discuss our experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' First, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1 shows the results of our ablation study where in total seven different feature configurations were consid- ered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='2 then builds on these results by looking at the performance of the best feature configuration per market sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Lastly, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='3 looks at predictive performance across dif- ferent implied volatility regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Recall that throughout the remainder of this section we may denote the 11 different stock market sectors by the symbol of their equivalent SPDR ETF tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This is solely done out of convenience and is not indicative of stocks only belonging to said ETF portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Refer to Table 1 for an overview of the sector symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Ablation Study The first objective of this study was to investigate to what extent daily movements in end- of-day implied volatility can be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' To this end, we obtained 11 different features from three different data sources (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='2) on which we built random forest classifiers to predict said target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' We assessed the effectiveness of the different data sources by performing an ablation study where 7 different scenarios were considered, shown in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In total, one-day ahead movements in implied volatility were predicted for 165 stocks (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1) spanning an out-of-sample period from January 1st, 2013 till March 1st, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 7: This tables shows the different feature scenarios considered in our ablation study together with their total number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Note that the third column indicates the usage of both original and derived features from the given feature source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Scenario Feature Source Features 1 Stock Price 2 2 Stock Price, Tweets 8 3 Implied Volatility 3 4 Implied Volatility, Tweets 9 5 Tweets 6 6 Stock Price, Implied Volatility 5 7 Stock Price, Implied Volatility, Tweets 11 We compared the predictive performance of different scenarios to that of a stratified dummy classifier, which makes the comparison more rigid than using a simple random classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Indeed, implied volatility tends to go down more often than it goes up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This causes a stratified dummy 9 classifier to achieve a median AUC of 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='8% across all 165 stocks versus a 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='0% achieved by a fully random one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 8 displays the median AUC achieved for each scenario averaged over the entire selected stock universe and the difference in AUC between our approach and the stratified dummy clas- sifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' These results provide empirical evidence that end-of-day movements in implied volatility can indeed be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' All possible feature scenarios perform better than a purely random classifier that achieves a median of 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='0% AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Moreover, 4 out of 7 scenarios outperform the stratified dummy classifier that achieves a median of 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='8% AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The commonality among these improved scenarios is the use of implied volatility features, indicating that this is an important source of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Moreover, including features derived from tweets always yielded a better median performance (S2 versus S1, S4 versus S3, and S7 versus S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This implies there is indeed a predictive interplay between information on Twitter and future implied volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Lastly, using all possible features (S7) yielded the best result overall, suggesting there are predictive patterns among all three feature sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 8: This table displays the median predictive performance across all 165 stocks per feature configuration obtained by predicting daily end-of-day movements in implied volatility over the period of January 1st, 2013 to March 1st, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Moreover, the second row shows how much the proposed approach does better than the stratified dummy classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' S1 S2 S3 S4 S5 S6 S7 Median AUC 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='6 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='6 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1 Improvement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='9 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='8 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='7 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Predictive Performance across Sectors In this section we look at the best performing feature configuration (S7) and its performance variability across 11 different stock market sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Figure 1 shows a box plot where the per- formance improvement of our proposed approach versus the stratified dummy classifier for each individual stock is grouped by sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The results were obtained on an out-of-sample period spanning January 1st, 2013 till March 1st, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' It is clear that our proposed methodology is generally able to beat the stratified dummy clas- sifier across all different sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The approach beats the dummy classifier on 148 out of 165 stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' However, there is a considerable amount of variability present across different sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Indeed, the approach does significantly better on XLRE and XLU, but predictions on XLC, XLY, and XLK also do better comparatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In contrast, XLI and XLB seem to lack in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The next two subsections will showcase a preliminary attempt to partially explain this variability in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The Effect of Option Liquidity The results from the previous section indicate that predictions on stocks from both XLRE and XLU do significantly better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Remarkably, it turns out that stocks in these two sectors are also significantly less liquid compared to other sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Figure 2 respectively shows a box plot of median option liquidity per sector, measured by the average daily dollar amount traded in options, and a regression plot where the relationship between liquidity and performance improvement versus the dummy classifier is outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' These results seem to suggest that there is indeed a weak negative correlation between pre- dictive performance and option liquidity, implying that less liquid stocks are easier to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 10 Figure 1: This box plot shows the performance improvement of our proposed methodology using feature configuration S7 versus a stratified dummy classifier on individual stocks grouped by sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The red dotted line represents the min- imum threshold necessary to beat the stratified dummy classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The blue dotted line represents the overall median improvement of our proposed approach versus the stratified dummy classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Figure 2: This figure presents a box plot (left) where the median of daily stock option liquidity is grouped per sector and a regression plot (right) where the relationship between stock option liquidity and performance improvement versus the dummy classifier is outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' We hypothesize that the relatively undersized liquidity of both XLRE and XLU in the options market is possibly accompanied by a less efficient price discovery process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This in turn might cause these markets to reflect new information more slowly, making them easier to predict with the information at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' However, we note that is only one possible explanation and many other factors might lie at the basis of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 11 dummythreshold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='125 medianimprovement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='100 (AUC) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='075 Improvement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='050 XLC XLY XLP XLE XLF XLV XLI XLB XLRE XLK XLU Sector16 median (overall) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='100 Improvement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='075 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='025 10 Daily 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='000 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='050 XLCXLYXLPXLEXLFXLVXLIXLBXLREXLKXLU 8 10 12 14 16 Sector DailyDollarVolume (log)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The Effect of Twitter Attention Lower liquidity might partially explain why sectors such as XLRE and XLU seem easier to predict, but it certainly does not tell the whole story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Indeed, predictions on stocks from XLC, XLY, and XLK, examples of very liquid sectors, also do better comparatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Here, we hypoth- esize that this might be due to the attention they receive on Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Figure 3 respectively shows a box plot of the median daily tweets published on stocks grouped per sector and a regression plot where the relationship between liquidity and daily tweets is outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Figure 3: This figure presents a box plot (left) where the median of daily tweet publication of stocks is grouped per sector and a regression plot (right) where the relationship between a stock its daily tweets and option liquidity is outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The box plot on the left of Figure 3 shows that stocks in XLC, XLY, and XLK receive signif- icantly more attention on Twitter than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Note that the plot demonstrates a striking resem- blance with the box plot showing liquidity per sector in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Indeed, the regression plot on the right shows a very strong correlation between attention on Twitter and liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' These find- ings seem to suggest that prediction is easier on stocks that are more popular on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' We investigated this phenomenon further by looking at the improvement in predictive performance caused by features extracted from Twitter per sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' More concrete, we looked at the difference in performance between feature configuration S6, which uses stock and options features, and S7, which combines features from S6 and Twitter features (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Figure 4 shows the median improvement of using Twitter features per stock grouped by sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' With exception of XLE, most sectors seem to have a sizeable number of stocks that benefit from using social media features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This is no surprise considering the results presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' However, it seems that sectors that receive more social media attention seem to benefit the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' For example, the three sectors XLC, XLY, and XLK that are most popular also benefit more consistently followed closely by XLV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Two reasons might explain these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' First, pre- vious research has hinted at social media inciting herd behavior and emotional reactions among investors, possibly driving inefficiency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Wang and Wang, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Groß- Klußmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' If this is true, it makes sense that Twitter features provide more added value for popular stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Second, our sentiment extraction technique on tweets is imperfect and might impact these results as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Without carefully filtering out tweets that are advertisements or spam, we rely on the law of large numbers to correctly estimate a stock its sentiment for a 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='median(overall) 16 (601) Tweets 5 12 4 10 Daily 3 8 n : XLC XLY XLP XLE XLE XLY XLI XLBXLREXLKXLU 2 3 4 5 6 7 Sector Daily Tweets (log)Figure 4: This figure shows the performance improvement of using Twitter features together with features from S6, versus only using features from S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The red dotted line represents the minimum threshold necessary to beat the feature configuration of S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' given day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Naturally, daily sentiment estimation will therefore be better on stocks that are more heavily tweeted about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Performance across Implied Volatility Regimes In this section, we look at the best-performing feature configuration (S7) and its performance variability across four different market regimes in implied volatility, identified by using a hidden Markov model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 9 shows the median of all results across 165 stocks where, for each of the four regimes, the columns respectively show how many days were spent in a regime, what the average implied volatility was, the AUC of a stratified dummy classifier, and how much better our approach did compared to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 9: This table shows the median number of days a stock resided in one out of four implied volatility regimes, together with each regime’s mean implied volatility, stratified dummy performance and the improvement of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In total, 165 stocks were considered over a period spanning January 1st, 2013 till March 1st, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Frequency Implied Volatility Dummy AUC Improvement Low 392 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='6 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='75 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='15 Medium 452 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='65 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='83 High 411 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='12 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='84 Very High 231 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='3 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='92 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='47 In general, stocks seem to reside more in the lower implied volatility environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The stratified dummy classifier also appears to perform worse here, implying that up and down move- ments are about equal in occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' However, the dummy classifier performance picks up as implied volatility increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This makes sense from a theoretical perspective, as implied volatil- ity is deemed to be a mean reverting process characterized by big up movements after which the variable slowly trails back down, resulting in more down movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Although our approach 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='04 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='--improvementthreshold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='02 (AUC) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='01 Improvement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='04 XLC XLY XLP XLE XLF XLV XLI XLB XLRE XLK XLU Sectorseems to outperform the dummy classifier in all regimes, the added value seems to be most sig- nificant in the low to high regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This may suggest that distressed markets are harder to predict than their calmer counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' However, note that the improvement of our approach seems to be slightly correlated with regime frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' From a data science perspective, this might imply that the model performs worse in less frequent regimes because it simply had fewer examples to learn from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Lastly, Table 10 offers a more finer-grained analysis where the median improvement for each regime per sector is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table 10: This table shows the median improvement of our approach compared to a stratified dummy classifier for stocks grouped per market sector and implied volatility regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' The best and worst regime for each sector are respectively indicated by bold and underlined text styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Low Medium High Very High XLB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='77 XLC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='57 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='76 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='26 XLE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='55 XLF 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='7 XLI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='5 XLK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='91 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='70 XLP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='83 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='53 XLRE 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='93 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='25 XLU 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='63 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='95 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='79 XLV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='13 XLY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='66 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='85 Again, we remark a significant amount of variability in performance across both regimes and sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Optimal implied volatility regimes seem to differ significantly for different sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' In contrast, with exception of XLK, sectors seem to comparatively do worse when implied volatility is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' As all sectors share roughly the same regime frequency distribution, no clear reason emerges to explain the performance variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' One potential reason might be due to sector idiosyncrasies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' For example, defensive sectors such as XLRE, XLU, and XLV seem to have lower optimal regimes than cyclical sectors such as XLC, XLK, and XLY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' However, this is not always the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Lastly, we have no explanation for the significant difference between XLK and the other sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Here, our approach performs best in the highest implied volatility regime and worst in the lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This is in stark contrast with the other sectors where the opposite is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Perhaps the speculative nature of technology stocks is more sensitive to herd behaviour and therefore investor irrationality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Conclusion In this study, we presented the first empirical evidence that one-day ahead movements of end-of-day stock implied volatility can be predicted to a certain extent, and that attention and sentiment features extracted from Twitter improve the performance of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' These alter- native features were not able to predict implied volatility in isolation, but improved the approach significantly when combined with predictors extracted from stock and options data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' This sug- gests that the interplay between these sources gives rise to predictive patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' By conducting 14 our experiments on a diversified universe of 165 US stocks, we were able to assess the predic- tive performance across 11 traditional stock market sectors and found that stocks in real estate, utilities, consumer discretionary, communications, and technology were easier to predict than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Further analysis indicated that these differences could potentially be explained by market inefficiencies caused by low option liquidity in the real estate and utilities sector, and excessive Twitter attention in the consumer discretionary, communications, and technology sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Lastly, using hidden Markov models, we evaluated the predictive performance of our approach across four different implied volatility regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Although it outperforms the dummy classifier in all four regimes, we found that it yields the least improvement in the regime associated with the highest average implied volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Moreover, we discovered that different stock market sectors have different optimal regimes for the application of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' By analyzing performance through the usage of regimes, we showed that this alternative quantification of time provides additional insight into the performance of models which in turn could help better anticipate their future performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' 15 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Selected Stock Universe This appendix contains more detailed information on the US stock universe used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='11 outlines which stocks were chosen for each traditional market sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='11: This table presents the US stock universe that was used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Note that for each traditional market sector we chose 15 stocks based on option liquidity.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', Krauss, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Separating the signal from the noise – financial machine learning for twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' Journal of Economic Dynamics and Control 114, 103895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtAyT4oBgHgl3EQfavdq/content/2301.00248v1.pdf'} +page_content=' doi:https://doi.' metadata={'source': 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